scholarly journals A NOTE ON CONTROL THEOREMS FOR QUATERNIONIC HIDA FAMILIES OF MODULAR FORMS

2012 ◽  
Vol 08 (06) ◽  
pp. 1425-1462 ◽  
Author(s):  
MATTEO LONGO ◽  
STEFANO VIGNI
Keyword(s):  
Modular Forms ◽  
Quaternion Algebras ◽  
Modular Symbols ◽  
Definite Case ◽  
Hida Families

We extend a result of Greenberg and Stevens on the interpolation of modular symbols in Hida families to the context of non-split rational quaternion algebras. Both the definite case and the indefinite case are considered.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras

1991 ◽  
Vol 121 ◽  
pp. 35-96 ◽  
Author(s):  
Siegfried Böcherer ◽  
Rainer Schulze-Pillot
Keyword(s):  
Modular Forms ◽  
Orthogonal Group ◽  
Automorphic Forms ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Theta Correspondence ◽  
Quaternion Algebras ◽  
Metaplectic Group

The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras II

1997 ◽  
Vol 147 ◽  
pp. 71-106 ◽  
Author(s):  
S. Böcherer ◽  
R. Schulze-Pillot
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Automorphic Forms ◽  
Multiplicative Group ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Quaternion Algebras

AbstractWe continue our study of Yoshida’s lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.

scholarly journals ON MODULAR SYMBOLS AND THE COHOMOLOGY OF HECKE TRIANGLE SURFACES

2009 ◽  
Vol 05 (01) ◽  
pp. 89-108 ◽  
Author(s):  
GABOR WIESE
Keyword(s):  
Commutative Ring ◽  
Modular Forms ◽  
Group Cohomology ◽  
Complex Numbers ◽  
Modular Curves ◽  
Modular Symbols ◽  
Algebraic Treatment ◽  
Holomorphic Modular Forms ◽  
Coefficient Ring

The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalized from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. A general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers.

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 Computing Jacobi forms

2016 ◽  
Vol 19 (A) ◽  
pp. 205-219 ◽  
Author(s):  
Nathan C. Ryan ◽  
Nicolás Sirolli ◽  
Nils-Peter Skoruppa ◽  
Gonzalo Tornaría
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Ambient Space ◽  
Jacobi Forms ◽  
Modular Symbols ◽  
Hecke Eigenforms ◽  
Large Index ◽  
Integral Weight

We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows a Jacobi eigenform to be generated directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms, it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.

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.

Sign in / Sign up

Export Citation Format

Share Document