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Self-conjugate vector partitions and the parity of the spt-function

George E. Andrews, Frank G. Garvan, Jie Liang (2013)

Acta Arithmetica

Let spt(n) denote the total number of appearances of the smallest parts in all the partitions of n. Recently, we found new combinatorial interpretations of congruences for the spt-function modulo 5 and 7. These interpretations were in terms of a restricted set of weighted vector partitions which we call S-partitions. We prove that the number of self-conjugate S-partitions, counted with a certain weight, is related to the coefficients of a certain mock theta function studied by the first author,...

Several-variable $p$ -adic families of Siegel-Hilbert cusp eigensystems and their Galois representations

J. Tilouine, E. Urban (1999)

Annales scientifiques de l'École Normale Supérieure

Siegel varieties and $p$ -adic Siegel modular forms.

Tilouine, J. (2007)

Documenta Mathematica

Slopes of modular forms and congruences

Douglas L. Ulmer (1996)

Annales de l'institut Fourier

Our aim in this paper is to prove congruences between on the one hand certain eigenforms of level $p N$ and weight greater than 2 and on the other hand twists of eigenforms of level $p N$ and weight 2. One knows a priori that such congruences exist; the novelty here is that we determine the character of the form of weight 2 and the twist in terms of the slope of the higher weight form, i.e., in terms of the valuation of its eigenvalue for $U_{p}$ . Curiously, we also find a relation between the leading terms of...

Slopes of modular forms and congruences

Douglas L. Ulmer (1996)

Annales de l'institut Fourier

Some congruence property of modular forms.

Shoyu Nagaoka (1997)

Manuscripta mathematica

Sturm type theorem for Siegel modular forms of genus 2 modulo p

Dohoon Choi, YoungJu Choie, Toshiyuki Kikuta (2013)

Acta Arithmetica

Suppose that f is an elliptic modular form with integral coefficients. Sturm obtained bounds for a nonnegative integer n such that every Fourier coefficient of f vanishes modulo a prime p if the first n Fourier coefficients of f are zero modulo p. In the present note, we study analogues of Sturm's bounds for Siegel modular forms of genus 2. As an application, we study congruences involving an analogue of Atkin's U(p)-operator for the Fourier coefficients of Siegel modular forms of genus 2.

Sur une condition suffisante pour l’existence de mesures $p$ -adiques admissibles

Alexei Panchishkin (2003)

Journal de théorie des nombres de Bordeaux

On donne une nouvelle condition suffisante pour l’existence des mesures $p$ -adiques admissibles $μ$ obtenues à partir de suites de distributions $Φ_{j} (j \geq 0)$ à valeurs dans les espaces de formes modulaires. On utilise la projection caractéristique sur le sous-espace primaire associé à une valeur propre non nulle $α$ de l’opérateur $U$ d’Atkin. Notre condition est exprimée en termes des congruences entre les coefficients de Fourier des formes modulaires $Φ_{j}$ . On montre comment vérifier ces congruences, et on traite plusieurs...

Symmetric square $L$ -functions and Shafarevich-Tate groups.

Dummigan, Neil (2001)

Experimental Mathematics

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