Supersingular elliptic curves, theta series and weight two modular forms

Emerton, Matthew

doi:10.1090/S0894-0347-02-00390-9

Supersingular elliptic curves, theta series and weight two modular forms
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by Matthew Emerton

J. Amer. Math. Soc. 15 (2002), 671-714

DOI: https://doi.org/10.1090/S0894-0347-02-00390-9

Published electronically: February 27, 2002

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Abstract:

Let $p$ be a prime, and let $\mathcal {M}$ denote the space of weight two modular forms on $\Gamma _{0}(p)$ all of whose Fourier coefficients are integral, except possibly for the constant term, which should be either integral or half-integral. We prove that $\mathcal {M}$ is spanned as a $\mathbb {Z}$-module by theta series attached to the unique quaternion algebra that is ramified at $p$, at infinity, and at no other primes.

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Coleman’s $\mathcal {L}$-invariant and families of modular forms