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Crystalline cohomology and $$GL(2,\mathbb{Q})$$. (English) Zbl 0854.14010
The aim of the paper is to study modular Galois representations attached to automorphic forms $$\text{mod } p$$ of weight $$k\geq 2$$. For a fixed level $$N\geq 3$$ let $$\Gamma (N) \subset \text{SL} (2, \mathbb{Z})$$ be the subgroup of matrices which are congruent to the identity modulo $$N$$. With the usual notation one has the open modular curve $$Y(N)$$ classifying elliptic curves with full level $$N$$ structure, and its compactification (i.e. adding the cusps) $$X(N)$$. There is a universal elliptic curve $$\pi: E\to Y= Y(N)$$ extending to a semi-abelian variety (the semi-stable compactification) $$\overline {\pi}\to \overline {E}\to X= X(N)$$. This is all defined over $$\mathbb{Q}_p (e^{2\pi i/N})$$. For a prime $$p$$, $$(p,N)=1$$, set $$\mathbb{V}= R^1 \pi_{*,\text{ét}} (\mathbb{Z}_p)$$. Similarly, for a congruence subgroup $$\Gamma \supseteq \Gamma (N)$$ one has the moduli curves $$Y(\Gamma)= Y(N)/ (\Gamma/ \Gamma (N))$$ and its completion $$X(\Gamma)$$ defined over some number field contained in $$\mathbb{Q} (e^{2\pi i/N})$$. Write $$K$$ for the completion of this number field at a prime lying over $$p \nmid N$$ and write $$H^1_{\text{par}}= \text{Image} (H^1_!\to H^1)$$, where $$H^1_!$$ denotes cohomology with compact support. As a first result one has the following theorem
$$(*)$$: With notations as above, let $$k> p$$. Then the $$V= H^1_{\text{par}} (Y (\Gamma )_{\overline {K}}, \text{Symm}^{k- 2} (\mathbb{V}))$$, $$\text{Gal} (\overline {K}/k)$$-representation, is crystalline, and the $$F$$-crystal corresponding to the dual of $$V$$ as a canonical Frobenius filtration $$M= F^0 \supset F^{k-1} \supset 0$$, with $$F^{k-1} \cong S_k (\Gamma)$$ (cusp forms of weight $$k$$ for $$\Gamma$$) and $$F^0/ F^{k-1} \cong S_k (\Gamma)^*$$.
In particular, for $$\Gamma= \Gamma_1 (N)$$ the corresponding moduli curves $$Y= Y(\Gamma)= Y_1 (N)$$ and $$X= X(\Gamma)= X_1 (N)$$ are defined over $$\mathbb{Q}$$. $$Y$$ admits correspondences $$\langle d\rangle$$, $$d\in (\mathbb{Z}/ n\mathbb{Z} )^\times$$ given by $$(E, X) \mapsto (E, dx)$$. They extend to $$X$$. One defines Hecke correspondences $${\mathcal T}_\ell \subset X\times X$$, $$\ell\neq p$$, and the induced operators on cohomology, denoted $$T_\ell$$. For primes $$r$$ dividing $$N$$ one has Hecke correspondences $${\mathcal U}_r$$ as usual with induced operators $$U_r$$. More interesting are the Hecke operators $$T_p$$. On étale, resp. de Rham, cohomology in characteristic zero one has $$T_p^{\text{ét}}$$ and $$T_p^{\text{DR}}$$. One also has $$T_p^{\text{crys}}= F_p+ \langle p\rangle F^t_p$$ with Frobenius $$F_p$$. The various $$T$$’s are related as follows:
Assume $$p> k$$ and $$p\nmid N$$. Then (i) for a prime $$\ell \nmid pN$$, $$T_\ell^{\text{crys}}$$ is associated to $$T_\ell^{\text{ét}}$$. Similarly for $$U_r$$, $$r\mid N$$, $$r\neq p$$; (ii) $$T_p^{\text{crys}}= T_p^{\text{DR}}$$, and these are associated to $$T_p^{\text{ét}}$$.
Fix $$N\geq 3$$ and $$k\geq 2$$ and, for a prime $$p$$, write $$\vartheta_p$$ for the étale sheaf $$\text{Symm}^{k -2} (\mathbb{V})$$ on $$Y= Y_1 (N)/ \mathbb{Q}$$. The various correspondences on $$Y$$ induce endomorphisms on $$H^1_{\text{par}} (Y):= H^1_{\text{par}} (Y_1 (N)_{\overline {\mathbb{Q}}}, \vartheta_p)$$. Let $$\mathbb{T}_0$$ denote the $$\mathbb{Z}$$-algebra of endomorphisms of $$H^1_{\text{par}} (Y)$$ generated by the $$T_\ell$$, $$\ell\nmid N$$, and the $$\langle d\rangle$$, $$d\in (\mathbb{Z}/ N\mathbb{Z} )^\times$$. Also, write $$\mathbb{T}$$ for the $$\mathbb{Z}$$-subalgebra of $$\text{End} (H^1_{\text{par}} (Y))$$ generated by $$\mathbb{T}_0$$ and the $$U_r$$, $$r\mid N$$. Let $${\mathbf m} \subset \mathbb{T}$$ be a maximal ideal with residue field $${\mathbf k}= {\mathbf k} ({\mathbf m})= \mathbb{T}/{\mathbf m}$$ of characteristic $$p$$. One has a unique (up to isomorphism) semi-simple representation $$\rho_{\mathbf m}: \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})\to \text{Gal} (2,{\mathbf k})$$ such that $$\rho_{\mathbf m}$$ is unramified for all primes $$r\nmid pN$$ and such that for such primes $$\text{Tr} (\rho_{\mathbf m} (\text{Frob}_r))= T_r\bmod {\mathbf m}$$, and $$\text{det} (\rho_{\mathbf m} (\text{Frob}_r))= \langle r\rangle r^{k-1} \bmod {\mathbf m}$$. By Eichler-Shimura one knows that the $$T_\ell$$ act as the geometric Frobenius (and its adjoint) on $$H^*_{\text{par}}$$ whereas $$\text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})$$ act as the arithmetic Frobenius, i.e. one must work with the dual $$H^1_{\text{par}} (Y)^\vee$$. For irreducible $$\rho_{\mathbf m}$$, $$(*)$$ implies:
Let $$k>p$$. Then (i) $$h^1_{\text{par}} (Y_1 (N)_{\overline {\mathbb{Q}}}, \vartheta_p/ p\vartheta_p )[{\mathbf m}]^\vee$$ is isomorphic to the $${\mathbf k} [\text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})]$$-module corresponding to $$\rho_{\mathbf m}$$. In particular, $$\dim_{\mathbf k} H^1_{\text{par}} (Y_1 (N)_{\overline {\mathbb{Q}}}, \vartheta_p/ p\vartheta_p )[{\mathbf m}] =2$$; (ii) The local ring $$\mathbb{T}_{\mathbf m}$$ is Gorenstein. When $$\rho_{\mathbf m}$$ is reducible one says that the maximal ideal $${\mathbf m} \subset \mathbb{T}$$ is reducible. The reducible $$\rho_{\mathbf m}$$ are harder to analyse. However, one has: Suppose $$\rho_{\mathbf m}$$ is irreducible. Then $$\rho_{\mathbf m}= \alpha \oplus \beta \chi^{k -1}$$, where $$\alpha, \beta: \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})\to {\mathbf k}^\times$$ are characters and $$\chi: \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})\to \mathbb{F}_p \subseteq {\mathbf k}^\times$$ is the $$p$$-cyclotomic character. $$\alpha$$ and $$\beta$$ are unramified outside $$pN$$. If $$p> k$$, then $$\alpha$$ and $$\beta$$ are unramified at $$p$$.
Reducible $$\rho_{\mathbf m}$$ lead to the study of Eisenstein series and Eisenstein ideals. A detailed review of the related function theory is presented, leading to the classification of the $${\mathbf m} \subset \mathbb{T}$$ for reducible $$\rho_{\mathbf m}$$. Taking the sum of the zero-th coefficients over the so-called oriented cusps of the $$q$$-expansions of a modular form $$f\in M_k (\Gamma_1 (N))$$ at these oriented cusps leads to a homomorphism $$\overline {a}_0: M_k (R)\to H^0 (\text{cusps}, \omega^k \otimes R)$$, where $$R\subset \mathbb{C}$$ is a commutative ring with $$1/N\in R$$ and where $$\omega$$ is the bundle of differentials on the universal curve. One has $$S_k (R)= \text{Ker} (\overline {a}_0) \subset M_k (R)$$, where $$M_k (R)$$ (resp. $$S_k (R)$$) is the $$R$$-module of modular forms (resp. cusp forms) for $$\Gamma_1 (N)$$ defined over $$R$$. The space of Eisenstein series $$E_k (\mathbb{C}) \subset M_k (\mathbb{C})$$ is the orthogonal complement (with respect to the Petersson inner product) of $$S_k (\mathbb{C})$$. Explicit calculation shows how the $$a_0$$’s of $$f\in M_k (R)$$ at the various cusps transform under $$T_\ell, U_\ell, \langle d\rangle, \dots\;$$. Eisenstein series furnish examples of reducible maximal ideals. Assume $$R$$ contains the values of all Dirichlet characters of conductor dividing $$N$$, and write $$\mathbb{T}'= \mathbb{T} \otimes R$$, similarly for $$\mathbb{T}_0'$$. Consider the $$q$$-expansion $$\sum^\infty_{n=0} a_n (E) q^n$$ at $$\infty$$ of a normalized weight $$k$$ Eisenstein series $$E$$ for $$\Gamma_1 (N)$$ such that $$E$$ is an eigenfunction of $$\mathbb{T}$$. Thus $$T_\ell(E)= a_\ell(E) E$$ for primes $$\ell \nmid N$$, and $$U_\ell (E)= a_\ell(E) E$$ for primes $$\ell \mid N$$. One knows that $$a_\ell (E) \in R$$ for all primes $$\ell$$ and $$a_0 (E)\in K[\text{oriented cusps}]$$, where $$K$$ is the quotient field of $$R$$. For a prime ideal $${\mathfrak p} \subseteq R$$ such that the associated valuation of $$a_0 (E)$$ at each oriented cusp of $$X_1 (N)$$ is positive, one puts $${\mathbf m}_0' (E, {\mathfrak p}) \subseteq \mathbb{T}_0'$$ for the ideal $$(T_\ell- a_\ell (E)$$, $$\ell \nmid N$$; $$\langle d\rangle- \varepsilon (d)$$, $$(d, N) =1$$; $${\mathfrak p}$$) ($$\varepsilon$$ is a primitive Dirichlet character with conductor $$N$$) and $${\mathbf m}'= ({\mathbf m}_0', U_\ell- a_\ell (E), \ell\mid N) \subseteq \mathbb{T}'$$, and finally, $${\mathbf m}_0' (E,{\mathfrak p})= {\mathbf m}_0' \cap \mathbb{T}_0$$, $${\mathbf m} (E,{\mathfrak p})= {\mathbf m}' \cap \mathbb{T}$$. In this situation, $${\mathbf m}_0' (E,{\mathfrak p})$$ (resp. $${\mathbf m}' (E, {\mathfrak p}))$$ is called an Eisenstein (maximal) ideal of $$\mathbb{T}_0'$$ (resp. $$\mathbb{T}'$$). Analogous terminology for the contracted $${\mathbf m}_0$$ and $${\mathbf m}$$.
It is conjectured that: A maximal ideal $${\mathbf m}_0' \subseteq \mathbb{T}_0'$$ is reducible if and only if it is Eisenstein. Same conjecture without the subscript $${}_0$$.
A maximal ideal $${\mathbf m}_0' \subseteq \mathbb{T}_0'$$ is called new in case the semi-simple representation $$\rho_{{\mathbf m}_0'}$$ does not occur in cusp forms of any strictly lower level. Analogously for the other cases. For the classification of reducible modular Galois representations associated to cusp forms it is enough to classify those which are new. The following result with respect to the conjecture is settled:
Let $$\mathbb{T}$$ be the weight $$k$$ Hecke algebra for $$\Gamma_1 (N)$$, and let $${\mathbf m} \subseteq \mathbb{T}$$ be a new reducible maximal ideal of residue characteristic $$p$$ with $$p> k+1$$, $$p \nmid N$$. Then $${\mathbf m}$$ is Eisenstein (of level $$N$$).
Next, one studies the Hecke algebra for $$\Gamma_0 (N)$$. The same notation is used as before. Reducible $$\rho_{\mathbf m}= \alpha \oplus \beta$$, $$\{\alpha, \beta\}= \{\chi_0= \text{trivial}, \chi^{k -1}\}$$ are the subject of study. First, for $$\chi_1, \chi_2= \alpha$$ or $$\beta$$ satisfying certain local requirements, one studies crystalline extensions annihilated by $$p$$ of the form $$0\to \chi_1\to \bullet\to \chi_2\to 0$$. Then, with $$M_1= \alpha$$ and $$N_1= \beta$$ one defines non-trivial extensions (if they exist) by $$0\to \alpha\to M_n\to N_{n-1}\to 0$$ and $$0\to \beta\to N_n\to M_{n-1}\to 0$$ leading to a family of modules $$\{M_i, N_j\}$$ (possibly ending somewhere). The $$M_n$$ and $$N_n$$ are indecomposable $$p$$-torsion $$\text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})$$-modules, and actually, they are all such indecomposables satisfying the local requirements above. Using this result one can derive the following multiplicity one theorem for Eisenstein ideals:
Let $$N$$ be prime and $$\mathbb{T}$$ the weight $$k$$ Hecke algebra for $$\Gamma_0 (N)$$. Suppose $${\mathbf m} \subseteq \mathbb{T}$$ is a new reduced maximal ideal of residue characteristic $$p> \max (3, k)$$. Assume $$p\not\equiv 1 \bmod N$$, $$U_N \not\equiv 1\bmod {\mathbf m}$$. Moreover if $$2< k< p-1$$, suppose $$p\nmid L(0, \omega^{k-1} \omega^{p-k})$$ $$(\omega$$ is the Teichmüller character). Then $$\dim_{\mathbb{F}_p} H^1_{\text{par}} (Y_0 (N)_{\overline {\mathbb{Q}}}, \vartheta_p/ p\vartheta_p) [{\mathbf m}]= \dim_{\mathbb{F}_p} H^1_{\text{par}} (Y_0 (N)_{\overline {\mathbb{Q}}}, \vartheta_p/ p\vartheta_p)/ {\mathbf m} H^1_{\text{par}} (Y_0 (N)_\mathbb{Q}, \vartheta_p/ p\vartheta_p) =2$$, and under these hypotheses the ring $$\mathbb{T}$$ is Gorenstein.
The paper ends with a section on the application of crystalline methods to prove the existence of companion forms for $$\Gamma_1 (N)$$. This solves a conjecture of Serre. It was first proved by Gross using different methods. Let $$f= \sum a_n q^n$$ be a normalized cusp form for $$\Gamma_1 (N)$$ with coefficients in the finite field $${\mathbf k}$$ (notations as before). Assume $$\rho_{\mathbf m}$$ is reducible. Then the theorem says:
Let $$p> \max (3,k)$$ and $$a_p\neq 0$$. Then $$\rho_{\mathbf m}$$ is tamely ramified at $$p$$ if and only if there is a normalized eigenform $$g= \sum b_n q^n$$ of weight $$k'= p+1- k$$ for $$\Gamma_1 (N)$$ over $${\mathbf k}$$ such that $$n^k b_n= na_n$$ for all $$n\geq 1$$.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 11F80 Galois representations 11R32 Galois theory 11F11 Holomorphic modular forms of integral weight 14G20 Local ground fields in algebraic geometry
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