On $l$-adic representations and congruences of modular forms: following Swinnerton-Dyer

Abstract

We review Swinnerton-Dyer's paper [11], which discusses congruence relations on modular forms with integral coefficients.

Using Serre-Deligne Theorem, we can associate each modular form $f$ with $\rho_l: \mathrm{Gal}(K_l/\mathbb{Q}) \rightarrow \mathrm{GL}_2(\mathbb{Z}_l)$ with the property $\mathrm{Tr}(\mathrm{Frob}(p))=a_p(f)$. We define $l$ to be an exceptional prime if the image of $\rho_l$ does not contain $\mathrm{SL}_2(\mathbb{Z}_l)$, which for $l \geq 5$ is equivalent to the image of $\rho_l$ under $\bmod$ $l$ reduction, $\overline{\rho_l}$, not containing $\mathrm{SL}_2(\mathbb{F}_l)$.

We determine all possible images of $\overline{\rho_l}$ in $\mathrm{GL}_2(\mathbb{F}_l)$ and obtain three types of congruence relations that may arise. Then, we develop the theory of modular forms $\bmod$ $l$. From this setting, we obtain a necessary condition for $l$ to be an exceptional prime of type 1 and 2.

This enables us to obtain a complete list of exceptional primes of type 1 and 2 for modular forms with integral coefficients. We also eliminate all but one candidate prime, 59, which may be an exceptional prime of type 3 for the unique cusp form of weight 16.

All the contents of this thesis can easily be derived from the original paper of Swinnerton-Dyer's [11], and we do not claim any originality in either mathematical or expositional aspects.