scholarly journals Universal Fourier expansions of Bianchi modular forms

2019 ◽  
Vol 16 (04) ◽  
pp. 731-746
Author(s):  
Tian An Wong
Keyword(s):  
Modular Forms ◽  
Hecke Operators ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Modular Symbols ◽  
Fourier Expansions

We generalize Merel’s work on universal Fourier expansions to Bianchi modular forms over Euclidean imaginary quadratic fields, under the assumption of the nondegeneracy of a pairing between Bianchi modular forms and Bianchi modular symbols. Among the key inputs is a computation of the action of Hecke operators on Manin symbols, building upon the Heilbronn–Merel matrices constructed by Mohamed.

scholarly journals -adic -functions for

2019 ◽  
Vol 71 (5) ◽  
pp. 1019-1059
Author(s):  
Daniel Barrera Salazar ◽  
Chris Williams
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Modular Symbol ◽  
Modular Symbols ◽  
General Number ◽  
Critical Slope ◽  
Totally Real ◽  
Made In

AbstractSince Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct $p$-adic $L$-functions for non-critical slope rational modular forms, the theory has been extended to construct $p$-adic $L$-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the $L$-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the $p$-adic $L$-function of the eigenform to be this distribution.

Coefficients of half-integral weight modular forms and indivisibility of class numbers of imaginary quadratic fields

2016 ◽  
Vol 12 (02) ◽  
pp. 561-565
Author(s):  
Haobo Dai
Keyword(s):  
Modular Forms ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields ◽  
Half Integral Weight ◽  
Integral Weight

In this paper, we investigate the nonvanishing coefficients of half-integral weight modular forms modulo a prime. We also consider the indivisibility of class numbers of imaginary quadratic fields.

scholarly journals Euler systems for modular forms over imaginary quadratic fields

2015 ◽  
Vol 151 (9) ◽  
pp. 1585-1625 ◽  
Author(s):  
Antonio Lei ◽  
David Loeffler ◽  
Sarah Livia Zerbes
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Field ◽  
Euler System ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Selmer Groups ◽  
Imaginary Quadratic Fields ◽  
Euler Systems

We construct an Euler system attached to a weight 2 modular form twisted by a Grössencharacter of an imaginary quadratic field $K$, and apply this to bounding Selmer groups.

scholarly journals Power Residues of Fourier Coefficients of Modular Forms

2005 ◽  
Vol 57 (5) ◽  
pp. 1102-1120 ◽  
Author(s):  
Tom Weston
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Galois Representation ◽  
Density Theorem ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Abelian Extensions ◽  
Numerical Investigations ◽  
Residue Modulo ◽  
Reciprocity Laws

AbstractLet ρ: GQ → GLn(Qℓ) be a motivic ℓ-adic Galois representation. For fixed m > 1 we initiate an investigation of the density of the set of primes p such that the trace of the image of an arithmetic Frobenius at p under ρ is an m-th power residue modulo p. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals 1/m whenever the image of ρ is open. We further conjecture that for such ρ the set of these primes p is independent of any set defined by Cebatorev-style Galois-theoretic conditions (in an appropriate sense). We then compute these densities for certain m in the complementary case of modular forms of CM-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem(which does apply in the CM case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at p in abelian extensions of imaginary quadratic fields unramified away from p.

l-adic representations associated to modular forms over imaginary quadratic fields

1993 ◽  
Vol 112 (1) ◽  
pp. 377-411 ◽  
Author(s):  
Michael Harris ◽  
David Soudry ◽  
Richard Taylor
Keyword(s):  
Modular Forms ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields
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