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arXiv · 1807.03883

Ap\'ery-like numbers and families of newforms with complex multiplication

Alexis Gomez·Dermot McCarthy·Dylan Young
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Abstract

Using Hecke characters, we construct two infinite families of newforms with complex multiplication, one by $\mathbb{Q}(\sqrt{-3})$ and the other by $\mathbb{Q}(\sqrt{-2})$. The values of the $p$-th Fourier coefficients of all the forms in each family can be described by a single formula, which we provide explicitly. This allows us to establish a formula relating the $p$-th Fourier coefficients of forms of different weights, within each family. We then prove congruence relations between the $p$-th Fourier coefficients of these newforms at all odd weights and values coming from two of Zagier's sporadic Ap\'ery-like sequences.

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Alexis Gomez, Dermot McCarthy, Dylan Young. 2018-07-10. Ap\'ery-like numbers and families of newforms with complex multiplication. https://arxiv.org/abs/1807.03883

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