L-functions of GL2n: p-adic properties and non-vanishing of twists

2020 ◽  
Vol 156 (12) ◽  
pp. 2437-2468
Author(s):  
Mladen Dimitrov ◽  
Fabian Januszewski ◽  
A. Raghuram
Keyword(s):  
Critical Points ◽  
Critical Value ◽  
Abelian Extension ◽  
Real Field ◽  
Type Condition ◽  
Modular Symbol ◽  
Automorphic Representations ◽  
Dirichlet Characters ◽  
Cuspidal Automorphic Representations ◽  
Totally Real

The principal aim of this article is to attach and study $p$ -adic $L$ -functions to cohomological cuspidal automorphic representations $\Pi$ of $\operatorname {GL}_{2n}$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive because we draw heavily upon the methods used in the recent and separate works of all three authors. By construction, our $p$ -adic $L$ -functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$ . Moreover, we work under a weaker Panchishkine-type condition on $\Pi _p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$ -adic $L$ -functions at all critical points. This has the striking consequence that, given a unitary $\Pi$ whose standard $L$ -function admits at least two critical points, and given a prime $p$ such that $\Pi _p$ is ordinary, the central critical value $L(\frac {1}{2}, \Pi \otimes \chi )$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$ -power conductor.

scholarly journals On the image of complex conjugation in certain Galois representations

2016 ◽  
Vol 152 (7) ◽  
pp. 1476-1488 ◽  
Author(s):  
Ana Caraiani ◽  
Bao V. Le Hung
Keyword(s):  
Symmetric Spaces ◽  
Galois Representations ◽  
Real Field ◽  
Complex Conjugation ◽  
Automorphic Representations ◽  
Locally Symmetric Spaces ◽  
Cuspidal Automorphic Representations ◽  
Totally Real ◽  
Totally Real Field

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$.

L-Functions

2019 ◽  
pp. 84-125
Author(s):  
Günter Harder ◽  
A. Raghuram
Keyword(s):  
Critical Points ◽  
The Other ◽  
Automorphic Representations ◽  
Special Values ◽  
Combinatorial Lemma ◽  
The One ◽  
Cuspidal Automorphic Representations

This chapter turns to L-functions. It first covers motivic and cohomological L-functions. There is a well-known conjectural dictionary between cohomological cuspidal automorphic representations of GLn and pure rank n motives. The chapter briefly reviews this dictionary while recasting it in the context of strongly inner Hecke summands on the one hand and pure effective motives on the other. Afterward, the critical points for L-functions and the combinatorial lemma are explored. In particular, the chapter reviews the Rankin–Selberg L-functions. A proof of combinatorial lemma is also given. The chapter then provides the main result on special values of L-functions. It concludes with some remarks.

scholarly journals The sign of Galois representations attached to automorphic forms for unitary groups

2011 ◽  
Vol 147 (5) ◽  
pp. 1337-1352 ◽  
Author(s):  
Joël Bellaïche ◽  
Gaëtan Chenevier
Keyword(s):  
Number Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Automorphic Representation ◽  
Unitary Groups ◽  
Complex Conjugate ◽  
Automorphic Representations ◽  
Group A ◽  
Cuspidal Automorphic Representations ◽  
Totally Real

AbstractLet K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.

scholarly journals Explicit application of Waldspurger’s theorem

2013 ◽  
Vol 16 ◽  
pp. 216-245
Author(s):  
Soma Purkait
Keyword(s):  
Modular Form ◽  
Cusp Form ◽  
Modular Forms ◽  
Critical Value ◽  
Automorphic Representations ◽  
Half Integral Weight ◽  
Dirichlet Characters ◽  
Integral Weight ◽  
Quadratic Twist

AbstractFor a given cusp form $\phi $ of even integral weight satisfying certain hypotheses, Waldspurger’s theorem relates the critical value of the $\mathrm{L} $-function of the $n\mathrm{th} $ quadratic twist of $\phi $ to the $n\mathrm{th} $ coefficient of a certain modular form of half-integral weight. Waldspurger’s recipes for these modular forms of half-integral weight are far from being explicit. In particular, they are expressed in the language of automorphic representations and Hecke characters. We translate these recipes into congruence conditions involving easily computable values of Dirichlet characters. We illustrate the practicality of our ‘simplified Waldspurger’ by giving several examples.

TWISTED TRIPLE PRODUCT -ADIC L-FUNCTIONS AND HIRZEBRUCH–ZAGIER CYCLES

2019 ◽  
Vol 19 (6) ◽  
pp. 1947-1992
Author(s):  
Iván Blanco-Chacón ◽  
Michele Fornea
Keyword(s):  
Quadratic Extension ◽  
Number Fields ◽  
Triple Product ◽  
Quadratic Number ◽  
Automorphic Representations ◽  
Quadratic Number Field ◽  
Central Values ◽  
Cuspidal Automorphic Representations ◽  
Totally Real ◽  
Real Quadratic

Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a $p$-nearly ordinary family of unitary cuspidal automorphic representations of $\text{Res}_{L\times F/F}(\text{GL}_{2})$. Furthermore, when $L/\mathbb{Q}$ is a real quadratic number field and $p$ is a split prime, we prove a $p$-adic Gross–Zagier formula relating the values of the $p$-adic $L$-function outside the range of interpolation to the syntomic Abel–Jacobi image of generalized Hirzebruch–Zagier cycles.

scholarly journals Effective Log-Free Zero Density Estimates for Automorphic L-Functions and the Sato–Tate Conjecture

2017 ◽  
Vol 2019 (22) ◽  
pp. 6988-7036
Author(s):  
Robert J Lemke Oliver ◽  
Jesse Thorner
Keyword(s):  
Short Interval ◽  
Prime Number Theorem ◽  
Similar Estimate ◽  
Automorphic Representations ◽  
Density Estimates ◽  
Tate Conjecture ◽  
Zero Density ◽  
Dirichlet Characters ◽  
Ramanujan Conjecture ◽  
Cuspidal Automorphic Representations

Abstract Let $K/\mathbb{Q}$ be a number field. Let π and π′ be cuspidal automorphic representations of $\textrm{GL}_{d}(\mathbb{A}_{K})$ and $\textrm{GL}_{d^{\prime }}(\mathbb{A}_{K})$. We prove an unconditional and effective log-free zero density estimate for all automorphic L-functions L(s, π) and prove a similar estimate for Rankin–Selberg L-functions L(s, π × π′) when π or π′ satisfies the Ramanujan conjecture. As applications, we make effective Moreno’s analog of Hoheisel’s short interval prime number theorem and extend it to the context of the Sato–Tate conjecture; additionally, we bound the least prime in the Sato–Tate conjecture in analogy with Linnik’s theorem on the least prime in an arithmetic progression. We also prove effective log-free density estimates for automorphic L-functions averaged over twists by Dirichlet characters, which allows us to prove an “average Hoheisel” result for GLdL-functions.

scholarly journals On Goren–Oort stratification for quaternionic Shimura varieties

2016 ◽  
Vol 152 (10) ◽  
pp. 2134-2220 ◽  
Author(s):  
Yichao Tian ◽  
Liang Xiao
Keyword(s):  
Line Bundle ◽  
Real Field ◽  
Necessary Condition ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Maximal Level ◽  
Totally Real ◽  
Totally Real Field

Let $F$ be a totally real field in which a prime $p$ is unramified. We define the Goren–Oort stratification of the characteristic-$p$ fiber of a quaternionic Shimura variety of maximal level at $p$. We show that each stratum is a $(\mathbb{P}^{1})^{r}$-bundle over other quaternionic Shimura varieties (for an appropriate integer $r$). As an application, we give a necessary condition for the ampleness of a modular line bundle on a quaternionic Shimura variety in characteristic $p$.

scholarly journals Asymptotic generalized Fermat’s last theorem over number fields

2019 ◽  
Vol 16 (05) ◽  
pp. 907-924
Author(s):  
Yasemin Kara ◽  
Ekin Ozman
Keyword(s):  
Number Fields ◽  
Real Field ◽  
Quadratic Number ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Imaginary Quadratic Number ◽  
Fermat Equation ◽  
Totally Real ◽  
Totally Real Fields ◽  
Generalized Fermat Equation

Recent work of Freitas and Siksek showed that an asymptotic version of Fermat’s Last Theorem (FLT) holds for many totally real fields. This result was extended by Deconinck to the generalized Fermat equation of the form [Formula: see text], where [Formula: see text] are odd integers belonging to a totally real field. Later Şengün and Siksek showed that the asymptotic FLT holds over number fields assuming two standard modularity conjectures. In this work, combining their techniques, we show that the generalized Fermat’s Last Theorem (GFLT) holds over number fields asymptotically assuming the standard conjectures. We also give three results which show the existence of families of number fields on which asymptotic versions of FLT or GFLT hold. In particular, we prove that the asymptotic GFLT holds for a set of imaginary quadratic number fields of density 5/6.

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