Algebraic Hecke characters and strictly compatible systems of abelian mod p Galois representations over global fields

In this article we explain how the results in our previous article on ‘algebraic Hecke characters and compatible systems of mod$p$Galois representations over global fields’ allow one to attach a Hecke character to every cuspidal Drinfeld modular eigenform from its associated crystal that was constructed in earlier work of the author. On the technical side, we prove along the way a number of results on endomorphism rings of${\it\tau}$-sheaves and crystals. These are needed to exhibit the close relation between Hecke operators as endomorphisms of crystals on the one side and Frobenius automorphisms acting on étale sheaves associated to crystals on the other. We also present some partial results on the ramification of Hecke characters associated to Drinfeld modular eigenforms. An important phenomenon absent from the case of classical modular forms is that ramification can also result from places of modular curves of good but non-ordinary reduction. In an appendix, jointly with Centeleghe we prove some basic results on$p$-adic Galois representations attached to$\text{GL}_{2}$-type cuspidal automorphic forms over global fields of characteristic $p$.

In this paper we prove a version of Deligne’s conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.

Sturm-type bounds for modular forms over function fields

Compatible systems of mod $p$ Galois representations and Hecke characters

p-adic Eisenstein–Kronecker series and non-critical values of p-adic Hecke L-function of an imaginary quadratic field when the conductor is divisible by p

Existence of algebraic Hecke characters

This is an expository article that concerns the various related notions of algebraic idèle-class characters, the Größencharaktere of Hecke, and cohomological automorphic representations of \(\text {GL}(1)\), all under the general title of algebraic Hecke characters. The first part of the article systematically lays the foundations of algebraic Hecke characters. The only pre-requisites are: basic algebraic number theory, familiarity with the adelic language, and basic sheaf theory. Observations that play a crucial role in the arithmetic of automorphic L-functions are also discussed. The second part of the article, on the ratios of successive critical values of the Hecke L-function attached to an algebraic Hecke character, concerns certain variations on a theorem of Harder [6], especially drawing attention to a delicate signature that apparently has not been noticed before.

In this paper, we construct a theory of geometric Euler systems, complementary to the arithmetic theory of Rubin, Kato and Perrin-Riou. We show that geometric Euler systems can be used to prove the finiteness of certain Galois representations of weight zero and we discuss a conjectural framework for the existence of geometric Euler systems for motivic Galois representations. We give applications to adjoint Selmer groups of certain classical and Drinfeld modular forms.

Artin conjectured that certain Galois representations should give rise to entire L-series. We give some history on the conjecture and motivation of why it should be true by discussing the one-dimensional case. The first known example to verify the conjecture in the icosahedral case did not surface until Buhler's work in 1977. We explain how this icosahedral representation is attached to a modular elliptic curve isogenous to its Galois conjugates, and then explain how it is associated to a cusp form of weight 5 with level prime to 5. In 1917, Erich Hecke (10) proved a series of results about certain characters which are now commonly referred to as Hecke characters; one corollary states that one-dimensional complex Galois representations give rise to entire L-series. He re- vealed, through a series of lectures (9) at Princeton's Institute for Advanced Study in the years that followed, the relationship between such representations as gen- eralizations of Dirichlet characters and modular forms as the eigenfunctions of a set of commuting self-adjoint operators. Many mathematicians were inspired by his ground-breaking insight and novel proof of the analytic continuation of the L-series. In the 1930's, Emil Artin (1) conjectured that a generalization of such a result should be true; that is, irreducible complex projective representations of finite Ga- lois groups should also give rise to entire L-series. He came to this conclusion after proving himself that both 3-dimensional and 4-dimensional representations of the simple group of order 60, the alternating group on five letters, might give rise to L-series with singularities. It is known, due to the insight of Robert Langlands (16) in the 1970's relating Hecke characters with Representation Theory, that in order to prove the conjecture it suffices to prove that such representations are associated to cusp forms. This conjecture has been the motivation for much study in both Algebraic and Analytic Number Theory ever since. In this paper, we present an elementary approach to Artin's Conjecture by con- sidering the problem over Q. We consider Dirichlet's theorem which preceeded Hecke's results, and sketch a proof by introducing theta series. We then introduce Langland's program to exhibit cusp forms. We conclude by studying a specific ex- ample which is associated to an elliptic curve. We assume in the final sections that the reader is somewhat familiar with the basic properties of elliptic curves.

Shimura and Taniyama proved that if A A is a potentially CM abelian variety over a number field F F with CM by a field K K linearly disjoint from F, then there is an algebraic Hecke character λ A \lambda _A of F K FK such that L ( A / F , s ) = L ( λ A , s ) L(A/F,s)=L(\lambda _A,s) . We consider a certain converse to their result. Namely, let A A be a potentially CM abelian variety appearing as a factor of the Jacobian of a curve of the form y e = γ x f + δ y^e=\gamma x^f+\delta . Fix positive integers a a and n n such that n / 2 > a ≤ n n/2 > a \leq n . Under mild conditions on e , f , γ , δ e, f, \gamma , \delta , we construct a Chow motive M M , defined over F = Q ( γ , δ ) F=\mathbb {Q}(\gamma ,\delta ) , such that L ( M / F , s ) L(M/F,s) and L ( λ A a λ ¯ A n − a , s ) L(\lambda _A^a\overline {\lambda }_A^{n-a},s) have the same Euler factors outside finitely many primes.

Algebraic hecke characters

Let $\Pi$ be a cohomological cuspidal automorphic representation of ${\rm GL}_{2n}(\Bbb{A})$ over a totally real number field $F$. Suppose that $\Pi$ has a Shalika model. We define a rational structure on the Shalika model of $\Pi_f$. Comparing it with a rational structure on a realization of $\Pi_f$ in cuspidal cohomology in top-degree, we define certain periods $\omega^{\epsilon}(\Pi_f)$. We describe the behavior of such top-degree periods upon twisting $\Pi$ by algebraic Hecke characters $\chi$ of $F$. Then we prove an algebraicity result for all the critical values of the standard $L$-functions $L(s,\Pi\otimes\chi)$; here we use the recent work of B. Sun on the non-vanishing of a certain quantity attached to $\Pi_\infty$. As applications, we obtain algebraicity results in the following cases: Firstly, for the symmetric cube $L$-functions attached to holomorphic Hilbert modular cusp forms; we also discuss the situation for higher symmetric powers. Secondly, for certain (self-dual of symplectic type) Rankin-Selberg $L$-functions for ${\rm GL}_3\times{\rm GL}_2$; assuming Langlands Functoriality, this generalizes to certain Rankin-Selberg $L$-functions of ${\rm GL}_n\times{\rm GL}_{n-1}$. Thirdly, for the degree four $L$-functions attached to Siegel modular forms of genus $2$ and full level. Moreover, we compare our top-degree periods with periods defined by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.

In this paper we prove a nonvanishing theorem for central values of L-functions associated to a large class of algebraic Hecke characters of CM number fields. A key ingredient in the proof is an asymptotic formula for the average of these central values. We combine the nonvanishing theorem with work of Tian and Zhang [TiZ] to deduce that infinitely many of the CM abelian varieties associated to these Hecke characters have Mordell–Weil rank zero. Included among these abelian varieties are higher-dimensional analogues of the elliptic $${{\mathbb Q}}$$ -curves A(D) of B. Gross [Gr].

In this paper we study the central values of L-functions associated to a large class of algebraic Hecke characters of imaginary quadratic fields. When these central values are nonzero, the Bloch–Kato conjecture predicts an exact formula for the algebraic parts of the central values in terms of periods and arithmetic data, most notably the Selmer groups corresponding to the Hecke characters. We investigate the nonvanishing of these central values, and prove the p-part of the Bloch–Kato conjecture in these cases for primes p which split in K.

We describe the generic blocks in the category of smooth locally admissible mod $2$ representations of $\mathrm{GL}_2(\mathbb{Q}_2)$. As an application we obtain new cases of Breuil--M\'ezard and Fontaine--Mazur conjectures for $2$-dimensional $2$-adic Galois representations.