34 problems

Let $C$ be a smooth projective curve of genus $g$. Assume that $C$ is in the generic family. For each $k=1,2,\dots,g$, let $a_k(m;g)$ denote the $m$-th moment of the corresponding…

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 $k$ 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 $A$ be a nonsingular projective $g$-dimensional variety over $\mathbb C$. For each prime $\ell$, let $G_{A,\ell}$ be the Zariski closure of the image of the $\ell$-adic represe…

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 $M$ 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…

Under the algebraic Sato–Tate setup, let $\operatorname{ST}_K(V,\psi)$ be the associated compact Sato–Tate group. For a prime $v$ of $K$, let $s_v$ be the semisimple part of the no…

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…

Let $A$ be an abelian variety with rational symplectic representation $(V,\psi)$, and let $K$ be a field over which $A$ is defined. Write…

Let $A$ be an abelian variety over a number field $F$, let $K/F$ be the extension appearing in the definition of the Chebotarev-Sato-Tate group, and let…

Algebraic Sato–Tate conjecture. There is an algebraic subgroup $\operatorname{AST}(A)$ of $\operatorname{GSp}_{2g}$ over $\mathbb Q$, called the algebraic Sato–Tate group, such tha…

Let $X$ be as above, let $i$ be as above, and let $S$ be the finite set of bad primes. For each good prime $p\in\mathbb P\setminus S$, let…

Let $E$ be an elliptic curve for which the Sato–Tate conjecture holds. If $\theta_p\in[0,\pi]$ are the Sato–Tate angles and $\rho(\theta)$ is their density, define the unfolded var…

Sato–Tate conjecture. The angles $\varphi_p$ are equidistributed in $[0,\pi]$ with respect to the Sato–Tate density measure

Let $A$ be a generic abelian $g$-fold over $\mathbb{Q}$, meaning that the image of its adelic Galois representation is open in $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. Let…

Let $C_{2p}$ be the genus $g$ curve … where $p$ is prime. For the characteristic polynomials $P_{d,j}(T)$ associated with the component representatives indexed by $d$ and $j$, the…

Let $C_p$ be the genus $g$ curve … where $p=2g+1$ is prime. For the characteristic polynomial $P_{\gamma^d}(T)$ associated with the indicated component representative, the characte…

Let $C_1$ be the hyperelliptic curve … where $g=2k$ is its genus and $c$ is the constant appearing in the defining equation. Let $\operatorname{ST}^0(C_1)$ denote the identity comp…

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…

Let $\mathcal{S}$ be a finite set of entire analytic $L$-functions that is stable under conjugation, so that $\overline{\mathcal{S}}=\mathcal{S}$. Let $(a_f)_{f\in\mathcal{S}}$ be…

Let $A$ be an abelian variety of dimension $g$ over a number field, and let $G_\ell^{1,\rm zar}\subseteq \operatorname{Sp}_{2g}$ be the Zariski closure of the determinant-one part…

Let $A$ be an abelian variety over a number field $K$, and let $\operatorname{ST}(A)$ be its Sato--Tate group. For each prime $\mathfrak{p}$ of good reduction, let…

Let $A$ be an abelian variety of dimension $g\geq 1$ over a number field $K$. For each prime ideal $\mathfrak{q}$ of good reduction, let $\overline{P}_{\mathfrak{q}}(T)$ be the ren…