Classical Modular Symbols, Modular Forms, L-functions

2021 ◽  
pp. 119-151
Author(s):  
Joël Bellaïche
Keyword(s):  
Modular Forms ◽  
Modular Symbols

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.

scholarly journals Overconvergent Eichler–Shimura isomorphisms for quaternionic modular forms over ℚ

2017 ◽  
Vol 13 (10) ◽  
pp. 2687-2715
Author(s):  
Daniel Barrera Salazar ◽  
Shan Gao
Keyword(s):  
Modular Forms ◽  
Weight Space ◽  
Shimura Curves ◽  
Open Disk ◽  
Modular Symbols ◽  
Finite Slope

In this work, we construct overconvergent Eichler–Shimura isomorphisms over Shimura curves over [Formula: see text]. More precisely, for a prime [Formula: see text] and a wide open disk [Formula: see text] in the weight space, we construct a Hecke–Galois-equivariant morphism from the space of families of overconvergent modular symbols over [Formula: see text] to the space of families of overconvergent modular forms over [Formula: see text]. In addition, for all but finitely many weights [Formula: see text], this morphism provides a description of the finite slope part of the space of overconvergent modular symbols of weight [Formula: see text] in terms of the finite slope part of the space of overconvergent modular forms of weight [Formula: see text]. Moreover, for classical weights these overconvergent isomorphisms are compatible with the classical Eichler–Shimura isomorphism.

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 Wach modules and critical slope p-adic L-functions

2013 ◽  
Vol 2013 (679) ◽  
pp. 181-206 ◽  
Author(s):  
David Loeffler ◽  
Sarah Livia Zerbes
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Iwasawa Theory ◽  
Modular Symbols ◽  
Critical Slope ◽  
Local Properties

Abstract We study Kato and Perrin-Riou's critical slope p-adic L-function attached to an ordinary modular form using the methods of A. Lei, D. Loeffler and S. L. Zerbes, Wach modules and Iwasawa theory for modular forms, Asian J. Math. 14 (2010), 475–528. We show that it may be decomposed as a sum of two bounded measures multiplied by explicit distributions depending only on the local properties of the modular form at p. We use this decomposition to prove results on the zeros of the p-adic L-function, and we show that our results match the behaviour observed in examples calculated by Pollack and Stevens in “Overconvergent modular symbols and p-adic L-functions”, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 1, 1–42.

scholarly journals Overconvergent Eichler–Shimura isomorphisms

2014 ◽  
Vol 14 (2) ◽  
pp. 221-274 ◽  
Author(s):  
Fabrizio Andreatta ◽  
Adrian Iovita ◽  
Glenn Stevens
Keyword(s):  
Finite Subset ◽  
Modular Forms ◽  
Weight Space ◽  
Analytic Family ◽  
Open Disk ◽  
Modular Symbols

AbstractGiven a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$, to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$-adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope $\leq h$.

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