p-adic congruences and modular forms of half integer weight
type: Item
instance of: scholarly article
title: p-adic congruences and modular forms of half integer weight
zbMATH Open document ID: 0571.10030
DOI: 10.1007/BF01457070
author: Neal Koblitz
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164095
review text: This paper investigates p-adic interpolation of modular forms of half integer weight, and gives evidence that under certain circumstances p- adic congruences between forms of integer weight descend via the Shimura lifting to congruences between corresponding forms of half integer weight. First, an example is given: the Ramanujan congruence descends to weight 13/2; this gives a mod 691 factorization into a product of Bernoulli numbers for the algebraic part of the central critical value of the twisted L-function of the discriminant form. Next, it is shown that it makes sense to speak of half integer weight forms of p-adic weight. Then the generalized class number Eisenstein series of half integer weight are interpolated p-adically; this motivates certain conjectures concerning congruences between central critical values of Hecke L-series corresponding to elliptic curves. These conjectures are then shown to be compatible with the Katz-Manin-Vishik interpolation of two-variable Eisenstein series. Finally, numerical evidence in support of the conjectures is given in the case of two families of Hecke series corresponding to certain elliptic curves with complex multiplication by third and fourth roots of unity.
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 3910491
zbMATH Keywords: p-adic interpolation of modular forms
zbMATH Keywords: p-adic congruences
zbMATH Keywords: Shimura lifting
zbMATH Keywords: half integer weight forms of p-adic weight
zbMATH Keywords: generalized class number Eisenstein series
zbMATH Keywords: congruences between central critical values of Hecke L-series
zbMATH Keywords: modular forms of half-integral weight
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3910491