Automorphic forms, 𝐿-functions, and 𝑝-adic analysis

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

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Sums of k-th powers and the Whittaker–Fourier coefficients of automorphic forms

The Ramanujan Journal ◽

10.1007/s11139-020-00332-4 ◽

2021 ◽

Author(s):

Wenzhi Luo

Keyword(s):

Fourier Coefficients ◽

Automorphic Forms

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Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

Compositio Mathematica ◽

10.1112/s0010437x09004266 ◽

2009 ◽

Vol 146 (1) ◽

pp. 21-57 ◽

Cited By ~ 1

Author(s):

Harald Grobner

Keyword(s):

Algebraic Group ◽

Eisenstein Series ◽

Symplectic Group ◽

Automorphic Forms ◽

General Theorem ◽

Simple Algebraic Group ◽

Ramanujan Conjecture ◽

Eisenstein Cohomology ◽

Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

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