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Cohomological theory of crystals over function fields and applications. (English) Zbl 1386.11076
Bars, Francesc (ed.) et al., Arithmetic geometry over global function fields. Selected notes based on the presentations at five advanced courses on arithmetic geometry at the Centre de Recerca Matemàtica, CRM, Barcelona, Spain, February 22 – March 5 and April 6–16, 2010. Basel: Birkhäuser/Springer (ISBN 978-3-0348-0852-1/pbk; 978-3-0348-0853-8/ebook). Advanced Courses in Mathematics – CRM Barcelona, 1-118 (2014).
Summary: This lecture series introduces in the first part a cohomological theory for varieties in positive characteristic with finitely generated rings of this characteristic as coefficients developed jointly with Richard Pink. In the second part various applications are given.
For the entire collection see [Zbl 1305.11001].

##### MSC:
 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11F52 Modular forms associated to Drinfel’d modules 14F30 $$p$$-adic cohomology, crystalline cohomology
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  , which was later reproved by Diaz-Vargas  and by Thakur . Theorem 8.29(Riemann hypothesis). Let q = p. Then the following hold: (a) For any n∈ N 0 one has (n)d q− 1 and s d(n) =deg t g j(q−1). j=1 (b) The sequence (deg t g j(q−1))j≥0 is strictly decreasing, and thus the Newton polygon of ζA(−n, T ) has (n) q−1 distinct slopes all of width one. (c) For any m∈ Z p, the entire function L A(m, T ) in T has a Newton polygon whose slopes are all of width one and thus the roots of T→ L A(m, T ) lie in K∞; they are simple and of pairwise distinct valuation. Part (c) for q = 4 was proved by Poonen and for arbitrary q by Sheats in . Recall that the Newton polygon of the polynomial deg TζA(−n, T ) ∈ A[T ] ⊂ K∞[T ] is the lower convex hull of the points (d,− deg t(S d(n)))d≥0. By (a), the slope between d and d + 1 is−s d+1(n)− (−s d(n)) =− deg t g d(q−1). By the ﬁrst assertion of (b), the sequence deg t g j(q−1)is strictly decreasing and it follows that the points (d,− deg t(S d(n)))d≥0 lie all on the lower convex hull and are break points, i.e., points where the slope changes. Thus the second part of (b) follows, once (a) and the ﬁrst part is shown. Part (c) is a simple formal consequence of (b) as explained in . To prove the theorem, our ﬁrst aim will be to compute the degrees of the polynomials g j. Let us ﬁrst recall a lemma of Lucas. Lemma 8.30.For integers a 0, . . . , a r and l 0, . . . , l r in the interval [0, p− 1] one has a ir a =i=0 r i p i. l i i=0 l i p i i=0 m An analogous formula holds for multinomial coeﬃcients. m 1 m 2... m Lucas’ formula follows easily from expanding both sides of the following equality by the binomial theorem: r ar 1 + T i=0 i p i=1 + T p ia i. i=0 Lucas’ lemma holds for arbitrary q; however the formulation for p implies the lemma also for all p-powers. Note that by the lemma of Lucas the coeﬃcient n modulo p is non-zero l if and only if, considering the base p expansions of n and l, each digit of the expansion of l is at most as large as the corresponding digit of n. Lemma 8.31. rn g j= (−1)j a i t l 0+l 1 q+···+l r q r= (−1)j n t l. l i l i=0 l=0 i l i=d−j(l)=d−j For each m∈ N 0 the sum contains at most one summand of degree m in t. Proof. The ﬁrst equality in Lemma 8.31 is proved by a multiple application of the binomial theorem and collecting all the coeﬃcients of θj in g≥0 g jθj= ! i≥0(T q i− θ)a i. The second equality follows from Lucas’ lemma. Set μ(D) = r and mμ(D)= 0 and for j∈ 0, . . . , D − 1 deﬁne μ(j) ∈ 0, . . . , r and mμ(j)∈ 1, . . . , aμ(j) by (D− j)(q − 1) = a r+· · · + aμ(j)+1+ mμ(j). Lemma 8.32.Suppose q = p. Then for each j∈ 0, . . . , D one has deg g j= a r q r+· · · + aμ(j)+1 qμ(j)+1+ mμ(j)qμ(j).  !ra l i j are all non zero. The formulas are now immediate from the deﬁnitions of μ and mμ. Suppose q = p. Then if j increases, D− j decreases and thus deg g j is strictly decreasing in j. Thus we have proved Theorem 8.29(b) once we have proved part (a). Moreover if  and from0, . . . , D satisfy ≥  + (q − 1), then μ()≥ μ() + 1. This yields a precise result on the rate of the decrease of the deg t g j: Lemma 8.33.Suppose p = q. Then for 0≤ , ≤ D − 1 and ≥  + (q − 1) one has 0 < deg g− deg g+1= qμ()≤· qμ()=1·deg g q q− deg g+1. Proof of Theorem 8.29(a). Our aim is to compute the degree in t of the coeﬃcients of the T i in the expression  det(1− T M) =sign πδ1π(1)− T m 1π(1)· · · · ·δdπ(d)− T m dπ(d). π∈Σd Let us expand the inner products by the distributive law. If a product contributes to T i, then in the distributed term we need d−j occurrences of terms not involving T , i.e., of 1’s. The latter can only come from the diagonal. We deduce: Up to sign, the terms contributing to T i are those (j× j)-minors of M which are symmetric, i.e., in which the same rows and columns were deleted from M . Let J be a subset of1, . . . , d (which may be empty) and let π ∈ Σd be a permutation of the set1, . . . , d which is the identity on J. Then for the pair J, π we deﬁne deg J,π:=deg mπ(j),j. j∈J For ﬁxed J , we shall show that the identity permutation is the unique one for which deg J,πis maximal. The following lemma is the key step. Lemma 8.34.Suppose q = p and ﬁx J⊂ 1, . . . , d. Then for all π ∈ Σd id ﬁxing J one has degπ,J< deg id,J. In particular ord(det(M )) = ord id and thus det(M ) is non-zero. Proof. We only give the proof for J =∅. The other cases are analogous. To simplify notation, we abbreviate degπ:= deg∅,π. Assume, contrary to the assertion of the lemma, that for some π∈ Σd id we have degπ= maxdegτ| τ ∈ Σd. Since π is not the identity, the permutation matrix representing π has some entry above the diagonal. Let j 0∈ 1, . . . , d be maximal such that π(j 0) < j 0. In row j 0 let j 1 be the column which contains the non-zero entry of the permutation matrix of π, i.e., j 1= π−1(j 0). By the maximality of j 0 we have j 1< j 0.(8.1) Consider the matrix ⎛⎞ . .. ⎜⎜⎟⎟ ⎜⎜0 mπ(j 0),j 0⎟ ⎜⎜. ..⎟⎟⎟ ⎝m j 0,j 1 0⎟⎠ . .. with entries mπ(j),j at (π(j), j) for j = 1, . . . , d and zero otherwise. Deﬁne the permutation πby  π(j) = π(j) for j= j 0, j 1,πj 0= j 0,πj 1= πj 0. Then  degπ−degπ= deg mπ(j 0),j 1+ deg m j 0,j 0− deg m j 0,j 1− deg mπ(j 0),j 0  = deg g j 1 q−π(j 0)− deg g j 1 q−j 0+ deg g j 0 q−j 0− deg g j 0 q−π(j 0) j 0 =deg(g j 1 q−i−deg g j 1 q−i+1−deg(g j 0 q−i−deg g j 0 q−i+1 i=π(j 0)+1 j 0 ≥q− 1≥ (q − 1), i=π(j 0)+1 where the last inequality follows from formula (8.1) and Lemma 8.33(a). We reach the contradiction degπ> degπ. For ﬁxed J , the lemma tells us that id is the unique permutation for which deg J,id is maximal. Moreover from the deﬁnition of deg J,π, we see that deg J,id=deg t m jj=deg t g j(q−1). j∈J j∈J Since the degrees of the g i are strictly decreasing, we ﬁnd that among those J for which #J is ﬁxed there is also a unique one for which deg J,id is maximal, namely J =1, 2, . . . , #J. It follows that the degree of the coeﬃcient of T i is equal to deg1,2,,...,i,id= deg t g j(q−1). This completes the proof of Theorem 8.29(a) and thus of j=1,...,i the theorem itself. Let us add some further observations regarding the degrees of the coeﬃcients S d(n). Say we write a r a r−1. . . a 1 a 0 p for the base p digit expansion of n. By the deﬁnition of μ(j) and mμ(j), we have that for 0 0 . . . 0 aμ(j)− mμ(j)aμ(j)−1. . . a 1 a 0  positionμ(j) the sum over its digits in base p is exactly j. We deﬁne  n 1:=0 0 . . . aμ(p−1)− mμ(p−1)aμ(p−1)−1. . . a 1 a 0,  pos. μ(p−1)  n 2:=0 0 . . . aμ(2 p−2)− mμ(2 p−2)aμ(2 p−2)−1. . . mμ(p−1)0 . . . 0,    pos. μ(2 p−2)pos. μ(p−1)  n 3:=0 0 . . . aμ(3(p−1)− mμ(3(p−1))aμ(3(p−1))−1. . . mμ(2(p−1))0 . . . 0, etc.  pos. μ(3(p−1))pos. μ(2(p−1)) and m d by n = m d+ n d+ n d−1+· · · + n 1. Thus n 1 is formed from the p− 1 lowest digits of n; next n 2 is formed from the next p− 1 lowest digits of n that have not been used in forming n 1, etc. We can now rephrase Theorem 8.29(a) as follows:  s 1(n) = n− n 1, s 2(n) =n− n 1+n− n 1− n 2, . . . ,  s d(n) =n− n 1+n− n 1− n 2+· · · +n− n 1− · · · − n d, etc. and s(n) =−∞ for all  ≥ 0+ 1 where 0 is smallest so that n−n 1−· · ·−n0 has digit sum less than p− 1 in its p-digit, or if no such 0> 0 exists, we set 0= 0. We obtain an alternative proof of the following recursion from : Corollary 8.35(Thakur). s d(n) = s 1(n) + s d−1(s 1(n)). Proof.  s d(n) =n− n 1+n− n 1− n 2+· · · +n− n 1− · · · − n d  =n− n 1+n− n 1− n 2+(n− n 1− n 2− n 3  +· · · +n− n 1− · · · − n d  = s 1(n) +s 1(n)− n 2+s 1(n)− n 2− n 3+· · · +s 1(n)− n 2− · · · − n d  = s 1(n) + s d−1 s 1(n). Note that to deﬁne n 2, n 3, . . . , n d it is not necessary to know n. It suﬃces to know n− n 1= s 1(n) as deﬁned above. Open questions regarding the zero distributions ofζA(−n, T ) for A diﬀerent from F q[T ] Due to some simple but remarkable examples, Thakur  showed that Theo rem 8.29(c) cannot hold for general A. It is known that ζA(−n, T ) has a zero T = 1 if q− 1 divides n. Such zeros are called trivial zeros. For F q[t] all trivial zeros are simple and if n is not divisible by q− 1 then T = 1 is not a root of ζA(−n, T ). In  Thakur shows by explicit computation that for some even n, i.e., n divisible by q− 1, the root T = 1 is a double root! Let us say that ζA(−n, T ) has an extra zero at −n if either (T − 1)2 divides it, or if (T− 1) divides it but n is not a multiple of p − 1. The following patterns for negative integers−n were observed when comput ing, using a computer algebra package, the Newton polygons for several rings A for the function ζA(−n, T ) :=d≥0 T d a∈A a n under the hypotheses p = q d+ and d∞= 1. Note that the Newton polygons all lie under or on the x-axis and start at (0, 0). Moreover all slopes are less than or equal to zero. Deﬁne B⊂ N 0 as B :=0∪d∈ N 0| dim k A(d+1)> dim A d. In particular, B contains every integer d≥ 2 g, where g is the genus of A. The missing integers (in0, . . . , 2 g) are precisely the Weierstrass gaps for ∞. • The x-coordinates of all the break and end points of all Newton polygons are in the set B and at every x-coordinate in B (along the Newton polygon) there is a break point. • In particular, all slopes beyond the g th one have width 1. • There are no extra zeros for n not divisible by p − 1. Thus extra zeros can only occur at horizontal slopes of width larger than one, i.e., among the ﬁrst g slopes. • Even if a slope is horizontal and of length 2, there may not be an extra zero. • The degree in T of ζA(−n, T ) is determined by the following rule: The number of slopes of the Newton polygon is exactly equal to(n). p−1 Lecture 9 Relation to ´Etale Sheaves Throughout this lecture we assume that A is a ﬁnite k-algebra, and so in par ticular A is ﬁnite. For such A, we shall set deﬁne a functor  from A-crystals to constructible ´etale sheaves of A-modules that is an equivalence of categories. The correspondence is modeled at the Artin-Schreier sequence in ´etale cohomology and Deligne’s [11, Fonctions L]. One reason why one is interested in ﬁnite rings A is to study geometric ques tions in positive characteristic via ´etale mod p cohomology. Another reason is the following. Suppose ϕ is a Drinfeld A-module and M(ϕ) its associated A-motive. Then for all ﬁnite primes p of A, the p n-torsion of ϕ provides us with a Galois representation, or a lisse ´etale sheaf of (A/p n)-modules on the base. On the side of A-motives, A/p n-torsion is described by M(ϕ) ⊗A A/p n. Using the equivalence of categories  introduced in Section 9.1 and the relation between the torsion points of ϕ and the above quotient of M(ϕ), the following result is straightforward. Hom Aϕ[p n], A/p n ∼= M(ϕ) ⊗A A/p n. That is, the dual of the module of p n-torsion points is naturally associated to the motive modulo p n. In Section 9.1 we deﬁne the functor from A-crystals to constructible ´etale sheaves of A-modules and discuss its basic properties. Some proofs are given. In the subsequent Section 9.2 we use these results to reprove (in many but not all cases) a result of Goss and Sinnott which links properties of class groups to special values of Goss’ L-functions. 9.1 An equivalence of categories Our ﬁrst aim is to deﬁne a functor   :QCohτX, A−→ ´Et X, A. For this, we consider a τ -sheaf F. Using adjunction, we assume that it is given by a pair (F, τF:F → (σ × id)∗F). Let u: U → X be any ´etale morphism. Pullback of τ along u× id induces a homomorphism ∗∗∗∗ ∗F ∼=σ× id∗u× id F. Taking global sections on U× C and observing that σ × id is a topological iso morphism, we obtain a homomorphism of A-modules ∗∗∗ ∗u× id FU× C =u× id∗FU× C. By slight abuse of notation, let us denote this homomorphism by τet. Then one veriﬁes that ∗∗ u : U→ X−→ Ker 1− τet:u× id F U× C−→u× id F)(U × C for (u : U→ X) varying over the ´etale morphisms to X deﬁnes a sheaf of A-mod ules on the small ´etale site of X, denoted (F). A more concise way of deﬁning  is as follows. Let F et denote the ´etale sheaf associated to pr 1∗F by change of sites – this is what is done above, if one forgets about τ . Then τF induces a homomorphism τet:F et→ F et and  F:= Ker id−τet:F et−→ F et.(9.1) Clearly this construction is functorial in F, that is, to every homomorphism ϕ :F → G it associates a homomorphism (ϕ): (F) → (G). Thus it deﬁnes an A-linear functor  :QCohτX, A−→ ´Et(X, A).(9.2) Following its construction one ﬁnds that  is left exact. Example 9.1.Let 1l X,A denote the τ -sheaf consisting of the structure sheaf O X×C together with its obvious τ given by  ∗O X×C. The ´etale sheaf associated to O X×C is simply O X et⊗ A with τet the morphism (u⊗ a) → u q⊗ a. Therefore (1l X,A) ∼= A X, the constant ´etale sheaf on X with stalk A. In the special case A = k we recover the sequence 0−→ k X−→ O X et−→ O 1−σX et from Artin-Schreier theory. Lemma 9.2.Let ϕ :F → G be a nil-isomorphism in Cohτ(X, A). Then the induced (ϕ) : (F) → (G) is an isomorphism. Proof. Observe ﬁrst that regarding τ as a homomorphism of τ -sheaves, we have (τ ) = id. This is so because  is precisely the operation on the ´etale sheaf as sociated to F of taking ﬁxed points under τ. Clearly τ is the identity on the set of these ﬁxed points. Having clariﬁed this, the proof of the proposition follows immediately from applying τ to the diagram (2.2). The assertion of the lemma also holds for τ -sheaves whose underlying sheaf is only quasi-coherent. The proof is however much more subtle. In  it is shown that  factors via the category of ind-coherent τ -sheaves, i.e., τ -sheaves which can be written as inductive limits of coherent τ -sheaves. Then an argument involving direct limits reduces one to the already proved case of the lemma. In total one obtains: Proposition 9.3.The functor  induces a unique left exact A-linear functor   :QCrys X, A−→ ´Et X, A. It is shown in  that the isomorphism  deﬁned for all pairs (X, A) (with A ﬁnite) is compatible with the formation of functors on crystals and on the ´etale site, respectively: Proposition 9.4.For f : Y→ X a morphism, j : U → X an open immersion and h : C→ Ca base change homomorphism, one has the following compatibilities: ◦ f∗∼= f∗◦ ,  ◦⊗ ∼=⊗◦ ,  ◦⊗A A ∼=⊗A A◦ , ◦ f∗∼= f∗◦ ,  ◦ j!∼= j!◦ . Except for the very ﬁrst compatibility, i.e., that of inverse image, the proofs are rather straightforward. Note that to the left of◦  the functors are functors on ´etale sheaves and to the right of ◦ they are functors on A-crystals. Let ´Et c(X, A)⊂ ´Et(X, A) denote the subcategory of constructible ´etale sheaves. Recall that an ´etale sheaf of A-modules is constructible if X has a ﬁnite stratiﬁcation by locally closed subsets U i such that the restriction of the sheaf to each U i is locally constant. This in turn means that there exists a ﬁnite ´etale morphism V i→ U i such that the pullback to V i is a constant sheaf on a ﬁnite A-module. Proposition 9.5.The image of Crys(X, A) under  lies in ´Et c(X, A). Sketch of proof. Since being constructible is independent of the A-action, we may restrict the proof to Crys(X, k). We also may assume that X is reduced; cf. The orem 3.9. Let F be a coherent τ-sheaf on X over k. Since we have the functors j!, f∗and f∗at our disposal we can apply noetherian induction on X in order to show that (F) is constructible. Thus it suﬃces to ﬁx a generic point η of X and to prove that there exists an open neighborhood U of η such that (F|U) is locally constant. We ﬁrst choose a neighborhood U of η which is regular as a scheme. By [30, Theorem 4.1.1], it suﬃces to show that after possibly further shrinking U one can ﬁnd a τ -sheaf G which is nil-isomorphic to F and such that G is locally free and τ is an isomorphism on it. At the generic point both properties can be achieved by replacing F by Im(τF m) for m suﬃciently large. And then one shows, using A = k, that this extends to an open neighborhood of η. A main theorem of [8, Ch. 10] is the following: Theorem 9.6.For A a ﬁnite k-algebra, the functor  : Crys(X, A)→ ´Et c(X, A) is an equivalence of categories. Since it is compatible with all functors, the deﬁnition of ﬂatness for both categories implies that  induces an equivalence between the full subcategory of ﬂat A-crystals and the full subcategory of ﬂat constructible ´etale sheaves of A-mod ules:  : Crysﬂat X, A−→ ´∼=Etﬂat cX, A. For ﬂat A-crystals we have a deﬁnition of L-functions if X is of ﬁnite type over k. Under the same hypothesis on X, for ﬂat constructible ´etale sheaves of A-modules such a deﬁnition is given in [11, Fonctions L, 2.1]. At a closed point x, it is the following: Deﬁnition 9.7.The L-function of F ∈ ´Etﬂat c(x, A) is L x,F, t:= det A id−t d x· Frob−1 x F x¯−1∈ 1 + t d x At d x. The obvious extensions to schemes X of ﬁnite type over k and to complexes representable by bounded complexes of objects in ´Etﬂat c(X, A) are left to the reader. It is a basic result that  is compatible with the formation of L-functions: Proposition 9.8.For any F ∈ Crysﬂat(X, A)) we have  L X, F, t= L crys X,F, t. As a consequence of Theorem 6.14 we ﬁnd: Theorem 9.9.Let f : Y→ X be a morphism of schemes of ﬁnite type over k and F•∈ D b(´Et c(Y, A))ftd a complex representable by a bounded complex with objects in ´Etﬂat c(X, A). Then one has  L Y,F•, t∼ L X, Rf!F•, t, i.e., their quotient is a unipotent polynomial. For reduced coeﬃcient rings A, the above result was ﬁrst proved by Deligne in [11, Fonctions L, Theorem 2.2]. In [14, Theorem 1.5], Emerton and Kisin give a proof for arbitrary ﬁnite A of some characteristic p m. By an inverse limit pro cedure, in [14, Corollary 1.8] they give a suitable generalization to any coeﬃcient ring A which is a complete noetherian local Z p-algebra with ﬁnite residue ﬁeld. Remark 9.10.The category ´Et c(X, A) has no duality and f!, f∗and an internal Hom are either not all deﬁned or not well behaved. Thus for the theory of A-crys tals we cannot hope for this either. 9.2 A result of Goss and Sinnott In what follows we shall use the correspondence between ´etale sheaves and crystals of the previous section to reprove a result of Goss and Sinnott – in many, but so far not all cases. The original proof of the result of Goss and Sinnott is based on the comparison of classical L-functions for function ﬁelds and Goss-Carlitz type L-functions for function ﬁelds. Our proof avoids all usage of classical results but uses the results from the previous section instead. Class groups of Drinfeld-Hayes cyclotomic ﬁelds We consider the following situation. Let K be a function ﬁeld with constant ﬁeld k, let∞ be a place of K and A the ring of regular functions outside ∞. Let H ⊂ H+ be the (strict) Hilbert class ﬁeld with ring of integers O ⊂ O+. By the theory of Drinfeld-Hayes modules, see [24, Ch. 7] or Appendix A.3, there exist [H+: K] many sign-normalized rank 1 Drinfeld-Hayes modules  ϕ : A−→ O+τ,a−→ ϕa. Let p be a maximal ideal of A. Then the p-torsion points ϕ[p]( ¯K) of ϕ over ¯K form a free (A/p)-module of rank 1 carrying an A-linear action of Gal( ¯K/H+). If H p+ denotes the ﬁxed ﬁeld of the kernel of this representation, then G := Gal(H p+/H+) is isomorphic to (A/p)∗. The extension H p+/H+is totally ramiﬁed at the places of O+above p and unramiﬁed above all other ﬁnite places. For all places above ∞ the decomposition and inertia groups agree and are isomorphic to the subgroup k∗⊂ (A/p)∗. H p+ G H+ K. For K = k(t) and A = k[t] one has H = K and the Drinfeld module ϕ is simply the Carlitz module. Let us denote by χ : G→ (A/p)∗the character of G that arises from the action of ϕ[p]. This is the analog of the mod p cyclotomic character in classical number theory. We introduce the following notation. By Jac K,p we denote the Jacobian variety of the smooth projective geometrically irreducible curve C K,p with constant ﬁeld k∞and function ﬁeld H+, and we let Cl(H p+) denote the class group of the ﬁeld H p+. Then the p-torsion subgroup of Cl(H p+) is isomorphic to the group of invariants under Gal(¯k∞/k∞) of the p-torsion group Jac K,p[p](¯k). For w∈ Z (it suﬃces w ∈ 1, 2, . . . , #(A/p)∗) we deﬁne the χw components of the above groups as ¯ pχw,C(w) :=Jac K,p p k⊗F p A/pχw. Note that ¯kH+/H+is a constant ﬁeld extension, while H p+/H+is purely geomet ric. Hence these extensions are linearly disjoint. Moreover the group G is of order prime to p and thus its action on the p-group Jac K,p[p](¯k) is exact, so that C(w)Gal(¯k∞/k∞)= C(w). ∗ Let h+A denote the number of places of H+above∞, so that h+A= h A##k k∞∗. The following result (even under more general hypotheses) is due to Goss and Sinnott. Theorem 9.11(Goss, Sinnott). Let w∈ N. For a, b ∈ N deﬁne δa|b to be 1 if a is a divisor of b and zero otherwise. Then the following hold: (a) dim A/p C(w) = deg T(L Spec O+(w, T ) mod p) − h+Aδ(q−1)|w. (b) C(w)= 0 if and only if ord T =1(L Spec O+(w, T ) mod p) > h+Aδ(q−1)|w. (c) dim A/p C(w) is the multiplicity of the eigenvalue 1 of the action of τ on H 1(C K,p, (M(ϕ)⊗w)max)⊗A A/p. (For the superscript max, see Deﬁnition 9.12.) The proof in  uses congruences between L-functions of τ -sheaves and classical Hasse-Weil L-functions. This comparison is replaced by comparing the cohomology of a τ -sheaf and that of the ´etale sheaf associated to its mod p reduc tion. Before we can give the proof of the theorem, we need to introduce the concept of maximal extension of a τ -sheaf. Once this is understood, we shall present a proof of the above theorem diﬀerent from that in . The maximal extension of Gardeyn The material on maximal extensions is based on work and ideas of F. Gardeyn from [16,§ 2]. We follow the approach in . By B we denote a k-algebra which is essentially of ﬁnite type. Typically it is equal to A or to A/n for some ideal n ⊂ A. We omit almost all proofs. They can be found either in [16,§ 2] or in [4, Ch. 8]. Throughout the discussion of maximal extensions, we ﬁx an open immersion j : U → X and a complement Z ⊂ X of U. Deﬁnition 9.12(Gardeyn). Suppose F ∈ Cohτ(U, B). (a) A coherent τ -subsheaf G of j∗F with j∗G = F is called an extension of F. (b) The union of all extensions of F is denoted by j#F ⊂ j∗F. (c) If j#F is coherent, it is called the maximal extension of F. It is also denoted by F max. The assignment F → j#F deﬁnes a functor Cohτ(U, B)→ QCohτ(X, B). Note that if j∗F is not coherent, the same holds for j#(F, 0) = (j∗F, 0) - consider for instance the case Spec R → Spec K where R is a discrete valuation ring with fraction ﬁeld K. We state some basic properties: Proposition 9.13.Any τ -sheaf F has an extension to X which represents the crys tal j!F. Proof. This follows from the part of the proof of Theorem 3.10 giving the existence of the crystal j!F. The τ -sheaf j#F has the following intrinsic characterization modeled after the N´eron mapping property: Proposition 9.14.Suppose F ∈ Cohτ(U, B) and G ∈ QCohτ(X, B) are such that j∗G ∼=F. Then G is isomorphic to j#F if and only if the following conditions hold: (a)G is an inductive limit of coherent τ-sheaves, and (b) for all H ∈ Cohτ(X, B), the following canonical map is an isomorphism:  ττ(U,B)j∗H, F. Proposition 9.14 motivates the following axiomatic deﬁnition of maximal extension for crystals: Deﬁnition 9.15.A crystal G ∈ Crys(X, B) is called an extension of F if j∗G ∼=F. It is called a maximal extension if, in addition, the canonical map  Hom Crys(X,B)H, G−→ Hom Crys(U,B)j∗H, F is an isomorphism for all H ∈ Crys(X, B). Proposition 9.16.Let F be in Cohτ(X, B). If j#F is coherent, then the crys tal represented by F possesses a maximal extension and the latter is represented by j#F. Proposition 9.17.The functor j#is left exact on τ -sheaves. Moreover, if one has a left exact sequence of crystals such that the outer terms have a maximal extension, then so does the central term, and the induced sequence of the maximal extensions is left exact. We now impose the following conditions suﬃcient for our intended appli cations. Under these, the main result on the existence of a maximal extension, Theorem 9.22, is due to Gardeyn. (a) The ring B is ﬁnite over k or over A. (b) X is a smooth geometrically irreducible curve over k and U⊂ X is dense. Proposition 9.18.For F ∈ Cohτ(U, B) and G an extension of F, the following assertions are equivalent: (a)G is the maximal extension of F. (b) For any x∈ Z and j x: Spec O X,x→ X the canonical morphism, the τ-sheaf j x∗G is the maximal extension of i∗ηF; here η is the generic point of X This proposition allows one to reduce the problem of the existence of a max imal extension to the situation where X is a discrete valuation ring. The proof is a simple patching argument. Deﬁnition 9.19(Gardeyn). Let G be a locally free τ-sheaf on X over B. Then G is called good at x∈ X if τ is injective on i∗x G. It is called generically good if it is good at the generic point of X. Note that if G is generically good, then it is good for all x in a dense open subset. Proposition 9.20.If G ∈ Cohτ(X, B) is an extension of F ∈ Cohτ(U, B) such that G is good at all x ∈ Z, then G = j#F. The point is that after pulling back the situation to any Spec O X,x for x∈ Z, the fact that G is good at x easily implies that it is a maximal extension. Now one can apply Proposition 9.18. Corollary 9.21.The unit τ -sheaf 1l X,A is good at all x∈ X. Suppose now that G ∈ Cohτ(X, B) is an extension of F ∈ Cohτ(X, B) such that i∗x G ∼= 1l x,A for all x∈ Z. Then G = j#F and moreover in Crys(X, B) the following sequence is exact: 0−→ j!F −→ j#F −→ ⊕x∈Z 1l x,A−→ 0. The following are the central results on maximal extensions: Theorem 9.22(Gardeyn). If F is a locally free, generically good τ-sheaf on U over B, then j#F is locally free. Theorem 9.23.Suppose B is ﬁnite. Then j#: Crys(U, B)→ Crys(X, B) is a well deﬁned functor. Moreover one has ◦ j#∼= j∗◦  where : Crys(. . .) → ´Et c(. . .) is the functor in Theorem 9.6. Another simple assertion along the lines of Corollary 9.21 is the following: Proposition 9.24.Suppose F ∈ Coh(U, A) has a maximal extension to X. Then the canonical homomorphism of crystals max F max⊗A A/p →F ⊗A A/p is injective. If F max⊗A A/p has good reduction at all x ∈ Z, it is an isomorphism. Proof of Theorem 9.11 Proof. It is well known that the ﬁrst ´etale cohomology of a curve for the constant sheaf F p can be expressed in terms of the p-torsion group of its Jacobian. Denoting by the superscript∨the formation of the F p-dual, i.e, Hom F(,F p p), one has ¯ ∼∨ Jac K,p p k= H et 1 C K,p/¯k,F p. Both sides carry Galois actions of Gal(¯k/k∞) and of G. The extension H p+/H+ is totally ramiﬁed at all places above p. Therefore it is linearly disjoint from the constant ﬁeld extension ¯kH+/H+, and hence the two actions commute. We tensor both sides with A/p over F p. This allows to decompose them into isotypic compo nents for the semisimple action of G, whenever desired. Observe that the isotypic components on the left are the groups C(w). To analyze the right-hand term, we introduce some notation. Let o p denote the order of (A/p)∗. Denote by f p: C K,p→ C H+the G-cover of smooth projective geometrically irreducible curves over k∞corresponding to H p+/H+. Deﬁne U p to be Spec O+minus the ﬁnitely many places above p and let j p denote the open immersion of U p→ C H+. Over U p the representation of G on ϕ[p] is unramiﬁed, and thus it deﬁnes a lisse ´etale sheaf of rank one over A/p which we denote by ϕ[p]. This sheaf and all its tensor powers become, after pullback along the ﬁnite ´etale cover f p−1(U p)→ U p, isomorphic to the constant sheaf A/p on C K,p with generic ﬁber A/p. Using simple representation theory, one deduces that f p∗A/p∼=)ϕp⊗w. U p w∈Z/o p From adjunction of j∗and j∗we deduce a homomorphism ) f p∗A/p −→j p∗ϕ[p]⊗w. w∈Z/o p On stalks one can verify that the map is an isomorphism: At points where the representation ϕ[p]( ¯K)⊗w is ramiﬁed, the sheaf j p∗ϕ[p]⊗w is zero and so is the corresponding summand on the left. At the other (unramiﬁed) points above p, ∞, the sheaf j p∗ϕ[p]⊗w is lisse, as is the corresponding summand on the left. Using H et 1(C K,p,) ∼= H et 1(C H+, f p∗), we deduce ¯)∨ Jac K,p p k⊗F A/p ∼= H et 1 C H+/¯k,j p∗ϕp⊗w. p w∈Z/o p Now we decompose the isomorphism into isotypic components – note that∨ changes the sign of w. This yields C(w) ∼= H et 1 C H+/¯k, j p∗ϕ[p]⊗(−w)∨. Our next aim is to relate the coeﬃcient sheaf to a tensor power of the τ -sheaf M(ϕ) attached to the Drinfeld module ϕ. We observed earlier that (M(ϕ) ⊗A A/p) on Spec O+is dual to ϕ[p]. Since ϕ[p]( ¯K) is totally ramiﬁed at p, the same holds for tensor powers w, except if w is a multiple of o p – here (M(ϕ) ⊗A A/p)⊗w may be zero above p, while the representation deﬁned by ϕ[p]( ¯K)⊗w is trivial and hence lisse. Let j : Spec O+→ C H+denote the canonical open immersion. Using Theorem 9.23 for w not a multiple of o p, we ﬁnd ⊗(−w)∼⊗w j p∗ϕp= j#M(ϕ) ⊗A A/p. One can either use that the representation deﬁned by ϕ[p]( ¯K)⊗w is unramiﬁed at the places above∞ if and only if (q − 1) divides w – the ramiﬁcation group at those places is k∗⊂ A/p∗∼= G - or a direct computation on the side of τ -sheaves to deduce from Corollary 9.21 that j!(M(ϕ) ⊗A A/p)⊗w→ j#(M(ϕ) ⊗A A/p)⊗w is an isomorphism whenever w is not a multiple of (q*− 1) and has cokernel ∞|∞1l∞,A/p otherwise – the sum is over all places of H+above∞. One can in fact also prove that j*!M(ϕ)⊗w→ j#(M(ϕ))⊗w is an isomorphism for (q− 1) | w and has cokernel∞|∞1l∞,A otherwise. Finally we use that  commutes with all functors deﬁned for crystals, so that to compute H et 1 we may ﬁrst compute H 1 for crystals and then apply . This yields  C(w) ∼= H 1(C H+/¯k, 1l C,A)⊗ A/p Spec ¯k for o p|w;(9.3) * C(w) ∼= H 1(C H+/¯k, j!M(ϕ)⊗w)/∞|∞1l Spec ¯k,A⊗ A/p(Spec ¯k) for (q− 1)|w, o p | w or w = 0;(9.4)  C(w) ∼= H 1(C H+/¯k, j!M(ϕ)⊗w⊗ A/p Spec ¯k for (q− 1) | w.(9.5) Note that without the evaluation (Spec ¯k) outside  we would have a sheaf on the right-hand side. In either case, the expression inside (. . .) is a τ -sheaf G on Spec ¯k over A/p. By Proposition 5.16 it can be written as G ∼=G ss⊕G nil where on the ﬁrst summand τ is bijective and on the second nilpotent. The underlying modules in both cases are ﬁnitely generated projective over k⊗k A/p. By the theory of the Lang torsor the τ -ﬁxed points of the ﬁrst summand form free A/p vector space of dimension equal to rank k¯⊗A/p G ss; those of the second summand are clearly zero. Moreover computing  in the case at hand, cf. (9.1), is precisely the operation of taking τ -ﬁxed points – the only relevant ´etale morphism to Spec ¯k is the identity. At the same time, the dual characteristic polynomial of  H 1(C H+, j!M(ϕ)⊗w⊗ A/p) has degree precisely equal to rank¯k⊗A/p G ss. By Theorem 8.10 and Remark 8.12 this rank is the degree of L O+(w, T ) mod p. Thus we have now proved The orem 9.11 (a). One may wonder about the case o p|w and w = 0. There are two answers why this case is covered as well. The formal answer is that the L-functions mod p for w and win−N 0 are equal whenever w≡ w(mod o p), and so it sufﬁces to understand the case w = 0. An answer obtained by looking closer at what is happening goes as follows: The places above p have L-factors congruent to 1 module p. So it does not matter whether we leave them in or not, i.e., whether we compute via the trace formula with H 1(C H+/¯k, j∗M(ϕ)⊗w⊗ A/p) or with H 1(C H+/¯k, j p∗M(ϕ)⊗w|U p⊗ A/p). To prove (b) and (c) observe that we obtain formulas for C(w) by taking invariants under Gal(¯k/k∞) in the isomorphisms (9.3) to (9.5). The eﬀect on the right-hand sides is that we replace the curve C H+/¯k by C H+/k∞= C H+ and that for the resulting sheaf we compute global sections over Spec k∞. This amounts to the same as computing the ﬁxed points under τ of the expressions inside the brackets (. . .). Since the τ -ﬁxed points being non-zero is the same as the assertion that 1 is an eigenvalue of the τ -action, part (b) is now clear. Note* that∞|∞1l Spec ¯k,A being a subcrystal of M := H 1(C H+/¯k, j!M(ϕ)⊗w) in (9.4) means that (T− 1)h+A is a factor of the L-function of M . Part (c) simply says that the dimension of the eigenspace of the eigenvalue 1 for the τ action is precisely the dimension of the space of τ -invariants, and the latter is C(w). Lecture 10 Drinfeld Modular Forms The aim of this lecture is to give a description of Drinfeld modular forms via the cohomology of certain universal crystals on moduli spaces of rank 2 Drinfeld modules. The basic deﬁnition of Drinfeld modular forms goes back to Goss [20, 21]. Many important contributions are due to Gekeler, e.g., . Moreover, Gekeler ob tains foundational results on Drinfeld modular curves in . The work of Gekeler and Goss gives a satisfactory description of Drinfeld modular forms as rigid an alytic functions on the Drinfeld analog of the upper half-plane. The important work  of Teitelbaum links this to harmonic cochains on the Bruhat-Tits tree underlying the Drinfeld symmetric space. As shown in , the latter provides the link to a description of modular forms via crystals. After introducing a moduli problem for Drinfeld modules of arbitrary rank (with a full level structure) in Section 10.1, in Section 10.2 we give equations for a particular example of such a moduli space. The universal Drinfeld module on it will give rise to a crystal via Anderson’s correspondence between A-modules and A-motives. Following the classical case, this crystal yields a natural candi date for a cohomological description of Drinfeld modular forms; cf. Section 10.3. The cohomological object so obtained plays the role of a motive for the space of forms of ﬁxed weight and level. It has various realizations: Its analytic realization is directly linked to Teitelbaum’s description of Drinfeld modular forms via har monic cochains. In Section 10.4 we consider its ´etale realizations. They allow one to attach Galois representations to Drinfeld-Hecke eigenforms as in the classical case. Unlike in the classical case, the representations are one-dimensional! The fol lowing Section 10.5 gives some discussion of ramiﬁcation properties of the Galois representations so obtained. In Section 10.6 we indicate in what sense these com patible systems of one-dimensional Galois representations associated to a cuspidal Drinfeld-Hecke eigenform arise from a (suitably deﬁned) Hecke character. So far the nature of these characters is still rather mysterious. We conclude with Sec tion 10.7, which contains an extended example of the computation of the crystals associated to some low-weight modular forms for F q[t] and a particular level. It pro vides exemplary answers to many natural questions and points to open problems. Due to lack of time and space, we omit many details and refer the reader to . We recall the following notation: By C = Spec A we denote an irreducible smooth aﬃne curve over k whose smooth compactiﬁcation is obtained by adjoining precisely one closed point∞. We deﬁne K as the fraction ﬁeld of A, K∞as the completion of K at∞, and C∞as the completion of the algebraic closure of K∞. Similarly, for any place v of K we denote by K v the completion of K at v; by O v the ring of integers of K v, and by k v the residue ﬁeld of K v. Often A will simply be k[t]. We ﬁx a non-zero ideal n of A. By A[1/n] we denote the localization of A at all elements which have poles at most at n. The weight of a form will usually be denoted by n (or n+2), the letter k being taken as the name of the ﬁnite base ﬁeld. 10.1 A moduli space for Drinfeld modules Let S be a scheme over Spec A[1/n]. Let ϕ := (L, ϕ) be a Drinfeld A-module on S of rank r, i.e., the line bundle L considered as a scheme of k-vector spaces over S is equipped with an endomorphism ϕ : A→ End(L), a → ϕa. For any a∈ A, the morphism ϕa: L→ L is ﬁnite ﬂat of degree #(A/a)r and hence its kernel  ϕ(a):= Kerϕa: L→ L is a ﬁnite ﬂat A-module scheme over S. If furthermore all prime factors of the ideal aA are prime factors of n, working locally on aﬃne charts, it follows that dϕa is a unit in A[1/n], and thus that ϕa(z) is a separable polynomial. This means that ϕ[(a)] is ´etale over S. As a consequence the subscheme + ϕn:=ϕ(a) a∈n0 is ﬁnite ´etale over S and of degree equal to #(A/n)r. A level n-structure on ϕ is an isomorphism r∼= ψ :A/n−→ ϕn S of ﬁnite ´etale group schemes over S, where (A/n)r denotes the constant group S scheme on S with ﬁber (A/n)r. Deﬁnition 10.1.Let M r(n) denote the functor on A[1/n]-schemes S given by  S−→ϕ, ψ| ϕ =L, ϕis a rank r Drinfeld A-module on S, ψ is a level n-structure on ϕ/ ∼=, i.e., we consider such triples up to isomorphism. One has the following important theorem from : Theorem 10.2(Drinfeld). Suppose 0= n  A. Then the functor M r(n) is rep resented by an aﬃne scheme M r(n) which is smooth of ﬁnite type and relative dimension r− 1 over Spec A[1/n] Remark 10.3.In , Drinfeld also deﬁnes level structures for levels dividing the characteristic of the Drinfeld module. Using these, he obtains a more general theorem as above: A universal Drinfeld module with level n-structure exists for A-schemes provided that n has at least two distinct prime divisors. The universal space is regular of absolute dimension r. Its pullback to Spec A[1/n] is the space M r(n). Remark 10.4.Let (L univ, ϕuniv, ψuniv) denote the universal object on M r(n) = Spec R univ. Then in fact L univ is the trivial bundle on Spec R univ. The reason is that the image under ψuniv of any non-zero element in (A/n)r is a section of L which is everywhere diﬀerent from the zero section, i.e., it is a nowhere vanishing global section. For this reason, we shall in the universal situation always assume that L univ= O M r(n). Moreover since M r(n) is smooth and hence reduced, the universal Drinfeld module is in standard form, i.e.,  A−→ R univτ,a−→ ϕa= α0(a) + α1(a)τ +· · · + αr deg(a)τr deg(a) with αr deg a(a)∈ (R univ)∗. Exercise 10.5. Let ϕ be a Drinfeld A-module of rank r in standard form with L =G a, i.e., a ring homomorphism  A−→ Rτ,a−→ ϕa= α0(a) + α1(a)τ +· · · + αr deg(a)τr deg(a) for some A-algebra R. Suppose s∈ R is an a-torsion point which is non-zero on any component of R, i.e., α0(a)s + α1(a)s q+· · · + αr deg(a)s q r deg(a)= 0. Show that a→ sϕa s−1 deﬁnes an isomorphic Drinfeld A-module such that 1 is an a-torsion point. 10.2 An explicit example To make the above example more explicit, we consider the following special case: Let A = k[t] and n = (t). Then any Drinfeld A-module of rank r over an aﬃne basis Spec R in standard form is described by the image of t∈ A in R[τ]. This image is a polynomial of degree r which we denote by α0+ αaτ +· · · + αrτr with αr∈ R∗. We assume that R is an A[1/t]-algebra, so that also α0∈ R∗, and hence ϕt is a separable polynomial. Thus over a ﬁnite ´etale extension of R the solution set of ϕt= 0 is an F q-vector space of dimension r. Suppose we have a basis of t-torsion points s 1, . . . , s r already deﬁned over Spec R. We trivialize the bundle L via the section s 1. This means that we have s 1= 1 on L(Spec R) which is isomorphic via s 1 to G a(Spec R) = (R, +). The set of all t-torsion points is thus the set r i=1 s iαi, where the αi range over all elements of k. Since these points are precisely the roots of ϕt, we ﬁnd that r ϕt(z) = c·z−s iαi.(10.1) α∈k r i=1 Recall the following result from [24, 1.3.7]: Proposition 10.6(Moore determinant). Suppose w 1, . . . , w r lie in an F q-algebra. Then w 1 w 1 q w 1 q 2. . .w q 1 r−1 w 2 w 2 q w 2 q 2. . .w 2 q r−1r .......... .....=w i+ i−1 w i−1+· · · + 1 w 1. w r w r q w r q 2. . .w r q r−1i=1(i−1,...,1)∈k i−1 By the theory of the Moore determinant, we obtain z z q z q 2. . .z q r s r s r q s r q 2. . .s r q r,s r s r q. . .s r q r−1 ϕt(z) = c·.......... ............ ..... s 1 2 s 1 2 q s 2 1 q 2. . .. . .s 1 2 q rs 1 2 s 1 2 q. . .. . .s 1 2 q r−1.(10.2) Since the constant term of ϕt, i.e., the coeﬃcient of z, is θ, the image of t under A[1/t]→ R, we can solve for c by computing the coeﬃcient of z on the right-hand side. It yields θ = c·....... .....,s...r s r...q. . .. ..s r...q r−1 1 1. . .11 2 s 2 1 q. . .. . .s 2 1 q r−1 s r s r q. . .s r q r−1q−1 = c·....... ..... s 2 s 2 q. . .s 2 q r−1. 1 1. . .1 Proposition 10.7.Let ⎢⎢s r s r q. . .s r q r−1−1⎤ R = k⎢⎢θ±1, s 2, . . . , s r,....... .....⎥⎥⎥ ⎣s 2 s 2 q. . .s 2 q r−1⎥⎦ 1 1. . .1 for indeterminates θ, s 2, . . . , s r, and let ϕ : k[t]→ R be the rank r Drinfeld module where ϕt is deﬁned by (10.2). Then M r(t) ∼= Spec R and the universal triple is (ϕ,G a,R, ψ) where ψ : (k[t]/(t))r→ ϕ[t] is deﬁned by mapping the ith basis vector on the left to s i with the convention that s 1= 1. Proof. Let (L, ϕ, ψ) be a Drinfeld module with a full level t-structure on a scheme S = Spec R. Assume ﬁrst that S is aﬃne. As in the above case we may take the section ψ(1, 0, . . . , 0) of L to trivialize it. By the construction of R, there is a homomorphism from R→ Rover F q[θ±1]-algebras sending s i to the torsion point si:= ψ(0, . . . , 0, 1, 0, . . . , 0), where 1 occurs in the i th place. The si determine in the same way as the s i the function ϕt. Hence (L, ϕ, ψ) is the pullback of (L, ϕ, ψ) under the morphism Spec R→ Spec R. Moreover the morphism R → R with this property is unique: The element s1 determines a unique isomorphism L→ G a. With respect to the coordinates of G a given by s1= 1, the sections s 2, . . . , s r are uniquely determined from (L, ϕ, ψ) and hence R→ Ris unique. Now, let S be arbitrary. Fix an aﬃne coverSpec R ii. By the preceding para graph we have unique morphisms Spec R i→ Spec R. However by the uniqueness it also follows that on any aﬃne subscheme of Spec R i∩ Spec R j, the two restric tion to this subscheme agree. This in turn means that the local morphisms patch to a morphism S→ Spec R under which (L, ϕ, ψ) is the pullback of (L, ϕ, ψ). The uniqueness is true locally, hence also globally. This completes the proof of the representability of the functor M r(n). Remark 10.8.Geometrically Spec R can be described as follows: It is the aﬃne space A r−1 k[θ±1]with all the #k r−1 hyperplanes with coordinates in k removed. To see this, observe that s 1= 1 and, by applying the Moore determinant, one has s r s r q. . .s r q r−1 ....... .....r s 2 s 2 q. . .s 2 q r−1=i=1(i−1,...,1)∈k i−1(s i+ i−1 s i−1+· · · + 1 s 1). 1 1. . .1 Proposition 10.9.We keep the notation of Proposition 10.7. The t-motive on Spec R corresponding to the universal Drinfeld module is isomorphic to the pair ⎛⎞ 0 0. . .0 t−θ ⎜⎜1 0. . .0−ααr 1⎟⎟ F :=R[t]r, τ =⎜0 1. . .0−ααr 2⎟⎟ ⎜⎝..... ......⎟⎟(σR× id t) ....⎠ 0 0. . .1−αr−1 αr where αi= θ·.... ......... .....,s...r s r...q. . .. ..s r...q r−1q s 2. . .s 2 q i−1 s 2 q i+1. . .s 2 q rs 2 s 2 q. . .s 2 q r−1 1. . .1 1. . .1 1 1. . .1 for i = 0, . . . , r. Proof. The shape of τ is determined as in Example 1.10. The formulas for the coeﬃcients result easily from (10.2) by ﬁrst eliminating c and then expanding the determinant in the numerator of (10.2) according to the ﬁrst row. Let us, for some computations below, describe the case n = 2 in greater detail. For simplicity, we write s := s 1. In this case, −1 R = kθ±1, s,s q− s, F =R[t]2, τ =0(t/θ− 1)(s − s q)q−1  1(s− s q 2)(s− s q)−1σR× id t. Substituting u := s q− s and observing that s − s q 2= u + u q, we obtain R u:= k[θ±1, u±1],F =R u[t]2, τ =0(t/θ− 1)u q−1  1 1 + u q−1σR u× id t. (10.3) The introduction of u corresponds to a cover Spec R→ Spec R u of degree q. The space Spec R u is a moduli space for Drinfeld modules with a level Γ1(t)-structure, where Γ1(t) is the set of matrices a c b d∈ GL 2(k[t]) such that a, d≡ 1 (mod t) and c≡ 0 (mod t). Note that the moduli correspond to a choice of two t-torsion points 1, u where u is only determined up to adding a multiple of 1. (Due to our choice of coordinates for the line bundle underlying the rank 2 Drinfeld module, the ﬁrst torsion point is 1.) In the sequel, the symmetric powers Sym n F and their extension by zero to a compactiﬁcation of M r(n) will play an important role. We make this explicit in the setting of (10.3). Here a smooth compactiﬁcation of Spec R u=G m/k[θ±1]is P 1 k[θ±1]. One simply has to extend F to 0 and ∞. (Taking symmetric powers is compatible with this extension process.) At u = 0, the matrix 0 (t/θ−1)u q−1 specializes to 0 0, i.e., the ex 1 1+u q−1 1 1 tension is deﬁned but not zero. At u =∞ specializing the matrix leads to poles. To analyze the situation, we introduce v = 1/u, so that the matrix describing τ becomes 0 (t/θ−1)v 1−q 1 1+v 1−q. Next we multiply the standard basis e 1, e 2 of R u[t]2 by v. Then the action of τ for this new basis is given by v−1 0(t/θ− 1)v 1−q  1 1 + v 1−qσR u× id t v = v−1 0 1(t/θ1 + v− 1)v q−1 q−1 v q  0(t/θ− 1) v 1−q v 1−q+ 1. The following result summarizes the above discussion. Proposition 10.10.Consider R u= k[θ±1, u±1] as an algebra over A[1/θ] = k[θ±1]. (a) The moduli space of rank 2 Drinfeld modules with level Γ1(t)-structure is isomorphic to Spec R u as a scheme over Spec A[1/θ]. (b) A relative smooth compactiﬁcation of Spec R u is the projective line P 1. A[1/θ] (c) The A-motive attached to the universal Drinfeld A-module over Spec R u is given by F =R u[t]2, τ =0 1(t/θ1 + u− 1)u q−1 q−1 σR u× id t.  P 1−1·[∞], τand j!F :=O⊕2 P 1  , the modular curve M 2(n)/F is ordinary for any n. By a result of Pink , there is a Grothendieck-Ogg-Shafarevich type formula for the F p-dimension of the cohomology of ´etale F p-sheaves on curves, provided the curve has an ordinary cover over which the monodromy of the ´etale sheaf is unipotent. In the case at hand, by Gekeler’s result we can take the cover M 2(np m). The formula of Pink shows that  p p n(n) for any m. From this the theorem follows easily – the proof is essentially the same as the proof that the n-torsion of a Drinfeld module away from the characteristic is equal to A/n r. It follows that S n(n)et,p/F deﬁnes a continuous homomorphism  n(n)A p. Moreover S n(n)et,p carries the Hecke action induced from S n(n). In  the following result is proved: Theorem 10.13.For any prime q diﬀerent from n and p, the actions of T q and of Frob q are the same on the reduction of S n(n)et,p from Spec A[1/p] to Spec A/q. Since the Hecke operators commute among each other and since the Frob q where q runs through all maximal ideal prime ideals of Spec A[1/np] are dense in Gal(F sep/F ), we deduce: Corollary 10.14.The image of ρA,n is abelian. The crystal S n(n) has given rise to two realizations, namely an analytic one and for each maximal ideal of A an ´etale one: anτoo_ _ __ _//_ S n n/K S n n S n n et,p. ∞ In each case, the realizations inherits a Hecke action. As examples show (see ), the Hecke action may not be semisimple. So we pass in both cases to the semisim pliﬁcation and decompose S n(n) into Hecke eigenspaces (if necessary after invert ing some elements in the coeﬃcient ring A). This yields a correspondence between Hecke eigensystems of Drinfeld modular forms and simple abelian Galois repre sentations. Before we give the precise statement from , we recall the following classical theorem due to Goss: Theorem 10.15(Goss). Let f be a Hecke eigenform of weight n and level n with Hecke eigenvalues a p(f ) for all p not dividing n. Then all a p(f ) are integral over A and the ﬁeld F f:= F (a p(f )| p ∈ Spec A[1/n]) is a ﬁnite extension of F . Denote by O f the ring of integers of F f. For any prime P of O f, the com P pletion at P will be 5 O f. Theorem 10.16(B.). Let f be as above and suppose that f is cuspidal. Then there exists a system of Galois representations P ρf,P: Gal(F sep/F )→ GL 1( 5 O f)P∈Max(O f) uniquely characterized by the condition that, for each ﬁxed P, one has for almost all q prime to Pn the equation  ρf,P Frob q= a q(f ), so that the right-hand side is independent of the prime P. Remark 10.17. (a) It is not clear whether there is a theory of old and new forms for Drinfeld modular forms. So one cannot proceed as in the classical case. (b) There are various counterexamples to a strong multiplicity one theorem, by Gekeler and Gekeler-Reversat; e.g., [19, Example 9.7.4] for an example in weight 2. (c) As far as we know there are no counterexamples to multiplicity one for S n(Γ0(p)) for ﬁxed n and a prime p of Spec A. (d) Despite the results of the following section, the ramiﬁcation locus of the f)is rather mysterious. 10.5 Ramiﬁcation of Galois representations associated to Drinfeld modular forms From a result of Katz  one easily deduces the following. Theorem 10.18.Let S n(n)max be the maximal extension in the sense of Gardeyn representing the same-named crystal. Deﬁne D as the support of the cokernel of the injective homomorphism τlin: (σ× id)∗S n(n)max→ S n(n)max. Let f be a Drinfeld modular form of level n and weight n, and let P be a maximal ideal of O f and p its contraction to A. Then ρf,P is unramiﬁed at all primes q of A such that (q, p) is not in D. Moreover for such q one has  det 1− zρf,P Frob q= 1− za q f,  det 1− zρA,n Frob q= det 1− zT q ss|S n n, where T q ss is the semisimpliﬁcation of T q acting on the analytic space of modular forms S n(n) of weight n and level n. Remark 10.19.After replacing the coeﬃcient ring A by a larger ring Awhich is a localization of A at a suitable element, one can in fact decompose S n(n) into components corresponding to generalized eigenforms under the Hecke action. By enlarging Ato a ring A, one may furthermore assume that Acontains all the Hecke eigenvalues of all eigenforms. Then, over A, to any eigenform f one has a corresponding subcrystal of S f of S n(n)⊗A A. Katz’ criterion then applies to the Gardeyn maximal model of S f. This gives, in theory, a precise description of the ramiﬁcation locus of ρf,P- provided that P is in Max(A) - given by a divisor D f on Spec A× Spec A. The deﬁnition of D in the previous theorem is coarser. It gives a bound on ramiﬁcation for all eigenforms f simultaneously. While Galois representations ρf,P for eigenforms f of level n tend to be ram iﬁed at the places above n, it is not clear how the additional places of ramiﬁcation are linked to P. In an abstract sense, the answer is that this link is given by D, or more precisely by D f. Concretely, we do not know how to determine D f from f or D from n and n. Some explicit examples are given below. One clue to the ramiﬁcation locus is given by Theorems 10.21 and 10.22. They describe a link be tween places of bad reduction of modular curves and the ramiﬁcation of modular forms. The following result might serve as a motivation. Let K be a local ﬁeld of characteristic p with ring of integers O and residue ﬁeld k. Let A/O be an abelian scheme with generic ﬁber A/K of dimension g and special ﬁber A/k. The p n-torsion subscheme of A (or A) is denoted by A[p n] (or A[p n], respectively) and for any ﬁeld L⊃ K we write A[p n](L) for the group of L-valued points of A[p n]. Consider the p-adic Tate module  Tate p A := lim A p n K sep ←− n of A. The module underlying Tate p A is free over Z p. One deﬁnes the p-rank of A as rank p A := rank Z Tate p A. It satisﬁes 0≤ rank p A≤ g. The action p of G K:= Gal(K sep/K) on Tate p A is Z p-linear and thus with respect to some Z p-basis it yields a Galois representation ρA,p: G K−→ Aut Z Tate p A ∼= GL rank p p A Z p. By Hensel’s Lemma any p n-torsion point of A/k will lift to a unique p n-torsion point of A/K. Thus rank p A/k ≤ rank p A, i.e., the p-rank can only decrease under reduction. The following result from  links the ramiﬁcation of ρA,p to the p-rank: Theorem 10.20.The p-rank is invariant under reduction if and only if the action of G K on Tate p A is unramiﬁed. Let us, after this short interlude come back to the ramiﬁcation of Drinfeld modular forms: Recall that for a Hecke eigenform f , we denote by a q(f ) the Hecke eigenvalue of f at q. Let J n denote the Jacobian of the Drinfeld modular curve for level n. Then the following two results are shown in  (in a more precise form): Theorem 10.21.Suppose that f is a doubly cuspidal Drinfeld-Hecke eigenform of weight 2 and level n. Then for a prime q not dividing n the following results hold: (a) If a q(f )= 0, then for any (or for all) P ∈ Max(O f) the representation ρf,P is unramiﬁed at q. (b) If a q(f ) = 0, then the Jacobian J n has non-ordinary reduction modulo q. Note that in the case of weight 2 there is a representation ρf from G K into GL 1 over a ﬁnite ﬁeld F contained in O such that ρf,P= ρf⊗F O P for all P ∈ Max(O f). This is similar to the case of classical weight 1 modular forms, and it explains why the condition in (a) is independent of P. Suppose now that f has weight larger than two, and consider a representation ρf,P for P ∈ Max(O f) over p ∈ Max(A). As ρf,P is associated to a Hecke character (see 10.25) and because it is known that such have square free levels, it follows that ρf,P is ramiﬁed at q if and only if its reduction mod P is so. As in the case of classical modular forms, the reduction mod P is congruent to the representation of a form of weight 2 and level np. To the latter one can apply the previous result. This yields: Theorem 10.22.Suppose f is a doubly cuspidal Drinfeld-Hecke eigenform of weight n≥ 3 and level n. Let P be in Max(O f) with contraction p ∈ Max(A). Then for a prime q not dividing nq the following hold: (a) If a q(f )= 0 (mod P), then the representation ρf,P is ramiﬁed at q. (b) If a q(f ) = 0 (mod P), then the Jacobian J np has non-ordinary reduction modulo q. Note that in known examples, e.g., , the places of ramiﬁcation of ρf,P which are prime to the level n do typically depend on P unlike in the case of weight 2. Question 10.23. For classical modular forms it is simple to list all the primes which are ramiﬁed for the associated Galois representations. By the theory of new forms these primes are those dividing the minimal level associated to the modular form together with the place p (or the places above p) if one considers p-adic Galois representations. Because of this simplicity one wonders if there is also a simple recipe in the case of Drinfeld modular forms. Numerical data seem insuﬃcient to make any predictions. This deserves to be studied more systematically. Because of Theo rem 10.21 this question is directly linked to the reduction behavior of Drinfeld modular curves (at primes of good reduction!) and their associated Jacobians. 10.6 Drinfeld modular forms and Hecke characters In  we introduce a notion of Hecke character that was more general than previous deﬁnitions due to Gross  and others. Our motivation was a question of Serre and independently Goss which asked whether Drinfeld modular forms are linked to Hecke characters. In  this question was answered in the aﬃrmative. We will brieﬂy indicate this result. Deﬁnition 10.24.Let F be a global function ﬁeld over F p. A homomorphism ∗ χ :A∗F−→ F p t, where A∗F denotes the ideles of F and F p(t) is discretely topologized, is a Hecke character (of type Σ) if (a) χ is continuous (i.e., trivial on a compact open subgroup of A∗F) and (b) there exists a ﬁnite subset Σ =σ1, . . . , σr of ﬁeld homomorphisms σi: F → F p(t) and n i∈ Z for i = 1, . . . , r, such that χ(α) = σ1(α)n 1· . . . · σr(α)n r, where on the left we regard F∗as being diagonally embedded in A∗F. Note that for any compact open subgroup U⊂ A∗F the coset space F∗\A∗F/U admits a surjective degree map to Z whose kernel is ﬁnite and may be interpreted as a class group. By an observation of Goss , the Hecke characters as deﬁned above have square free conductors. The main result on Hecke characters and modular forms is the following. Theorem 10.25.For any cuspidal Drinfeld-Hecke eigenform f with eigenvalues (a p(f ))p∈Max(A)there exists a unique Hecke character χf:A∗F−→ K f∗ such that a p f= χf 1, . . . , 1, p, 1, . . . , 1  at p  for almost all p ∈ Max A. Unfortunately, the set Σf for the Hecke character χf, as required by Deﬁnition 10.24, is rather mysterious. The proof of the theorem sheds no light on it. What is however not so hard to see is that the ramiﬁcation divisor Df introduced ∈Σf Graph(σ) where the σ∈ Σf are viewed as morphisms of algebraic curves. Example 10.26.The following Hecke characters are taken from . The are asso ciated to Drinfeld modular forms. Let F =F q(θ), let n be in2, . . . , p and consider σn:F q(θ)−→ F q(t) : θ→ (1− n)t. . Deﬁne 1∗ U :=1 + θF q[[θ]]×O v∗× F q. θ ν |0,∞ Then the natural homomorphism F q(θ)∗ −→ A∗F/U is an isomorphism. Hence there exists a unique Hecke character χn:A∗F→ F p(t)∗ such that χn is trivial on U and such that it agrees with σn on F q(θ). Remark 10.27.The Hecke character χf provides a compact way of storing essen tial information about the cuspidal Drinfeld-Hecke eigenform f . To explain this, suppose that A =F q[t], that the weight of f is n and that we have computed its Hecke eigenvalues a p(t) for many primes p not dividing n. The conductor of χf consists of those primes p | n for which T p acts as zero and some primes dividing n. Having computed many eigenvalues, we may thus hope to know the prime-to-n part of the conductor of χf and thus a lower bound for the conductor m f⊂ A of χf, by which we mean the largest square-free ideal such that χf is trivial on the group U f of all ideles congruent to 1 modulo m f. Suppose furthermore that we know the coeﬃcient ﬁeld K f of f . For theoretical reasons the character χf is trivial on F∞∗(the image of the decomposition group at∞ of any ρf,P is trivial). If the weight n is 2, then χf is of ﬁnite order and in particular Σ is empty. Knowing a bound on m f and that χf is trivial on F∞∗, by computing suﬃciently many Hecke eigenvalues, we can completely determine χf as a function on F∗\A∗F/U f F∞∗. If the weight n is larger than 2, it is necessary to ﬁnd the embeddings σi:F q(θ)→ K f, i = 1, . . . , r and their exponents n i. Denote by b i∈ K f the image of θ under σi. If g is any element of A =!F q[θ] which is congruent to 1! modulo m f, then T g acts on f as i g(b i)n i. At the same time, if (f ) =p p m p for exponents m p= ord p(g)∈ N 0, then we have the equation mn a p f p=g b i i. p i The number r, the exponents n i and the b i can be determined by the following algorithm. Let n run through the positive integers and let (n i) run through all (unordered) partitions of n. For each partition, determine the solution set of m pn a p f=g x i i, p i while g runs through many polynomials congruent to 1 modulo m f. The algorithm terminates if a solution is found. The algorithm will terminate because of the above theorem. If it terminates and if suﬃciently many g have been tested, the solution can assumed to be correct. In all explicitly known cases one has r = 1 and n 1= 1; but this may be due to the fact that not so many examples are known. 10.7 An extended example In this section we will carry out the explicit computation of the cohomology of certain crystals associated to Drinfeld cusp forms of low weight and level Γ1(t). We consider R u= k[θ±1, u±1] as an algebra over A[1/θ] = k[θ±1] as in Proposi tion 10.10 and let f be the morphism of the corresponding schemes. Let further more ¯f :P 1 A[1/θ]→ Spec A[1/θ] be the relative compactiﬁcation of Spec R u. We consider the τ -sheaf F =R u[t]2, τ =0 1(t/θ1 + u− 1)u q−1 q−1 σR u× id t and wish to compute R 1 f¯∗of  P 1 A[1/θ]− 1 ·∞, τ as a crystal. Abbreviating b := (t/θ−1)u q−1 and c := 1 + u q−1, the endomorphism Sym nτ on Sym n R u[t]2= R u[t]n+1 is given by ⎛ ⎜⎜ ⎜⎜ ⎜⎜ αn(σ× id), where αn:=⎜⎜ ⎜⎜ ⎜⎜ ⎝ ⎞ b n ·····⎟⎟⎟⎟ b 3· · ·b 3 c n−3n⎟⎟⎟ 3n⎟⎟. 2· · ·b 2 c n−2 2⎟ 23n⎟⎟ 1 1· · ·bc n−1 1⎠ 1 c c 2 c 3· · ·c n We denote the corresponding basis of R[u]n+1 by e j, j = 0, . . . , n. Next recall that we can compute the cohomology of a coherent sheaf G on P 1 S(over any aﬃne base S) as follows: Let A 1 S⊂ P 1 S be the standard aﬃne line contained in P 1 S. Let O∞,S be the aﬃne coordinate ring of the completion of P 1 S along the section∞ × S at ∞ and let K∞,S be the ring obtained from O∞,S by inverting the section at∞. Then one has the short exact sequence  ∞,S|K∞,S−→ H 1 P 1 S,G−→ 0. The sequence is obtained as the direct limit over U over the sequences for the computation of ˇCech cohomology where P 1 is covered A 1 and a second aﬃne set U containing∞ × S. We apply this to the base Spec A with A = k[θ±1, t] and the sheaf Sym n F. Disregarding τ we obtain the short exact sequence 1n 66 1 771 0−→ A[u]n+1⊕k⊗k A n+1−→ k⊗k A n+1−→ Coker n−→ 0. u u u This shows that Coker n= H 1(P 1, Sym n j#F) is a free A-module with basis u−i e j| i = 1, . . . , n − 1; j = 0, . . . , n. Let us write the elements of the cokernel as u−1 v 1+· · · + u 1−n v n−1, where the v j are column vectors over A of length n + 1. (We can write them in the basis e 0, . . . , e n.) Applying τ to the summands u−i v i yields  τ (u−i v i) = u−iqαn v i=u−i(q−1)αn u−i v i. Now observe that αn lies in A[u q−1]. Thus u−i(q−1)shifts the pole order at u = 0 (and u =∞) by multiples of (q − 1). We deﬁne matrices αn,i∈ M(n+1)×(n+1)(A) so that αn=αn,i u i(q−1). i≥0 Assumption 10.28. We now assume that the weight n lies in the interval0, . . . , q. Because the exponent of u−i lies in−1, . . . , 1 − n, the absolute value of the diﬀerence of two such i is at most q− 2. Therefore at most one summand in   τ (u−i v i) =u(i−i)(q−1)αn,iu−i v i i≥0 t θ− 1 and abbreviate x = u q−1. We enumerate the rows and columns by r, s∈ 0, . . . , n. We let r := n − r, so that this variable counts rows from the bottom starting at zero. Then the (r, s)-coeﬃcient of αn is  c r+s−n b n−r s= c s−r br s= (1 + x)s−r xrβrs n− r r r s−r =x+rβr s− r s. r =0 The (r, s)-coeﬃcient of αn,i is the coeﬃcient of x i in the previous line. Thus it is the summand for  = i− r, i.e.,  i− r r= βr(i− r)!(s − i)!(s− r)!(s− r)! r!s!  (i− r)! r!(s− i)!i!= βr i− r i s i.  Let w i be the transpose of the row vector 0, . . . , 0,iβi,iβi−1, . . . , ,iβ0 and  i i i n,i= w i⊗ x i and so  τu−i v i= u−i w i·x i v i. We deduce the following. As a τ -sheaf, Coker n is the direct sum of the sub-A modules W i spanned by u−i e j, j = 0, . . . , n. The τ -submodule W i contains itself the τ submodule Au−i w i, and because the image of W i under τ is contained in Au−i w i, it is nil-isomorphic to W i. Thus we ﬁnd that  ⊕n j=1−1 Au−i w i, τ|Au−i w i−→Coker n, τ is a nil-isomorphism. We compute the τ -action on u−i w i: mini,n−iin−  τu−i w i= u−i w i x i· (σ × id)w i=u−i w iβ i =0 t θq− 1. Set mini,n−iin− γn,i:==0iβand L n,i:=kθ±1, t, γn,iσ× id. Then (Au−i w i, τ|Aw) ∼= σ∗L and we have thus shown that as A-crystals we i n,i have: Proposition 10.29.Suppose 0≤ n ≤ q. Then )n−1 S n+2Γ1 t= R 1 f¯∗Sym n j#F ∼=L n,i. i=1 Moreover L n,i=L n,n−i and S n+2(Γ1(t)) = 0 for n = 0, 1. In Remark 10.33 we shall compare the above formula for γn,i to a similar expression in [35, Formula (7.3)]. 1n1n−1 Example 10.30.For i = 1 and 2≤ n ≤ q one has γn,1=0 1β0+1 1β = 1 + (n− 1)t. The corresponding ramiﬁcation divisor in the sense of Remark 10.19 θ is deﬁned by θ = (1− n)t on Spec k[θ±1, t]. This leads to the Hecke character described in Example 10.26. Example 10.31.Next we compute the local L-factors of L n,1. The base scheme is X :=A k0 in the coordinate θ. Let p be a place of X deﬁned by the irreducible polynomial h(θ)∈ k[θ]. We normalize it so that h(0) = 1. This is possible because p = 0. The residue ﬁeld at p is k p= k[θ]/(h(θ)) = k[¯θ] with ¯θ a root of h over ¯k. In k p[t] we have ttt θ¯·1−θ¯q· · · · ·1−θ¯q deg h−1. Thus  det(1− τL T deg p) = det 1−1 + (n− 1)t¯·1 + (n− 1)t θθ¯q t · · · · ·1 + (n− 1)¯T deg h θq deg h−1  = 1− h 1− n t T deg h. Thus, if we denote by g n,1 the cuspidal Drinfeld-Hecke eigenform corresponding to L n,1, then its eigenvalue at p = (h) is h(1− n)t. A similar but more involved computation yields the eigenvalue system for the form g n,i corresponding to L n,i and any 1≤ i ≤ n − 1. Example 10.32.For classical as well as Drinfeld modular forms, their automorphic weight, say n, is the exponent of the automorphy factor in the transformation formulas for the action of the congruence subgroup deﬁned by the level of the form. In the classical case there is naturally a weight attached to the motive associated with a cuspidal Hecke eigenform. This motivic weight is the exponent of the complex absolute values of the roots of the characteristic polynomials deﬁned by the Hecke action. By the proof of the Ramanujan-Petersson conjecture due to Deligne, this weight is (n−1)/2. This weight occurs in the formula for the absolute values of the p th Hecke eigenvalue of a classical modular form f : a pf≤ 2 p(n−1)/2. C It is therefore natural to also ask for a motivic weight of a cuspidal Drinfeld- Hecke eigenform. Does it exist and how is it related to the weight that occurs in the exponent of the automorphy factor in the transformation law of the form? In the Drinfeld modular case, the characteristic polynomial arising from Hecke operators at p is one-dimensional. Therefore the motivic weight of a Drinfeld- Hecke eigenform f is (if it exists) the exponent q∈ Q such that  v∞|a p f|=−q deg p for almost all p ∈ Max A. This weight is modeled after Anderson’s deﬁnitions of purity and weights for t-mot ives [1, 1.9 and 1.10]. The τ -sheaves L n,i deﬁned in Proposition 10.29 possess a motivic weight. It is equal to deg tγn,i, since by computations as in the previous example one shows that v∞(|a p(g n,i)|) = − deg γn,i· deg p. For q = p the formulas in Remark 10.33 yield deg n,i= minn − i, i. Thus, for a given n, any weight in 1, 2, . . . , n 2 occurs. For q= p, the possible weights are more diﬃcult to analyze because the expression m i n m−i can vanish for m = mini, n − i (and also for many values less than this minimum – see Lemma 8.30. We expect but have no proof that the motives for all cuspidal Drinfeld- Hecke eigenforms are pure, i.e., that they have a motivic weight. If this is true, it can be shown from the cohomological formalism in  that this weight lies, for a given n, in the interval0, . . . , [n]. That the range is optimal is shown by the 2 above examples. Moreover the example shows that it is not possible to compute the motivic weight from the automorphic weight. Remark 10.33.In [35, Formula (7.3)] a diﬀerently looking formula is given from which the Hecke eigenvalue systems for the forms g n,i are computed (for primes of degree one). From the following claim it follows that the formulas given there agree with those here (and thus with those given in ). i n− itm γn,i=, m mθ m≥0 mini,n−iin− where we recall that γn,i:==0βwith β =t− 1. The proof iθ follows by the use of generating series in a standard fashion. The key steps are γn,i X n=iβXn− X n− i n≥0≥0 n≥i+ =βX(1− X)−i X i=1 + βX i X i 1− X ≥0 t Xiitm =1 +θX i=X 1− X−m X i 1− X mθ m≥0 i tmn− i =X n m≥0 mθn≥i+m m m =X n i n− i t. m mθ n≥0 m≥0 Appendix Further Results on Drinfeld Modules In this appendix we collect as a reference some further results on Drinfeld modules which were used in parts of the lecture notes. Throughout the appendix we denote by ι the canonical embedding A → C∞. A.1 Drinfeld A-modules over C∞ Important examples of Drinfeld A-modules are obtainable over C∞via a uni formization theory modeled after that of elliptic curves. For Drinfeld modules over X = Spec C∞one has the following result of Drinfeld : Theorem A.1(Drinfeld). Let ϕ be a Drinfeld module over C∞with dϕ : A → C∞ equal to ι. Then there exists a unique entire function  eϕ:C∞−→ C∞,x−→ x +a i x q i a i∈ C∞ i≥1 such that for all a∈ A the following diagram is commutative: C∞eϕ// C∞(A.9) x→ax x→ϕa(x)  C∞eϕ// C∞ Moreover eϕis a k-linear epimorphism and its kernel is a projective A-module of rank equal to the rank of the Drinfeld A-module which is discrete in C∞. Conversely, to any discrete projective A-submodule Λ⊂ C∞of rank r one can associate a unique exponential function  eΛ(x) = x +a i x q i= x 1−x, λ i≥1λ∈Λ0 whose set of roots is the divisor Λ, and a unique Drinfeld A-module ϕΛof rank r over C∞such that (A.9) commutes for ϕ = ϕΛand eϕ= eΛ. The characteristic of ϕΛis the canonical inclusion A → C∞. Moreover, two Drinfeld A-modules ϕΛand ϕΛare isomorphic (over C∞) if and only if there is a scalar λ∈ C∗∞such that λΛ = Λ. They are isogenous if there exists λ∈ C∗∞and a∈ A  0 such that aΛ⊂ λΛ ⊂ Λ. In particular there exist Drinfeld A-modules of all ranks r∈ N = 1, 2, 3, . . .. For any rank r≥ 2 there exist inﬁnitely many non-isomorphic Drinfeld A-modules of rank r over C∞. In the rank 1 case, the number of isomorphism classes of Drinfeld A-modules (over C∞) is equal to the number of projective A-modules of rank 1, i.e., to the cardinality of the class group Cl(A) = Pic(A) of A. In fact within each isomorphism class there is one representative which is deﬁned over the class ﬁeld H of K with respect to∞. There will be more on this in Appendix A.3. Indication of proof of Theorem A.1. Let ϕ be given and ﬁx a∈ A  k and write t ϕa= a +j=1 u jτj(t = r deg a). Then (A.9) yields the recursion t a i a q i− a=u j a q i−j j j=1 for the coeﬃcients of eϕ, where we set a i= 0 for i < 0. Let v denote the valuation on C∞such that v(π) = 1 for π a uniformizer π of F at∞. Let C := min j=1,...,t v(u j). Then from v(a) < 0 one deduces that  ≥C− v(a)+ min v(a i−j). q i q i j=1,...,t q i−j Choose 0 < θ <−v(a) so that for i  0 one has C− v(a) ≥ θ. Setting B q i i:= min j=1,...,t v(a i−j)it follows that there is some i q i−j 0> 0 such that for all i≥ i 0 one has B i+1≥ B i and B i+t≥ B i+ θ. Thus (B i) converges to∞ and hence lim i→∞v(a i)=∞. This shows that e q iϕhas inﬁnite radius of convergence, i.e., that it is entire. (The uniqueness of eϕfor a follows from the initial condition deϕ= 1.) ≥0 a iτi as a formal power series in τ over C∞. Then for any b∈ A the expression e−1ϕϕb eϕis again such a power series. Since the image of A under ϕ is commutative, it follows that e−1ϕϕb eϕ commutes with a = e−1ϕϕa eϕas an element in the formal non-commutative power series ring C∞[[τ ]] over τ . One deduces that e−1ϕϕb eϕis a constant and by taking derivatives that b = e−1ϕϕb eϕ. Hence (A.9) commutes for any b∈ A. That Ker(eϕ) is an A-module is immediate from the commutativity of (A.9). The discreteness follows as in complex analysis from the entireness and non constancy. That the roots have multiplicity one is deduced from the root at 0 having multiplicity one. The rank of Ker(eϕ) as an A-module is obtained by con sidering torsion points (introduced in the following section) and exploiting their relation to the rank. All further assertions are rather straightforward. A.2 Torsion points and isogenies of Drinfeld modules Theorem A.1 indicates that Drinfeld modules are characteristic p analogs of elliptic curves. This suggests that torsion points of Drinfeld modules carry an interesting Galois action. Formally one deﬁnes modules of torsion points (or torsion schemes) as follows. Fix a non-zero ideal a ⊂ A and a Drinfeld A-module ϕ on X. For any a∈ A  0 the kernel of ϕa: L→ L is a ﬁnite ﬂat group scheme over X of rank equal to q r deg a. (Passing to local coordinates, it suﬃces to verify this for Drinfeld modules of standard type – where it is rather easy.) From this one deduces that the a-torsion scheme of ϕ, deﬁned as + ϕa:=Kerϕa, a∈a is a ﬁnite ﬂat A-module scheme over X of rank q r deg(a). If a is prime to the characteristic of ϕ, then ϕ[a] is ﬁnite ´etale over X. In the case X = Spec F for a ﬁeld F , the ﬁnite group scheme ϕ[a] becomes trivial over a ﬁnite Galois extension L of F . As an A-module, the group ϕ[a](L) is isomorphic to (A/a)r provided that a is prime to the characteristic of ϕ. The Galois action and the A-action commute on ϕ[a]. This yields the following ﬁrst result: Theorem A.2.Let ϕ be a Drinfeld A-module of rank r on Spec F for a ﬁeld F with algebraic closure F . Suppose a is prime to the characteristic of ϕ. Then the action of Gal(F /F ) on ϕ[a](F ) induces a representation  Gal F /F−→ GL r A/a. If a is a non-zero principal ideal generated by a ∈ A, then ϕ[a] = Ker(ϕa). If one regards ϕa as an isogeny ϕ→ ϕ, then ϕ[(a)] is the kernel of this isogeny. More generally, for any non-zero ideal a of A, principal or not, there exists an isogeny ψ : ϕ→ ϕfor a suitable Drinfeld module ϕsuch that ϕ[a] = Ker(ψ), as subgroup schemes of ϕ: Suppose ﬁrst that X = Spec F for a ﬁeld F containing the ﬁnite ﬁeld k. We follow [24, Sec. 4.7]. Given a one considers the left ideal aϕ:=a∈a Fτϕa of Fτ. Since all left ideals are principal, the ideal has a monic generator ϕa that satisﬁes Fτ · ϕa=aϕ. Since aϕ· ϕb⊂ aϕfor all b∈ A, for any b ∈ A there is a unique ϕb∈ F τ such that ϕbϕa= ϕaϕb. Exercise A.3. (a) b→ ϕb deﬁnes a Drinfeld A-module ϕ: A→ F τ such that ϕa is an isogeny ϕ→ ϕ. Moreover Ker(ϕa) = ϕ[a] as subgroup schemes of G a/F. (b) Denote ϕfrom (a) by a ∗ ϕ. Then (a, ϕ) → a ∗ ϕ deﬁnes an action of the monoid of non-zero ideals of A under multiplication on the set of Drinfeld A-modules, i.e. (ab) ∗ ϕ = a ∗ (b ∗ ϕ). The action preserves the rank and the height of ϕ. It is trivial on principal ideals, and thus it induces an action of the class group Cl(A) of A on the set of Drinfeld A-modules. Suppose next that R is an integrally closed domain with fraction ﬁeld F and ϕ is a Drinfeld A-module over Spec R in standard form. Using the normality of R, one can verify that ϕa and a ∗ ϕ constructed above have coeﬃcients in R. From there it is easy to verify all assertions of the exercise over R as well. The case that R is a normal domain applies in particular to the case of Drinfeld moduli schemes with a level structure. For the general we refer to [34, Sec. 3.5]. In [13, Prop. 2.3], Drinfeld gives a more general criterion for the existence of an isogeny whose kernel is a ﬁnite A-subgroup scheme of G a/F. The criterion can easily be extended to the case where X = Spec R with R a normal domain. For later reference, we provide some explicit formulas: Suppose a = (a) is a principal non-zero ideal of A. Then ϕ(a)= μϕ(a)−1ϕa for μϕ(a) the leading coeﬃcient of ϕa.(A.10) From this, a short computation yields the following formula for the leading term of (a)∗ ϕ: μ(a)∗ϕ(b) = μϕ(b)· μϕ(a)1−q deg b= μϕ(b)q deg a∀b ∈ A.(A.11) In the particular case where the base is Spec C∞, the characteristic is the canonical embedding ι : A → C∞, and r = 1, one can say more. By Theorem A.1 any such module is given by a rank 1 A-lattice in C∞. Homotheties induce iso morphisms of Drinfeld modules and any two lattices are isogenous up to rescaling. Making all identiﬁcations explicit, one obtains the following result: Proposition A.4(Drinfeld, Hayes). The set of isomorphism classes of Drinfeld A-modules of rank 1 over C∞with dϕ = ι is a principal homogeneous space under the∗-operation of Cl(A). A.3 Drinfeld-Hayes modules In this section we collect some results on Drinfeld-Hayes modules. The reader is advised to recall the deﬁnitions of a sign-function and the corresponding strict class group from Deﬁnitions 8.1 and 8.3. As in (A.10), the leading term of a Drinfeld module ϕ is denoted μϕ. We ﬁx a sign-function sign throughout this section. Deﬁnition A.5.A rank 1 sign-normalized Drinfeld module or simply a Drinfeld- Hayes module (for sign) is a rank 1 Drinfeld module ϕ over C∞with dϕ = ι whose leading term μϕagrees with the restriction of a twisted sign function (sign) to A⊂ K∞. A good reference for the following result is [24, Theorem 7.2.15]. Theorem A.6(Hayes). Every rank 1 Drinfeld module over C∞with characteristic ι is isomorphic to a Drinfeld-Hayes module. Indication of proof. Deﬁne the Z-graded ring gr(K∞) =⊕i∈Z M i/M i−1 using the ﬁltration M i:=x ∈ K∞: v∞(x)≥ −i. Let L be a subﬁeld of C∞and let ϕ : A→ Lτ be a rank 1 Drinfeld module with dϕ = ι. Sublemma A.7.There exists a unique map λϕ: gr(K∞)→ L∗with the following properties: (a) For all a∈ A with image ¯a in M v∞(a)/M v∞(a)−1⊂ gr(K∞) one has λϕ(¯a) = μϕ(a). (b) For all α, β∈ gr(K∞) one has λϕ(αβ) = λϕ(α)q degβλϕ(β). One ﬁrst observes that μϕ(ab) = μϕ(a)q deg bμϕ(b) for any a, b∈ A. Then one uses property (b) to extend the deﬁnition of λϕon the images ¯a for a∈ A to all of gr(K∞). One also observes that λϕis the identity on k and a Galois automorphism when restricted to k∞. The details are left to the reader. The next result, whose proof we leave again to the reader, describes the change of λϕunder isomorphism. Sublemma A.8.Suppose ϕ= αϕα−1 for some α∈ L∗. Then λϕ(x) = λϕ(x)α(1−q− deg(x)). Now, given ϕ, choose α∈ C∗such that αq−1= λϕ(π), so that ϕ= αϕα−1 satisﬁes λϕ(π) = 1. Because λϕis given by some σ∈ Gal(k∞/k) when restricted to k∞, one deduces that ϕis sign-normalized. Denote by M 1 sign(C∞) the set of sign-normalized rank 1 modules over C∞. Since the number of isomorphism classes of Drinfeld A-modules of rank 1 over C∞ with characteristic ι is ﬁnite and equal to the class number of A, and since the number of choices for α in the previous proof is ﬁnite, the set M 1 sign(C∞) is ﬁnite. Recall the action of fractional ideals a on Drinfeld modules from the paragraphs above Proposition A.4. The following is from [24,§ 7.2]: Theorem A.9.The action of I A on rank 1 Drinfeld A-modules preserves the sign normalization and thus deﬁnes a well-deﬁned action  I A× M 1 sign C∞−→ M 1 sign C∞,a, ϕ−→ a ∗ ϕ. The action of P+is trivial. Under the induced action  Cl+(A)× M 1 sign C∞−→ M 1 sign C∞,a, ϕ−→ a ∗ ϕ, the set M 1 sign(C∞) becomes a principal homogeneous space under Cl+(A), i.e., the action is simply transitive and the stabilizer of any element is trivial. Indication of proof. It is easy to see that the∗-action preserves M 1 sign(C∞): For an ideal I, the∗-action on the leading coeﬃcient is of the form y → y q d for q d the rank of the set ϕ[I](C∞). For a principal ideal (a) the∗-action on the leading term was determined in (A.11). Thus if ϕ is sign-normalized and if a is positive the eﬀect on leading terms is trivial because of μϕ(a) = sign(a) = 1. In particular P+acts trivially. Next we show that Cl(A)+acts faithfully. Suppose a ∗ ϕ = ϕ. Since this implies in particular that a ∗ ϕ is isomorphic to ϕ, we deduce that a is principal, say equal to (a). In this case we can again use formula (A.11). It implies that for all b∈ A we must have 1−q deg b μϕa= 1. Since the gcd of the deg b is 1, it follows that μϕ(a)q−1= 1, i.e., α := μϕ(a)∈ k∗. But then aα−1 is a positive generator of a and the faithfulness of the action is shown. Finally by determining the cardinalities of Cl(A)+and of M 1 sign(C∞) the proof will be complete: By deﬁnition #Cl(A)+= #Cl(A)· #(P/P+). But all elements of k∞∗occur as signs of some α∈ K and principal ideals αA are positively generated if and only if α∈ k∗. Hence #Cl(A)+= #Cl(A)· #k∞∗/#k∗. Next we observe that all isomorphism classes of rank 1 Drinfeld A-modules over C∞of characteristic ι are represented in M 1 sign(C∞). We count the number of times a class occurs in M 1 sign(C∞). If ϕ is a Drinfeld-Hayes module for sign, and if the same holds for αϕα−1, then one shows that α∈ k∗∞. Moreover the two are equal if and only if α∈ k∗. Hence the cardinality of M 1 sign(C∞) is equal to the number of isomorphism classes of rank 1 Drinfeld A-modules over C∞of characteristic ι times #k∗∞/#k∗, and thus equal to #Cl(A)+= #Cl(A)·#k∞∗/#k∗ by Theorem A.4. One now argues as in the case of CM elliptic curves to deduce that every ϕ∈ M 1 sign(C∞) is deﬁned over H+. Let H⊂ C∞be the ﬁeld of deﬁnition of ψ. Since Aut(C∞/K) preserves M 1 sign(C∞), the extension H/K is ﬁnite. Consider ing the inﬁnite place it follows that H/K is separable. Using that the automor phisms commute with the∗-operation, one shows that the extension is abelian. The∗-action also shows that H is independent of ϕ. Next one shows that ϕ has its coeﬃcients in O, the normal closure of A in H. The Drinfeld module has potentially good reduction everywhere. But the leading coeﬃcient is a unit, and thus the Drinfeld module can be reduced without twist. This allows one to use reduction modulo any prime of O as a tool. It is not hard to see that to test equality of sign-normalized rank 1 Drinfeld A-modules it suﬃces to test it modulo any prime of O. This implies that the inertia group at any ﬁnite place is trivial. In particular the Artin symbol σp is deﬁned at any prime of O. One deduces the following Shimura type reciprocity law: Theorem A.10.If σI denotes the Artin symbol of a fractional ideal I of A, then σIϕ = I∗ ϕ. Thus H = H+. Moreover every Drinfeld-Hayes module is deﬁned over the ring of integers O+of H relative to A⊂ K. The reciprocity identiﬁes the Galois action with the∗-action on M 1 sign(C∞) (it is rather trivial to see that any type Galois action preserves sign normalization). One easily deduces: Corollary A.11.Gal(H+/H) ∼=F∗∞/F∗is totally and tamely ramiﬁed at∞. It is unramiﬁed outside∞. Remark A.12.One can show that any rank 1 Drinfeld module with characteristic ι can be deﬁned over the ring of integers of the Hilbert class ﬁeld H. However this representative within the isomorphism class is less canonical and its leading coeﬃcient has no simple description. A.4 Torsion points of Drinfeld-Hayes modules Fix a sign-normalized rank 1 Drinfeld module ϕ over C∞. For I∈ I A, let ϕI denote the isogeny ϕ→ I ∗ ϕ. Recall that ϕ[I] denotes the A-module of I-torsion points of ϕ. Denote by M 1 I,sign(C∞) the set of isomorphism classes of pairs (ϕ, λ) where ϕ∈ M 1 sign(C∞) and λ is a primitive I-torsion point of ϕ. Let I A(I) denote the set of fractional ideals prime to I. Theorem A.13.Let I⊂ A be an ideal. The action of I A on M 1 I,sign(C∞) given by  J∗ϕ, λ=J∗ ϕ, ϕJλ is well deﬁned and transitive. The stabilizer of any pair (ϕ, λ) is the subgroup P I+⊂ P+of positively generated fractional ideals prime to I. The set M 1 I,sign(C∞) is a principal homogeneous space under the induced action of Cl(A, I) :=I A(I)/P I+. One has an exact sequence ∗ 0−→A/I−→ Cl A, I−→ Cl+A−→ 0. The ﬁeld H+(ϕ[I]) is the ray class ﬁeld of K of conductor I at the ﬁnite places and which is totally split at∞ above H+. One has Gal(H+(ϕ[I])/K) ∼= Cl(A, I). Let σJ denote the Galois automorphism which under the Artin reciprocity map corresponds to J∈ I A(I). The Shimura type reciprocity law reads: For every J∈ I A(I), σJϕ, λ= J∗ϕ, λ. One has the following ramiﬁcation properties: Theorem A.14.Let P⊂ A be a prime ideal and λ ∈ ϕ[P ] be a primitive element. The extension H+(ϕ[P i])/H+is totally ramiﬁed at P and unramiﬁed at all other ﬁnite places. It is Galois with Galois group Gal(H+(ϕ[P i])/H+) ∼= (A/P i)∗. The action of this group on ϕ[P ] is given by the character  σ−→σ(λ). λ The extension H+/H is totally ramiﬁed at∞ and unramiﬁed at all other places. It is Galois with Gal(H+/H) ∼=F∗∞/F∗q. The decomposition D∞group at ∞ in Gal(H+(ϕ[P i])/H) is isomorphic to F∗∞. The action of α in F∗q∼= D∞∩ Gal(H+(ϕ[P ])/H+) on ϕ[P ] is given by  σαλ= μψ(α)−1λ= α−1λ. Bibliography  Anderson, G., t-motives, Duke Math. J. 53 (1986), 457-502. · Zbl 0679.14001  Anderson, G., An elementary approach to L-functions mod p, J. Number Theory 80(2) (2000), 291-303. · Zbl 0977.11036  B¨ockle, G., Global L-functions over function ﬁelds, Math. 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