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2006•International Journal of Number TheoryRequires access

p-ADIC AND COMBINATORIAL PROPERTIES OF MODULAR FORM COEFFICIENTS

Po‐Ru Loh, Robert C. Rhoades

Open publisher page 4 citations

Abstract

For two particular classes of elliptic curves, we establish congruences relating the coefficients of their corresponding modular forms to combinatorial objects. These congruences resemble a supercongruence for the Apéry numbers conjectured by Beukers and proved by Ahlgren and Ono in [1]. We also consider the trace Tr 2k(Γ0(N), n) of the Hecke operator Tn acting on the space of cusp forms S2k(Γ0(N)). We show that for (n, N) = 1, these traces interpolate p-adically in the weight aspect.

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For two particular classes of elliptic curves, we establish congruences relating the coefficients of their corresponding modular forms to combinatorial objects. These congruences resemble a supercongruence for the Apéry numbers conjectured by Beukers and proved by Ahlgren and Ono in [1]. We also consider the trace Tr 2k(Γ0(N), n) of the Hecke operator Tn acting on the space of cusp forms S2k(Γ0(N)). We show that for (n, N) = 1, these traces interpolate p-adically in the weight aspect.

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Available abstract

For two particular classes of elliptic curves, we establish congruences relating the coefficients of their corresponding modular forms to combinatorial objects. These congruences resemble a supercongruence for the Apéry numbers conjectured by Beukers and proved by Ahlgren and Ono in [1]. We also consider the trace Tr 2k(Γ0(N), n) of the Hecke operator Tn acting on the space of cusp forms S2k(Γ0(N)). We show that for (n, N) = 1, these traces interpolate p-adically in the weight aspect.

Key concepts: Modular form, Mathematics, Congruence relation, Cusp form, Cusp (singularity), TRACE (psycholinguistics), Hecke operator, Pure mathematics

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