Abstract
We investigate certain Eisenstein congruences, as predicted by Harder, for level p paramodular forms of genus 2. We use algebraic modular forms to generate new evidence for the conjecture. In doing this, we see explicit computational algorithms that generate Hecke eigenvalues for such forms.
Highlights
Congruences between modular forms have been found and studied for many years
The first interesting example is found in the work of Ramanujan
Proof Choose an isomorphism of quaternion algebras Dq ∼= M2(Qq ) that preserves the norm, trace, and integrality
Summary
Congruences between modular forms have been found and studied for many years. The first interesting example is found in the work of Ramanujan. He studied in great detail the Fourier coefficients τ (n) of the discriminant function (z) = q ∞ n=1(1 − qn) (where q = e2πiz). The significance of is that it is the unique normalized cusp form of weight 12. Amongst Ramanujan’s mysterious observations was a pretty congruence: τ (n) ≡ σ11(n) mod 691. Σ11(n) = d|n d11 is a power divisor sum. One wishes to explain the appearance of the modulus 691. The true incarnation of this is via the fact that the
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