Automorphic Representations Research Articles

A note on the standard zero-free region for [formula omitted]-functions

In this short note, we establish a standard zero-free region for a general class of L-functions for which their logarithms have coefficients with nonnegative real parts, including the Rankin–Selberg L-functions for unitary cuspidal automorphic representations.

An Elliptic Generalization of A1 Spherical DAHA at K = 2

Abstract We construct an algebra that is an elliptic generalization of $A_1$ spherical DAHA acting on its finite-dimensional module at $t=-q^{-K/2}$ with $K=2$. We prove that $PSL(2,\mathbb{Z})$ acts by automorphisms of the algebra we constructed, and provide an explicit representation of automorphisms and algebra operators alike by $3 \times 3$ matrices with matrix elements given by products of elliptic functions. A relation of this construction to the $K$-theory character of affine Laumon space is conjectured. We point out two potential applications, respectively to $SL(3,\mathbb{Z})$ symmetry of Felder–Varchenko functions and to new elliptic invariants of torus knots and Seifert manifolds.

On the Gross–Prasad conjecture with its refinement for (SO(5), SO(2)) and the generalized Böcherer conjecture

We investigate the Gross–Prasad conjecture and its refinement for the Bessel periods in the case of $(\mathrm {SO}(5), \mathrm {SO}(2))$. In particular, by combining several theta correspondences, we prove the Ichino–Ikeda-type formula for any tempered irreducible cuspidal automorphic representation. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two, which are Hecke eigenforms, to central special values of $L$-functions. The formula is regarded as a natural generalization of the Böcherer conjecture to the non-trivial toroidal character case.

Cyclic base change of cuspidal automorphic representations over function fields

Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.

Multiplicity of nontrivial zeros of primitive 𝐿-functions via higher-level correlations

We give universal bounds on the fraction of nontrivial zeros having given multiplicity for L L -functions attached to a cuspidal automorphic representation of G L m / Q \mathrm {GL}_m/\mathbb {Q} . For this, we apply the higher-level correlation asymptotic of Hejhal, Rudnick, and Sarnak in conjunction with semidefinite programming bounds.

On the cubic Shimura lift to $\operatorname{PGL}(3)$: The fundamental lemma

The classical Shimura correspondence lifts automorphic representations on the double cover of \mathrm{SL}_{2} to automorphic representations on \mathrm{PGL}_{2} . Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of \mathrm{SL}_{3} to \mathrm{PGL}_{3} , and also characterize the image of the lift. The characterization would be through the non-vanishing of a certain global period involving a function in the space of the automorphic minimal representation \Theta_{\mathrm{SO}_8} for split \mathrm{SO}_{8}(\mathbb{A}) , consistent with a conjecture of Bump, Friedberg and Ginzburg (2001). In this paper, we first analyze a global distribution on \mathrm{PGL}_{3}(\mathbb{A}) involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of \mathrm{SL}_{3}(\mathbb{A}) . We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the fundamental lemma.

On the coefficients of automorphic representations over polynomials

On the coefficients of automorphic representations over polynomials

On the boundedness of Euler-Stieltjes constants for the Rankin-Selberg \(L-\)function

Let \(E\) be a Galois extension of \(\mathbb{Q}\) of finite degree and let \(\pi \) and \(\pi'\) be two irreducible automorphic unitary cuspidal representations of \(GL_m(\mathbb{A}_E)\) and \(GL_{m'}(\mathbb{A}_E)\), respectively. Let \(\Lambda(s,\pi\times\widetilde{\pi}')\) be a Rankin-Selberg \(L-\)function attached to the product \(\pi\times\widetilde{\pi}'\), where \(\widetilde{\pi}'\) denotes the contragredient representation of \(\pi'\), and let its finite part (excluding Archimedean factors) be \(L(s,\pi\times\widetilde{\pi}')\). The Euler-Stieltjes constants of the Rankin-Selberg \(L-\)function are the coefficients in the Laurent (Taylor) series expansion around \(s=1+it_0\) of the function \(L(s, \pi \times \widetilde{\pi}')\). In this paper, we derive an upper bound for these constants.

Period Relations Between the Betti–Whittaker Periods for GLn Under Duality

Abstract In this paper, under some regularity conditions, we prove a period relation between the Betti–Whittaker periods associated to a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_{n}({{\mathbb{A}}})$ and its contragredient. As a consequence, we prove the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_{2n}({{\mathbb{A}}})$ of orthogonal type, which implies the algebraicity of the ratios of successive critical $L$-values for $\textrm{GSpin}_{2n}^{*} \times \operatorname{GL}_{n^{\prime}}$ by the result of Harder and Raghuram.

Global liftings between inner forms of GSp(4)

For reductive groups G over a number field we discuss automorphic liftings of cohomological cuspidal irreducible automorphic representations π of G(A) to irreducible cohomological automorphic representations of H(A) for the quasi-split inner form H of G, and other inner forms as well. We show the existence of nontrivial weak global cohomological liftings in many cases, in particular for the case where G is anisotropic at the archimedean places. A priori, for these weak liftings we do not give a description of the precise nature of the corresponding local liftings at the ramified places, nor do we characterize the image of the lifting. For inner forms of the group H=GSp(4) however we address these finer questions. Especially, we prove the recent conjectures of Ibukiyama and Kitayama on paramodular newforms of square-free level.

ARCHIMEDEAN NEWFORM THEORY FOR

Abstract We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of $\operatorname {\mathrm {GL}}_n(F)$ , where F is an archimedean local field, that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of $\operatorname {\mathrm {GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.

The Weyl law for algebraic tori

We give an asymptotic evaluation for the number of automorphic characters of an algebraic torus T with bounded analytic conductor. The analytic conductor is defined via the local Langlands correspondence for tori by choosing a finite-dimensional complex algebraic representation of the L -group of T . Our results therefore fit into a general framework of counting automorphic representations on reductive groups by analytic conductor.

Zeros of Rankin–Selberg L-functions in families

Let $\mathfrak {F}_n$ be the set of all cuspidal automorphic representations $\pi$ of $\mathrm {GL}_n$ with unitary central character over a number field $F$ . We prove the first unconditional zero density estimate for the set $\mathcal {S}=\{L(s,\pi \times \pi ')\colon \pi \in \mathfrak {F}_n\}$ of Rankin–Selberg $L$ -functions, where $\pi '\in \mathfrak {F}_{n'}$ is fixed. We use this density estimate to establish: (i) a hybrid-aspect subconvexity bound at $s=\frac {1}{2}$ for almost all $L(s,\pi \times \pi ')\in \mathcal {S}$ ; (ii) a strong on-average form of effective multiplicity one for almost all $\pi \in \mathfrak {F}_n$ ; and (iii) a positive level of distribution for $L(s,\pi \times \widetilde {\pi })$ , in the sense of Bombieri–Vinogradov, for each $\pi \in \mathfrak {F}_n$ .

The sup-norm problem beyond the newform

AbstractIn this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $\mathrm{GL}_2(\mathbb{Z}_p)$ -type V in $\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.

A p$p$‐adic approach to the existence of level‐raising congruences

AbstractWe construct level‐raising congruences between ‐ordinary automorphic representations, and apply this to the problem of symmetric power functoriality for Hilbert modular forms. In particular, we prove the existence of the symmetric power lift of a Hilbert modular eigenform of regular weight for each odd integer . In a future work with James Newton, we will use these results to establish the existence of the symmetric power lift for all .

Certain L2-Norms on Automorphic Representations of SL(2, R)

Let Γ be a non-uniform lattice in SL(2,R). In this paper, we study various L2-norms of automorphic representations of SL(2,R). We bound these norms with intrinsic norms defined on the representation. Comparison of these norms can help us understand the growth of L-functions in a systematic way.

Value Distribution of Logarithmic Derivatives of Quadratic Twists of Automorphic L-functions

Abstract Let $d\in\mathbb{N}$ and π be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-\frac{L^{\prime}}{L}(1+it,\pi\otimes\chi_D)$ as D varies over fundamental discriminants. Here, t is a fixed real number and χD is the real character associated with D. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $\left|\frac{L^{\prime}}{L}(1,\pi\otimes\chi_D)\right|$ when π is self-dual.

Local-global compatibility for regular algebraic cuspidal automorphic representations when

Abstract We prove the compatibility of local and global Langlands correspondences for $\operatorname {GL}_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let $r_p(\pi )$ denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n(\mathbb {A}_F)$ . We show that the restriction of $r_p(\pi )$ to the decomposition group of a place $v\nmid p$ of F corresponds up to semisimplification to $\operatorname {rec}(\pi _v)$ , the image of $\pi _v$ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of $\left .r_p(\pi )\right |{}_{\operatorname {Gal}_{F_v}}$ is ‘more nilpotent’ than the monodromy of $\operatorname {rec}(\pi _v)$ .

COUNTING DISCRETE, LEVEL- $1$ , QUATERNIONIC AUTOMORPHIC REPRESENTATIONS ON $G_2$

AbstractQuaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm {GL}_2$ . Here, we use ‘hyperendoscopy’ techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group $G_2$ , eventually getting an analog of the Eichler–Selberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard and Taïbi to compute dimensions of spaces of level- $1$ quaternionic representations. On the way, we prove a Jacquet–Langlands-style result describing them in terms of classical modular forms and automorphic representations on the compact-at-infinity form $G_2^c$ .The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being ‘regular’. A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series.We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discrete-at-infinity automorphic representations on arbitrary reductive groups instead of just classical ones.

GALOIS REPRESENTATIONS FOR EVEN GENERAL SPECIAL ORTHOGONAL GROUPS

Abstract We prove the existence of $\mathrm {GSpin}_{2n}$ -valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of ${\mathrm {GSO}}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb {H}}$ , arising from forms of ${\mathrm {GSO}}_{2n}$ . As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin L-functions and improve on the construction of ${\mathrm {SO}}_{2n}$ -valued Galois representations by removing the outer automorphism ambiguity.