scholarly journals Level 1 Hecke algebras of modular forms modulo 

2014 ◽  
Vol 151 (3) ◽  
pp. 397-415 ◽  
Author(s):  
Joël Bellaïche ◽  
Chandrashekhar Khare
Keyword(s):  
Recent Result ◽  
Modular Forms ◽  
Hecke Algebras ◽  
Local Algebras ◽  
Deformation Rings ◽  
Level 1

AbstractIn this paper, we study the structure of the local components of the (shallow, i.e. without $U_{p}$) Hecke algebras acting on the space of modular forms modulo $p$ of level $1$, and relate them to pseudo-deformation rings. In many cases, we prove that those local components are regular complete local algebras of dimension $2$, generalizing a recent result of Nicolas and Serre for the case $p=2$.

scholarly journals On Hecke algebras for Hilbert modular forms

2014 ◽  
Vol 134 ◽  
pp. 197-225
Author(s):  
Yoshio Hiraoka ◽  
Kaoru Okada
Keyword(s):  
Modular Forms ◽  
Hecke Algebras ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

scholarly journals Products of vector valued Eisenstein series

2017 ◽  
Vol 29 (1) ◽  
Author(s):  
Martin Westerholt-Raum
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Congruence Subgroups ◽  
Level 1 ◽  
Vector Valued ◽  
Level Aspect

AbstractWe prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin–Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context their previously mentioned result.

scholarly journals ON REPRESENTATION OF MODULAR FORMS AS HOMOGENEOUS POLYNOMIALS

2017 ◽  
Vol 21 (6) ◽  
pp. 40-49
Author(s):  
G.V. Voskresenskaya
Keyword(s):  
Modular Forms ◽  
Homogeneous Polynomial ◽  
Cusp Forms ◽  
Homogeneous Polynomials ◽  
Base Functions ◽  
Level 1 ◽  
Eta Products

In the article we study the spaces of modular forms such that each element of them is a homogeneous polynomial of modular forms of low weights of the same level. It is a classical fact that it is true for the level 1. N. Koblitz point out that it is true for cusp forms of level 4. In this article we show that the analogous situation takes place for the levels corresponding to the eta-products with multiplicative coefficients. In all cases under consideration the base functions are eta-products. In each case the base functions are written explicitly. Dimensions of spaces are calculated by the Cohen - Oesterle formula, the orders in cusps are calculated by the Biagioli formula.

Polynomials for projective representations of level one forms

2011 ◽  
Author(s):  
Johan Bosman
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Cusp Forms ◽  
Tau Function ◽  
Projective Representations ◽  
Level 1

This chapter explicitly computes mod-ℓ Galois representations attached to modular forms. To be precise, it looks at cases with l ≤ 23, and the modular forms considered will be cusp forms of level 1 and weight up to 22. The chapter presents the result in terms of polynomials associated with the projectivized representations. As an application, it will improve a known result on Lehmer's nonvanishing conjecture for Ramanujan's tau function.

scholarly journals A WEIGHT INDEPENDENCE RESULT FOR QUATERNIONIC HECKE ALGEBRAS

2013 ◽  
Vol 09 (08) ◽  
pp. 1895-1922
Author(s):  
LEA TERRACINI
Keyword(s):  
Modular Forms ◽  
Quaternion Algebra ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Hecke Operators ◽  
Langlands Correspondence ◽  
Independence Result ◽  
Relationship Of ◽  
The Relationship

Let p be a prime and B be a quaternion algebra indefinite over Q and ramified at p. We consider the space of quaternionic modular forms of weight k and level p∞, endowed with the action of Hecke operators. By using cohomological methods, we show that the p-adic topological Hecke algebra does not depend on the weight k. This result provides a quaternionic version of a theorem proved by Hida for classical modular forms; we discuss the relationship of our result to Hida's theorem in terms of Jacquet–Langlands correspondence.

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