7 problems

Serre's modularity conjecture. If $\overline{\rho}$ is continuous, irreducible, and odd, then there exist a weight $k$, a level $N$, and a modular eigenform $f\in S_k(\Gamma_1(N))$…

Let $\mathfrak p\mid p$ be a place of $F$, with ramification index $e$, residue degree $s$, and inertia group $I_{\mathfrak p}$. Let $W_{\mathfrak p}^?(\rho)$ denote the predicted…

Let $(G,X)$ be a Shimura datum of Hodge type. Let $\mathbb{T}$ be the spherical Hecke algebra and let $\mathfrak{m}\subseteq\mathbb{T}$ be a maximal ideal. For an algebraic represe…

Serre's strong modularity conjecture. Any odd, irreducible representation $\rho$ as above, with $l\geq5$, is strongly modular.

Let $F$ be a totally real field, let $p$ be a prime, and let $G_F$ denote its absolute Galois group. A representation is totally odd if complex conjugation at every real place has…

A modular Galois representation is a continuous two-dimensional representation of the absolute Galois group of a number field that arises from a modular form. A representation is u…

Let $n>1$, let $p$ be prime, and let … be an irreducible odd Galois representation. Let $N^?(\rho)$ denote the prime-to-$p$ analogue of Serre's level. Ash–Doud–Pollack–Sinnott conj…