Trivial \(L\)-functions for the rational function field. (English) Zbl 1318.11147

Gross, Benedict H.

J. Number Theory 133, No. 3, 970-976 (2013).

From the text: In this paper, the author describes a number of interesting \(l\)-adic representations \(V\) of the Galois group of the rational function field with trivial \(L\)-function: \(L(V,s)=1\).
Let \(k\) be a global function field, over the finite field \(E\) with \(q\) elements. Let \(k^s\) be a separable closure of \(k\), and let \(E^s\) be the separable closure of \(E\) in \(k^s\).
The \(L\)-function \(L(V, s)\) of a semi-simple \(l\)-adic Galois representation \(\mathrm{Gal}(k^s/k) \rightarrow \mathrm{GL}(V)\) contains very little of the local information involved in its definition. Indeed, the cancellation of the local terms in the infinite Euler product defining \(L(V, s)\) ultimately results in a function which \(a\) is quotient of two polynomials in \(q^{-s}\). This led Weil to define the notion of a (formal) Dirichlet series belonging to \(k\), which keeps track of the local terms.
In this paper, the author studies an extreme case of cancellation over the rational function field \(k = E(T)\). If we assume that the geometric Galois group \(\mathrm{Gal}(k^s/kE^s)\) has no non-trivial invariants on \(V\) and that the degree \(f(V)\) of the Artin conductor of \(V\) is twice the dimension of \(V\), then \(L(V, s)\) is a polynomial of degree 0 in \(q^{-s}\) with constant coefficient 1. Hence \(L(V, s) = 1\) is a constant function! We call these trivial \(L\)-functions for the rational function field (although they arise in many non-trivial situations).
After a brief introduction to the cohomological theory of Weil, Grothendieck, and Deligne, he presents several examples of Galois representations of the rational function field \(k = E(T)\) with trivial \(L\)-functions. In all of his irreducible examples, the representation \(V\) remains geometrically irreducible and the set \(S\) of ramified places is contained in \(\{\infty, 0, 1\} \subset \mathbb P^1(E)\). More precisely, \(V\) corresponds either to an irreducible representation of \(\pi_1(\mathbb G_m) = \pi_1(\mathbb P^1 - \{\infty, 0\})\) which is tamely ramified at \(T=0\), or to an irreducible representation of \(\pi_1(\mathbb P^1 - \{\infty, 0, 1\})\) which is tamely ramified at all three places.

Reviewer: Olaf Ninnemann (Berlin)

MSC:

11R42	Zeta functions and \(L\)-functions of number fields
11R58	Arithmetic theory of algebraic function fields

Keywords:

\(L\)-function; \(l\)-adic sheaf; Artin conductor; rigid local system

References:

[1]	Artin, M.; Grothendieck, A.; Verdier, J.-L., SGA 4: Théorie des topos et cohomologie étale des schémas, Springer Lecture Notes, vols. 269, 270, 305 (1972) · Zbl 0234.00007
[2]	Deligne, P., Application de la formule des traces aux sommes trigonometriques, (Cohomology Étale. Cohomology Étale, Springer Lecture Notes, vol. 569 (1977)), 168-239 · Zbl 0349.10031
[3]	Deligne, P., La conjecture de Weil II, Publ. Math. IHES, 52, 313-428 (1981)
[4]	Freitag, E.; Kiehl, R., Étale Cohomology and the Weil Conjecture (1988), Springer Ergebnisse · Zbl 0643.14012
[5]	Gross, B., Rigid local systems on \(G_m\) with finite monodromy, Adv. Math., 224, 2531-2543 (2010) · Zbl 1193.22001
[6]	B. Gross, Lectures on the conjecture of Birch and Swinnerton-Dyer, in: Arithmetic of \(L\); B. Gross, Lectures on the conjecture of Birch and Swinnerton-Dyer, in: Arithmetic of \(L\) · Zbl 1285.11096
[7]	Gross, B., Irreducible cuspidal representations with prescribed local behavior, American J. Math., 133, 1231-1258 (2011) · Zbl 1228.22017
[8]	Heinloth, J.; Ngo, B. C.; Yun, Z., Kloosterman sheaves for reductive groups · Zbl 1272.14012
[9]	Katz, N., Kloosterman sums, Gauss sums, and monodromy, Ann. of Math. Stud., 116 (1987) · Zbl 0675.14004
[10]	Katz, N., Exponential sums and differential equations, Ann. of Math. Stud., 124 (1990) · Zbl 0731.14008
[11]	Katz, N., Rigid local systems, Ann. of Math. Stud., 139 (1995)
[12]	Lusztig, G., From conjugacy classes in the Weyl group to unipotent classes II · Zbl 1263.20045
[13]	Raynaud, M., Caractéristique dʼEuler-Poincaré dʼun faisceau et cohomologie des variétés abéliennes, (Séminaire Bourbaki 286, vol. 9 (1995), Soc. Math. France)
[14]	Serre, J.-P., Local Fields, Springer GTM, vol. 67 (1995)
[15]	Weil, A., Dirichlet Series and Automorphic Forms, Springer Lecture Notes, vol. 189 (1970)
[16]	Yun, Z., Motives with exceptional Galois groups and the inverse Galois problem · Zbl 1374.14013