622 problems

Let $X$ be a smooth projective variety defined over a finitely generated subfield $F\subset\mathbb{C}$. Under the Artin comparison identification … let…

Let $f\in S_k(\Gamma_1(N))$ be a newform, let $\mathfrak{p}$ be a prime such that the associated mod $\mathfrak{p}$ representation $\rho_{f,\mathfrak{p}}$ is irreducible, and let…

Let $G_{\mathbb{Q}}$ denote the absolute Galois group of $\mathbb{Q}$, and let $\rho:G_{\mathbb{Q}}\rightarrow \mathrm{GL}_2(\overline{\mathbb{Q}}_p)$ be an irreducible $p$-adic Ga…

Let $S_k(\mathrm{SL}_2(\mathbb{Z}))$ be the space of cusp forms of weight $k$ and level $1$, and let the Hecke polynomial denote the characteristic polynomial of a Hecke operator o…

Let $L/\mathbb{Q}_p$ be a finite extension, let $E$ be a coefficient field, let $W_L$ be the set of Serre weights, and let $\overline{\rho}_L:G_L\to\mathrm{GL}_2(\mathbb{F})$ be a…

Let $k$ be a finite field of characteristic $\ell\ne p$, let $X$ be a geometrically connected smooth curve over $k$, and let … be the exact sequence for the arithmetic fundamental…

Let $K$ be a number field, let $g$ be a positive integer, and let $\ell$ be a rational prime. For an abelian variety $A$ over $K$, write $[A]$ for its $K$-isomorphism class, and de…

Frey–Mazur conjecture. For all sufficiently large primes $p$, every $p$-congruence of elliptic curves comes from an isogeny.

Let $A$ be an abelian variety of $GL(2,E)$-type over a number field $K$, with conductor $\mathfrak{n}$, and let $\rho_{A,\lambda}$ denote its Galois representation for a prime…

Let $E$ be the quadratic field under consideration, let $h$ be the Hecke algebra acting on the relevant cohomology, let $\mathfrak{m}\subset h$ be a non-Eisenstein maximal ideal, a…

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$. For a prime $p$, let … be the residual representation. Serre's uniformity question asks whether there is a constant $N$, depen…

Algebraic Sato–Tate conjecture. The groups $\mathcal{G}_\ell$ for varying $\ell$ are all interpolated by a single algebraic group defined over $\mathbb{Q}$, called the algebraic Sa…

Let $F$ be a totally real field, let $p$ be unramified in $F$, and let $\rho:\operatorname{Gal}(\overline{F}/F)\rightarrow\operatorname{GL}_2(\overline{\mathbb{F}}_p)$ be a continu…

Let $K_infty/K$ be a $\mathbb{Z}_p$-extension, let $\mathbf{A}(rho)$ be the $p$-divisible module attached to a $p$-adic Galois representation, and let $Y(\mathbf{A}(rho)/K_infty)$…

Let $(R,T,\zeta)$ be a triple as in the paper's setup, where $T$ is an $R$-representation of $G_{\mathbb Q_p}$ and $\zeta$ is a $\mathbb Z_p$-basis of…

Buzzard–Gee conjecture. Galois representations attached to $C$-algebraic automorphic forms should take values in a $C$-group.

Serre's uniformity question. Does there exist an absolute constant $c$ such that, for every elliptic curve $E/\mathbb{Q}$,

Let $F$ be a totally real field, let $p$ be a prime, and let $\rho$ be a mod $p$ Galois representation. For a regular weight $(k,l)$, geometric modularity means that $\rho$ is modu…

Mazur–Rubin's conjecture. If $E_1$ and $E_2$ are $p$-Selmer near-companions over $K$, then there exists a $G_K$-module isomorphism

Let $S$ be a finite set of places, and let $\rho$ be an irreducible $\ell$-adic Galois representation of $\mathbb Q$ that is unramified outside $S$ and de Rham locally at $\ell$. F…

Let $n$ be a positive integer, let $p$ be a prime, and let $\Gamma\subset\operatorname{SL}(n,\mathbb{Z})$ be a congruence subgroup. A Hecke eigenclass is an element…

Let $p$ be an odd prime, let $k$ be even, and let $a_p$ be the relevant crystalline Frobenius trace. Write $\overline V_{k,a_p}$ for the reduction modulo $p$ of the associated two-…

Langlands' automorphy conjecture. Operations such as induction on the Galois side should correspond to automorphic forms; in this case, the theta series is the corresponding form.

Let $K/\mathbb Q$ be a finite Galois extension with Galois group $G$, and let $\chi$ be a non-trivial irreducible character of $\operatorname{Gal}(K/\mathbb Q)$. Write $\chi(1)$ fo…

Let $K$ be a totally real field and $L$ a number field. Let $A/L$ be an abelian variety of $\mathrm{GL}_2$-type such that … For a prime $\mathfrak{p}$ of $K$ above a rational prime…