Adjoint L-Functions for GL(3) and U2,1

We show that the finite part of the adjoint $L$-function (including contributions from all non-archimedean places, including ramified places) is holomorphic in ${\textrm{Re}}(s) \ge 1/2$ for a cuspidal automorphic representation of ${\textrm{GL}}_3$ over a number field. This improves the main result of [21]. We obtain more general results for twisted adjoint $L$-functions of both ${\textrm{GL}}_3$ and quasisplit unitary groups. For unitary groups, we explicate the relationship between poles of twisted adjoint $L$-functions, endoscopy, and the structure of the stable base change lifting.

Simple zeros of automorphic -functions

We prove that the complete $L$-function associated to any cuspidal automorphic representation of $\operatorname{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$ has infinitely many simple zeros.

Transfer to Characteristic Zero of the Jacquet–Mao’s Metaplectic Fundamental Lemma

Jacquet conjectured that globally a cuspidal automorphic representation of $\mathrm{G}\mathrm{L}_r$ should be in the image of the metaplectic correspondence precisely when it is distinguished with respect to a split orthogonal similitude group. Jacquet–Mao have suggested an approach to solve this problem by using “relative” traces formula. One of the steps of this approach is the Jacquet–Mao’s metaplectic fundamental lemma. The author proved this fundamental lemma for any $r$ in the case of positive characteristic. The aim of this paper is to extend this result to the more general case.

Period relations for cusp forms of GSp4

Let F be a totally real number field and let π be a cuspidal automorphic representation of {\mathrm{GSp_{4}}(\mathbb{A}_{F})}, which contributes irreducibly to coherent cohomology. If π has a Bessel model, we may attach a period {p(\pi)} to this datum. In the present paper, which is Part I in a series of two, we establish a relation of these Bessel periods {p(\pi)} and all of their twists {p(\pi\otimes\xi)} under arbitrary algebraic Hecke characters ξ. In the appendix, we show that {(\mathfrak{g},K)}-cohomological cusp forms of {\mathrm{GSp_{4}}(\mathbb{A}_{F})} all qualify to be of the above type – providing a large source of examples. We expect that these period relations for {\mathrm{GSp_{4}}(\mathbb{A}_{F})} will allow a conceptual, fine treatment of rationality relations of special values of the spin L-function, which we hope to report on in Part II of this paper.

Fourth power moment of coefficients of automorphic L-functions for GL(m)

Let π be a unitary cuspidal automorphic representation for {\mathrm{GL}_{m}(\mathbb{A}_{\mathbb{Q}})}, and let {L(s,\pi)} be the automorphic L-function attached to π, which has a Dirichlet series expression in the half-plane {\Re s>1}, i.e.L(s,\pi)=\sum_{n=1}^{\infty}\frac{\lambda_{\pi}(n)}{n^{s}}.In this paper we are interested in the upper bound of the fourth power moment of {\lambda_{\pi}(n)}, i.e. {\sum_{n\leq x}\lambda_{\pi}(n)^{4}}. If {m=2}, we are able to consider the sixteenth power moment of {\lambda_{\pi}(n)}. As an application, we consider the lower bound of {\sum_{n\leq x}\lvert\lambda_{\pi}(n)\rvert}, which improves previous results.

HYPERTRANSCENDENCE OF -FUNCTIONS FOR

We generalise a result of Hilbert which asserts that the Riemann zeta-function${\it\zeta}(s)$is hypertranscendental over$\mathbb{C}(s)$. Let${\it\pi}$be any irreducible cuspidal automorphic representation of$\text{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$with unitary central character. We establish a certain type of functional difference–differential independence for the associated$L$-function$L(s,{\it\pi})$. This result implies algebraic difference–differential independence of$L(s,{\it\pi})$over$\mathbb{C}(s)$(and more strongly, over a certain field${\mathcal{F}}_{s}$which contains$\mathbb{C}(s)$). In particular,$L(s,{\it\pi})$is hypertranscendental over$\mathbb{C}(s)$. We also extend a result of Ostrowski on the hypertranscendence of ordinary Dirichlet series.

A Specialisation of the Bump–Friedberg L-function

We study the restriction of Bump–Friedberg integrals to affine lines {(s + α, 2s), s ∊ ℂ}. It has simple theory, very close to that of the Asai L-function. It is an integral representation of the product L(s + α, π)L(2s, Λ2, π), which we denote by Llin(s, π, α) for this abstract, when π is a cuspidal automorphic representation of GL(k, 𝔸) for 𝔸 the adeles of a number field. When k is even, we show that the partial L-function Llin,S(s, π, α) has a pole at 1/2 if and only if π admits a (twisted) global period. This gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that π has a twisted global period if and only if L(α + 1/2, π) ≠ 0 and L(1, Λ2 , π) = ∞. When k is odd, the partial L-function is holmorphic in a neighbourhood of Re(s) ≥ 1/2 when Re(α) is ≥ 0.

Automorphy and irreducibility of some l-adic representations

In this paper we prove that a pure, regular, totally odd, polarizable weakly compatible system of $l$-adic representations is potentially automorphic. The innovation is that we make no irreducibility assumption, but we make a purity assumption instead. For compatible systems coming from geometry, purity is often easier to check than irreducibility. We use Katz’s theory of rigid local systems to construct many examples of motives to which our theorem applies. We also show that if $F$ is a CM or totally real field and if ${\it\pi}$ is a polarizable, regular algebraic, cuspidal automorphic representation of $\text{GL}_{n}(\mathbb{A}_{F})$, then for a positive Dirichlet density set of rational primes $l$, the $l$-adic representations $r_{l,\imath }({\it\pi})$ associated to ${\it\pi}$ are irreducible.

The Rankin–Selberg integral with a non-unique model for the standard -function of

Let$\mathcal{L}^{S}\left (s,\pi ,{\mathfrak{st}}\right )$be a partial$\mathcal{L}$-function of degree$7$of a cuspidal automorphic representation$\pi $of the exceptional group$G_2$. In this paper we construct a Rankin–Selberg integral for representations having a certain Fourier coefficient.