Automorphic Representations Research Articles

Sato–Tate equidistribution for families of automorphic representations through the stable trace formula

In arXiv:1208.1945, Shin and Templier proved certain equidistribution bounds on local components of certain families of automorphic representations. We extend their weight-aspect results to families of automorphic representations where the Archimedean component is restricted to a single discrete-series representation instead of an entire $L$-packet. We do this by using a so-called "hyperendoscopy" version of the stable trace formula developed by Ferrari. The main technical difficulties are defining a version of hyperendoscopy that works for groups without simply connected derived subgroup and bounding the values of transfers of unramified functions. We also present an extension of Arthur's simple trace formula for test functions with Euler-Poincar\'e component at infinity to non-cuspidal groups since it does not seem to appear elsewhere in the literature.

Big principal series, -adic families and -invariants

In earlier work, the first named author generalized the construction of Darmon-style$\mathcal {L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic$\mathcal {L}$-invariants can be computed in terms of derivatives of Hecke eigenvalues in$p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic$\mathcal {L}$-invariants of Hilbert modular forms of parallel weight$2$are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur$\mathcal {L}$-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur$\mathcal {L}$-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.

Overconvergent cohomology, p-adic L-functions and families for GL(2) over CM fields

The study of overconvergent cohomology, initiated by Pollack and Stevens in the setting of classical modular forms, has now been used to construct p-adic L-functions in a number of settings. The method is conceptual and is very closely related to the recent constructions of eigenvarieties by Ash–Stevens, Urban and Hansen. In this note, we give an exposition of the ideas behind the use of overconvergent cohomology in constructing p-adic L-functions, and use it to construct p-adic L-functions attached to base-change families of automorphic representations for GL(2) over CM fields. As a corollary, we prove a p-adic Artin formalism result for base-change p-adic L-functions.

On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives

In this article, we study the Beilinson–Bloch–Kato conjecture for motives associated to Rankin–Selberg products of conjugate self-dual automorphic representations, within the framework of the Gan–Gross–Prasad conjecture. We show that if the central critical value of the Rankin–Selberg L-function does not vanish, then the Bloch–Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch–Kato Selmer group constructed from a certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin–Selberg L-function, then the Bloch–Kato Selmer group is of rank one.

Plus/minus p-adic L-functions for mathrm {GL}_{2n}

We generalise Pollack’s construction of plus/minus L-functions to certain cuspidal automorphic representations of mathrm {GL_{2n}} using the p-adic L-functions constructed in work of Barrera Salazar et al. (On p-adic l-functions for text {GL}_{2n} in finite slope shalika families, 2021). We use these to prove that the complex L-functions of such representations vanish at at most finitely many twists by characters of p-power conductor.

Galois representations for general symplectic groups

We prove the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard–Gee conjecture on the global Langlands correspondence in new cases. As an application we complete the argument by Gross and Savin to construct a rank 7 motive whose Galois group is of type G_2 in the cohomology of Siegel modular varieties of genus 3. Under some additional local hypotheses we also show automorphic multiplicity 1 as well as meromorphic continuation of the spin L -functions.

On the vanishing of adjoint Bloch–Kato Selmer groups of irreducible automorphic Galois representations

On the vanishing of adjoint Bloch–Kato Selmer groups of irreducible automorphic Galois representations

Density of automorphic points in deformation rings of polarized global Galois representations

Conjecturally, the Galois representations that are attached to essentially self-dual regular algebraic cuspidal automorphic representations are Zariski-dense in a polarized Galois deformation ring. We prove new results in this direction in the context of automorphic forms on definite unitary groups over totally real fields. This generalizes the infinite fern argument of Gouvêa–Mazur and Chenevier and relies on the construction of nonclassical p-adic automorphic forms and the computation of the tangent space of the space of trianguline Galois representations. This boils down to a surprising statement about the linear envelope of intersections of Borel subalgebras.

Chow groups andL-derivatives of automorphic motives for unitary groups, II.

AbstractIn this article, we improve our main results from [LL21] in two directions: First, we allow ramified places in the CM extension$E/F$at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla–Rapoport conjecture for exotic smooth Rapoport–Zink spaces. Second, we lift the restriction on the components at split places of the automorphic representation, by proving a more general vanishing result on certain cohomology of integral models of unitary Shimura varieties with Drinfeld level structures.

The second moment of Rankin–Selberg L-functions

Abstract We prove an asymptotic expansion of the second moment of the central values of the $\mathrm {GL}(n)\times \mathrm {GL}(n)$ Rankin–Selberg L-functions $L(1/2,\pi \otimes \pi _0)$ for a fixed cuspidal automorphic representation $\pi _0$ over the family of $\pi $ with analytic conductors bounded by a quantity that is tending to infinity. Our proof uses the integral representations of the L-functions, period with regularised Eisenstein series and the invariance properties of the analytic newvectors.

$p$-adic $L$-functions of Hilbert cusp forms and the trivial zero conjecture

We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\\operatorname{GL}\_2$ over a totally real field, which is Iwahori spherical at places above $p$. In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the $p$-adic $L$-function of a nearly finite slope family and on improved $p$-adic $L$-functions that we construct using automorphic symbols and overconvergent cohomology. For higher order zeros we develop a conceptually new approach studying the variation of the root number in partial families and establishing the vanishing of many Taylor coefficients of the $p$-adic $L$-function of the family.

Towards a 𝐺𝐿_{𝑛} variant of the Hoheisel phenomenon

Let π \pi be a unitary cuspidal automorphic representation of G L n \mathrm {GL}_n over a number field, and let π ~ \widetilde {\pi } be contragredient to π \pi . We prove effective upper and lower bounds of the correct order in the short interval prime number theorem for the Rankin–Selberg L L -function L ( s , π × π ~ ) L(s,\pi \times \widetilde {\pi }) , extending the work of Hoheisel and Linnik. Along the way, we prove for the first time that L ( s , π × π ~ ) L(s,\pi \times \widetilde {\pi }) has an unconditional standard zero-free region apart from a possible Landau–Siegel zero.

P-adic L-functions in universal deformation families

We construct examples of p-adic L-functions over universal deformation spaces for {{,mathrm{GL},}}_2. We formulate a conjecture predicting that the natural parameter spaces for p-adic L-functions and Euler systems are not the usual eigenvarieties (parametrising nearly-ordinary families of automorphic representations), but other, larger spaces depending on a choice of a parabolic subgroup, which we call ‘big parabolic eigenvarieties’.

Higher Hida theory and p-adic L-functions for GSp4

We use the "higher Hida theory" recently introduced by the second author to p-adically interpolate periods of non-holomorphic automorphic forms for GSp(4), contributing to coherent cohomology of Siegel threefolds in positive degrees. We apply this new method to construct p-adic L-functions associated to the degree 4 (spin) L-function of automorphic representations of GSp(4), and the degree 8 L-function of GSp(4) x GL(2).

A remark on conductor, depth and principal congruence subgroups

In this paper we study a relation between conductor, depth, and the level of principal congruence subgroups for irreducible admissible representations of GLn(F) for a non-archimedean local field F of characteristic zero. As a global application, we estimate conductors of unitary irreducible automorphic cuspidal representations of GLn(AQ) in terms of principal congruence subgroups. We also give an explicit formula for the dimension of fixed vectors with respect to principal congruence subgroups for irreducible admissible representations of GL2(F) as a local application.

Lifting and automorphy of reducible mod p Galois representations over global fields

We prove the modularity of most reducible, odd representations \({\bar{\rho }}: \Gamma _{{\mathbb {Q}}} \rightarrow \mathrm {GL}_2(k)\) with k a finite field of characteristic an odd prime p. This is an analogue of Serre’s celebrated modularity conjecture (which concerned irreducible, odd representations \({\bar{\rho }}: \Gamma _{{\mathbb {Q}}} \rightarrow \mathrm {GL}_2(k)\)) for reducible, odd representations. Our proof lifts \({\bar{\rho }}\) to an irreducible geometric p-adic representation \(\rho \) which is known to arise from a newform by results of Skinner–Wiles and Pan. We likewise prove automorphy of many reducible representations \({\bar{\rho }}:\Gamma _{F} \rightarrow \mathrm {GL}_n(k)\) when F is a global function field of characteristic different from p, by establishing a p-adic lifting theorem and invoking the work of L. Lafforgue. Crucially, in both cases we show that the actual representation \({\bar{\rho }}\), rather than just its semisimplification, arises from reduction of the geometric representation attached to a cuspidal automorphic representation. Our main theorem establishes a geometric lifting result for mod p representations \({\bar{\rho }}:\Gamma _{F} \rightarrow G(k)\) of Galois groups of global fields F, valued in reductive groups G(k), and assumed to be odd when F is a number field. Thus we find that lifting theorems, combined with automorphy lifting results pioneered by Wiles in the number field case and the results in the global Langlands correspondence proved by Drinfeld and L. Lafforgue in the function field case, give the only known method to access modularity of mod p Galois representations both in reducible and irreducible cases.

Square root p-adic L-functions, I : Construction of a one-variable measure

The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, has given explicit formulas for central critical values of a large class of Rankin-Selberg tensor products. Although the conjecture is not proved in full generality, there has been considerable progress, especially for $L$-values of the form $L(1/2,BC(\pi) \times BC(\pi'))$, where $\pi$ and $\pi'$ are cohomological automorphic representations of unitary groups $U(V)$ and $U(V')$, respectively. Here $V$ and $V'$ are hermitian spaces over a CM field, $V$ of dimension $n$, $V'$ of codimension $1$ in $V$, and $BC$ denotes the twisted base change to $GL(n) \times GL(n-1)$. This paper contains the first steps toward generalizing the construction of my paper with Tilouine on triple product $L$-functions to this situation. We assume $\pi$ is a holomorphic representation and $\pi'$ varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to $\pi$ uses recent work of Eischen, Fintzen, Mantovan, and Varma.

A Bombieri–Vinogradov Theorem for Higher-Rank Groups

Abstract We establish a result of Bombieri–Vinogradov type for the Dirichlet coefficients at prime ideals of the standard $L$-function associated to a self-dual cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n$ over a number field $F$ when $\pi$ is not a quadratic twist of itself. Our result does not rely on any unproven progress towards the generalized Ramanujan conjecture or the nonexistence of Landau–Siegel zeros. In particular, when $\pi $ is fixed and not equal to a quadratic twist of itself, we prove the first unconditional Siegel-type lower bound for the twisted $L$-values $|L(1,\pi \otimes \chi )|$ in the $\chi $-aspect, where $\chi $ is a primitive quadratic Hecke character over $F$. Our result improves the levels of distribution in other works that relied on these unproven hypotheses. As applications, when $n=2,3,4$, we prove a $\textrm {GL}_n$ analogue of the Titchmarsh divisor problem and a nontrivial bound for a certain $\textrm {GL}_n\times \textrm {GL}_2$ shifted convolution sum.

Zeros of Rankin–Selberg $L$-functions at the edge of the critical strip (with an appendix by Colin J. Bushnell and Guy Henniart)

Let \pi (respectively \pi_0 ) be a unitary cuspidal automorphic representation of \mathrm{GL}_m (respectively \mathrm{GL}_{m_0} ) over \mathbb{Q} . We prove log-free zero density estimates for Rankin–Selberg L -functions of the form L(s,\pi\times\pi_0) , where \pi varies in a given family and \pi_0 is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke–Maaß forms, the rarity of Landau–Siegel zeros of Rankin–Selberg L -functions, the Chebotarev density theorem, and \ell -torsion in class groups of number fields.

Level-raising for automorphic forms on 𝐺𝐿_{𝑛}

Let E E be a CM number field and F F its maximal real subfield. We prove a level-raising result for regular algebraic conjugate self-dual automorphic representations of G L n ( A E ) GL_{n}(\mathbb {A}_E) . This generalizes previously known results of Thorne [Forum Math. Sigma 2 (2014)] by removing certain hypotheses occurring in that work. In particular, the level-raising prime p p is allowed to be unramified as opposed to inert in F F , the field E / F E/F is not assumed to be everywhere unramified, and the field E E is allowed to be a general CM field.