60 problems

Let $X$ be an elliptic curve, and for each finite place $v$ of good reduction let $a_X(v)$ be defined by $|X(k_v)|=p_v^{f_v}+1-a_X(v)$, where $p_v^{f_v}=|k_v|$. Sato–Tate conjectur…

Let $C$ be a smooth projective curve over a number field $k$. Let $X_k$ be the set of conjugacy classes of $\operatorname{ST}(\operatorname{Jac}(C)_k)$, and let $\{p_i\}_{i\geq 1}$…

Let $L$ be an automorphic L-function with associated representation $\rho$ of the conjectural Langlands group, and let its Sato–Tate group be the compact subgroup…

Let $M$ be a motive and let $\operatorname{MS}_{\sim}(M)$ be its motivic Serre group. Motivic Sato–Tate conjecture. For any prime number $l$, … The paper obtains this formulation a…

Generalized Sato–Tate conjecture. Curves of fixed genus $g$ are classified into certain families, and the quantities $a_k(m;g)$ are the same for all curves in each family. Moreover…

Let $A$ be a nonsingular projective $g$-dimensional variety over $\mathbb C$, and suppose its Sato–Tate group $\operatorname{ST}(A)$ is defined as a maximal compact Lie subgroup of…

Let $(V,\psi)$ be a rational polarized Hodge structure and let $(V_l,\rho_l)$ be a family of $l$-adic representations satisfying conditions (D1), (D2), and (R1)–(R4). For every fin…

Birch's conjecture. As $p\to\infty$, the values $\operatorname{Bi}_p(a)$ become equidistributed according to the Sato–Tate measure as $a$ ranges over $\mathbb{F}_p^{\times}$.

Let $A$ be an abelian surface over a number field $k$, and let $\operatorname{ST}_{A}\subset\operatorname{USp}(4)$ be its Sato–Tate group. For each prime $\mathfrak{p}$ of good red…

For $i=1,2$, let $g_i=\sum_{n=1}^{\infty}b_i(n)q^n$ be primitive eigenforms of weight $2k_i$ and level $2N_i$, and define … Let $g_1,g_2$ be distinct non-CM primitive forms of weig…

For a prime $p$, let ${\mathbb F}_p$ be the finite field with $p$ elements. For $a\in{\mathbb F}_p^*$, define the Kloosterman sum … where ${\mathbf e}_p(z)=\exp(2\pi i z/p)$, and d…

Sato–Tate conjecture. For every such $E$ and every $0cle ctheta_1cle ctheta_2cle cpi$,

Let $p_k$ denote the $k$-th prime, let $\lambda_j$ denote a zero parameter, and let $\theta$ be defined by $2\theta=\lambda_j\log p_k$. For $0\leq\alpha\leq\theta\leq\beta\leq\pi$,…

For a finite field $\mathbb{F}_{q}$, let $\mathcal{C}$ be the multiplicative group $\mathbb{F}_{q}^{*}$ or the group of norm-one elements in a quadratic extension, and let…

Let $b_p^{(n)}$ be the Hecke eigenvalue of the Maass cusp form $\nu_n$ at a prime $p$, let $b_1^{(n)}$ be its first coefficient, and let $R_n$ denote the Laplace-eigenvalue paramet…

Let $\nu_n$ be Maass cusp forms with Laplace eigenvalues parametrized by $R_n$, and let $b_p^{(n)}$ denote their Hecke eigenvalues at a prime $p$. Horizontal Sato–Tate conjecture.…

Let $\phi_j$ be a family of Maass cusp forms with Langlands parameters $\alpha^{(j)}$, Hecke coefficients $A_j(M)$, normalizing factors $\mathcal L_j$, and let…

Let $k$ be an integer with $k\geq 3$, and let $D=p_1\cdots p_\ell$ be an odd discriminant. Let $S_k^{\mathrm{Ell}}(\Gamma_1(D),\omega_{E/\mathbb{Q}})$ denote the space of elliptic…

Let $K/K'$ be a finite extension, let $C$ be a conjugacy class of the relevant Galois group, and let $I$ be an interval for the Sato–Tate angle. The known statement in the solvable…

Let $F$ be a totally real number field, let $\pi=\bigotimes_v\pi_v\in\mathrm{RA}_{F}\cap\mathfrak{F}_{2}^{*}$, and let $S_{\pi}$ be the finite set of non-archimedean places where…

Let $N_{t,p,g}'$ be the number of genus $g$ curves over $\mathbb{F}_p$ with Frobenius trace $t$. Higher-genus Sato–Tate conjecture. For $g\ge 3$, each $N_{t,p,g}'$ converges to the…

Let $L$ be a fixed non-self-dual L-function of degree $3$, let $a_p$ be its coefficient at a prime $p$, and let $T$ be the curvilinear triangle of traces of elements of…

Let $\mathcal F_{N,k}$ be the family of forms under consideration, let $R_2(g,\rho)(f)$ denote its pair correlation function, and let $x$ tend to infinity. Write $\mathbb E$ for av…

Let $A$ be an abelian variety over $K$, let $\operatorname{G}_{\ell,K,1}^{\operatorname{alg}}$ be its normalized algebraic $\ell$-adic monodromy group, and let…

Let $A$ be an abelian variety over $K$, let $\operatorname{G}_{\ell,K,1}^{\operatorname{alg}}$ be its normalized algebraic $\ell$-adic monodromy group, and let…