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Well, this isn't explicitly diophantine, but here goes:

If $f$ is a level one weight $k$ eigenform with rational coefficients, the image of the attached Galois representation

$\rho_f:G_{\mathbb{Q}}\rightarrow GL_2(\hat{\mathbb{Z}})$

is open in the subgroup $G$ defined by demanding

$det(G)\subset \hat{\mathbb{Z}}^{\times{k-1}}$.

In particular, the image contains an open subgroup of $SL_2(\hat{\mathbb{Z}})$. This has the following arithmetic consequence:

For almost all prime numbers $p$, there exists a non-solvable Galois extension $K/\mathbb{Q}$ ramified only at $p$.

In fact, Serre shows that for the unique normalized weight 12 level 1 cuspform, the list of exceptional primes is 2,3,5,7,23,691. This theorem is now known for all $p$, p, although the last known case, p=2p=7, was resolved only a few years ago very recently by DembéléDieulefait.

1

Well, this isn't explicitly diophantine, but here goes:

If $f$ is a level one weight $k$ eigenform with rational coefficients, the image of the attached Galois representation

$\rho_f:G_{\mathbb{Q}}\rightarrow GL_2(\hat{\mathbb{Z}})$

is open in the subgroup $G$ defined by demanding

$det(G)\subset \hat{\mathbb{Z}}^{\times{k-1}}$.

In particular, the image contains an open subgroup of $SL_2(\hat{\mathbb{Z}})$. This has the following arithmetic consequence:

For almost all prime numbers $p$, there exists a non-solvable Galois extension $K/\mathbb{Q}$ ramified only at $p$.

In fact, Serre shows that for the unique normalized weight 12 level 1 cuspform, the list of exceptional primes is 2,3,5,7,23,691. This theorem is now known for all $p$, although the last known case, p=2, was resolved only a few years ago by Dembélé.