scholarly journals On Ramanujan and Dirichlet series with Euler products

1984 ◽  
Vol 25 (2) ◽  
pp. 203-206 ◽  
Author(s):  
S. Raghavan
Keyword(s):  
Modular Form ◽  
Dirichlet Series ◽  
Modular Forms ◽  
Hecke Operators ◽  
Euler Product ◽  
Arithmetical Functions ◽  
In Trans ◽  
Euler Products ◽  
Unpublished Manuscripts ◽  
Profound Insight

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.

scholarly journals Quotients of L-functions

2000 ◽  
Vol 160 ◽  
pp. 143-159
Author(s):  
Bernhard E. Heim
Keyword(s):  
Modular Form ◽  
Dirichlet Series ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Euler Product ◽  
Jacobi Forms ◽  
New Variant

AbstractIn this paper a certain type of Dirichlet series, attached to a pair of Jacobi forms and Siegel modular forms is studied. It is shown that this series can be analyzed by a new variant of the Rankin-Selberg method. We prove that for eigenforms the Dirichlet series have an Euler product and we calculate all the local L-factors. Globally this Euler product is essentially the quotient of the standard L-functions of the involved Jacobi- and Siegel modular form.

scholarly journals Construction of siegel modular forms of degree three and commutation relations of Hecke operators

1985 ◽  
Vol 100 ◽  
pp. 83-96 ◽  
Author(s):  
Yoshio Tanigawa
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Hecke Operators ◽  
Weil Representation ◽  
Commutation Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Shimura Correspondence

In connection with the Shimura correspondence, Shintani [6] and Niwa [4] constructed a modular form by the integral with the theta kernel arising from the Weil representation. They treated the group Sp(1) × O(2, 1). Using the special isomorphism of O(2, 1) onto SL(2), Shintani constructed a modular form of half-integral weight from that of integral weight. We can write symbolically his case as “O(2, 1)→ Sp(1)” Then Niwa’s case is “Sp(l)→ O(2, 1)”, that is from the halfintegral to the integral. Their methods are generalized by many authors. In particular, Niwa’s are fully extended by Rallis-Schiffmann to “Sp(l)→O(p, q)”.

scholarly journals p-adic modular forms of non-integral weight over Shimura curves

2012 ◽  
Vol 149 (1) ◽  
pp. 32-62 ◽  
Author(s):  
Riccardo Brasca
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Hecke Operators ◽  
Analytic Space ◽  
Shimura Curves ◽  
Rigid Analytic Space ◽  
Integral Weight ◽  
Set Up ◽  
Totally Real ◽  
Totally Real Fields

AbstractIn this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of kth invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, that acts compactly on the space of overconvergent modular forms. We also construct the eigencurve.

scholarly journals Non-vanishing of Dirichlet series without Euler products

2018 ◽  
Vol Volume 40 ◽  
Author(s):  
William D. Banks
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Riemann Zeta Function ◽  
Euler Product ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Euler Products ◽  
Riemann Zeta

International audience We give a new proof that the Riemann zeta function is nonzero in the half-plane {s ∈ C : σ > 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).

scholarly journals QUATERNIONIC MODULAR FORMS OF ANY WEIGHT

2014 ◽  
Vol 10 (01) ◽  
pp. 31-53 ◽  
Author(s):  
RICCARDO BRASCA
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Hecke Operators ◽  
Line Bundles ◽  
Analytic Variety ◽  
Test Objects ◽  
Finite Slope ◽  
Geometric Definition

In this work we give a geometric definition, as sections of line bundles, of p-adic analytic families of overconvergent modular forms attached to an indefinite quaternion algebra over ℚ. As a consequence of this, we obtain the existence of an eigencurve in this context. Our theory includes the interpretation of a modular form as a rule on test objects. We introduce the Hecke operators U and T l, both in families and for a single weight. We show that the U -operator acts compactly on the space of overconvergent modular forms. We finally construct the eigencurve, a rigid analytic variety whose points correspond to systems of eigenvalues associated to overconvergent eigenforms of finite slope with respect to the U -operator.

A magnetic modular form

2019 ◽  
Vol 15 (05) ◽  
pp. 907-924
Author(s):  
Yingkun Li ◽  
Michael Neururer
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Hecke Operators ◽  
Shimura Lift ◽  
Divisibility Property ◽  
Vector Valued

In this paper, we prove a conjecture of Broadhurst and Zudilin concerning a divisibility property of the Fourier coefficients of a meromorphic modular form using the generalization of the Shimura lift by Borcherds and Hecke operators on vector-valued modular forms developed by Bruinier and Stein. Furthermore, we construct a family of meromorphic modular forms with this property.

EULER PRODUCTS IN RAMANUJAN'S LOST NOTEBOOK

2013 ◽  
Vol 09 (05) ◽  
pp. 1313-1349 ◽  
Author(s):  
BRUCE C. BERNDT ◽  
BYUNGCHAN KIM ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Binary Quadratic Forms ◽  
Arithmetical Functions ◽  
Lost Notebook ◽  
Elementary Methods ◽  
Famous Paper ◽  
Expository Article ◽  
Euler Products ◽  
First Time

In his famous paper, "On certain arithmetical functions", Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan's tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for the Euler products appearing in his lost notebook, we provide or sketch proofs using elementary methods, binary quadratic forms, and modular forms.

The convergence of Euler products over p-adic number fields

2009 ◽  
Vol 52 (3) ◽  
pp. 583-606 ◽  
Author(s):  
Daniel Delbourgo
Keyword(s):  
Topological Space ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
Number Fields ◽  
Product Structure ◽  
Euler Product ◽  
Euler Products ◽  
Riemann Zeta ◽  
New Formula

AbstractWe define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.

CONGRUENCES RELATED TO MODULAR FORMS

2010 ◽  
Vol 06 (06) ◽  
pp. 1367-1390 ◽  
Author(s):  
H. A. VERRILL
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Congruence Relation ◽  
Mellin Transform ◽  
Congruence Subgroup ◽  
Modular Function ◽  
Euler Product ◽  
Mod P

Let f be a modular form of weight k for a congruence subgroup Γ ⊂ SL 2(Z), and t a weight 0 modular function for Γ. Assume that near t = 0, we can write f = ∑n≥0bn tn, bn ∈ Z. Let ℓ(z) be a weight k + 2 modular form with q-expansion ∑γnqn, such that the Mellin transform of ℓ can be expressed as an Euler product. Then we show that if [Formula: see text] for some integers ai, di, then the congruence relation bmpr -γpbmpr-1 + εppk+1bmpr-2 ≡ 0 ( mod pr) holds. We give a number of examples of this phenomena.

Computing coefficients of modular forms

2011 ◽  
Author(s):  
Bas Edixhoven
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Galois Representations ◽  
Theta Functions ◽  
Positive Definite ◽  
Hecke Operators ◽  
Linear Combinations ◽  
Tau Function ◽  
Arbitrary Weight

This chapter applies the main result on the computation of Galois representations attached to modular forms of level one to the computation of coefficients of modular forms. It treats the case of the discriminant modular form, that is, the computation of Ramanujan's tau-function at primes, and then deals with the more general case of forms of level one and arbitrary weight k, reformulated as the computation of Hecke operators Tⁿ as ℤ-linear combinations of the Tᵢ with i < k = 12. The chapter gives an application to theta functions of even, unimodular positive definite quadratic forms over ℤ.

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