Weight-dependent congruence properties of modular forms

Congruence Properties of Siegel Modular Forms

Let X35 be a Siegel cusp form of degree 2 and weight 35. Kikuta, Kodama and Nagaoka [4] proved that det T a(T, X35) ≡ 0 mod 23 for every half integral positive symmetric matrix T . In this paper, we give a finite number of examples of Hecke eigenforms of degree 2 and odd weights that have the same type of congruence relation above. We also introduce congruence relations for the Hecke eigenvalue...

Congruence Properties of Taylor Coefficients of Modular Forms

In their work, Serre and Swinnerton-Dyer study the congruence properties of the Fourier coefficients of modular forms. We examine similar congruence properties, but for the coefficients of a modified Taylor expansion about a CM point τ . These coefficients can be shown to be the product of a power of a constant transcendental factor and an algebraic integer. In our work, we give conditions on τ...

A Supersingular Congruence for Modular Forms

Let p > 3 be a prime. In the ring of modular forms with q-expansions defined over Z(p), the Eisenstein function Ep+1 is shown to satisfy (Ep+1) p−1 ≡ − −1 p ∆ 2−1)/12 mod (p, Ep−1). This is equivalent to a result conjectured by de Shalit on the polynomial satisfied by all the j-invariants of supersingular elliptic curves over Fp. It is also closely related to a result of Gross and Landweber use...

Modular Forms of Weight One

The decomposition into an Euler product and the functional equation of the Dirichlet series associated by Hecke to modular forms of weight one suggests that these correspond to Artin L-functions of degree 2 over Q, otherwise known as Galois representations Gal(Q/Q)→ GL2(C). It is this correspondence, conjectured by Langlands, which we establish here. The first three sections are preliminary. Th...

MAT1200: Modular Forms of Half Integral Weight