4 problems

Ribet's conjecture. All $\mathbb{Q}$-curves are modular: $\rho_{E,\ell}$ is the $\ell$-adic Galois representation attached to a cusp form on $\Gamma_1(N)$ for some $N$.

Let $Y_*(N)$ be the quotient of the elliptic modular curve $Y_0(N)$ by all Atkin–Lehner involutions, where $N$ is square-free. A point of $Y_*(N)(\bQ)$ is called non-CM if the corr…

Let $X_0^*(N)$ denote the quotient of the modular curve $X_0(N)$ by the full group of Atkin–Lehner involutions, and let $g_N^*$ be its genus. A rational point on $X_0^*(N)$ paramet…

Let $d$ be a prime. A quadratic $\mathbb{Q}$-curve of degree $d$ is an elliptic curve related to its Galois conjugate by an isogeny of degree $d$. Elkies's conjecture. For all suff…