324 problems

Goldfeld's conjecture. For every elliptic curve $E/\mathbb Q$, those $\chi\rein \Sigma_2$ for which

For every prime power $q$ and every primitive imaginary quadratic Dirichlet character $\chi$ of $\mathbb{F}_q(T)$, one has $L\left(\tfrac{1}{2},\chi\right)\neq 0$.

For a family of tempered cuspidal automorphic representations $\pi$ of $\mathrm{GL}_2(\mathbb{Q})$ with bounded archimedean parameters and conductor $q(\pi)$ tending to infinity, d…

Let $M$ be a motive with $L$-function $L(M,s)$ and $p$-adic Selmer group $\mathrm{Sel}_{p^\infty}(M)$. Suppose its functional equation is … with $s=\frac{1}{2}$ a critical value. B…

Hasse–Weil conjecture. The function $\zeta_X(s)$ extends to a meromorphic function of $\mathbf C$. There exists a positive real number $A\in\mathbf R^{>0}$, non-zero rational funct…

Let $\chi$ be a multiplicative character of order $c$ over a finite field, let $f(x)=x^d+\lambda x^e$ with $(d,e)=1$, and let $p$ be the characteristic. Write…

Birch and Swinnerton-Dyer conjecture. One has

Let $M$ be a $\mathbb{Q}$-motive with coefficients in a number field $F$ and odd weight. Let $H_B^+(M)$ be the $+1$-eigenspace of complex conjugation, let $\alpha_M^+$ be the compa…

Let $\chi$ be a Dirichlet character modulo $q greater than or equal to 1$, and let ${\mathcal N}(\frac{1}{2},\infty,\chi)$ denote the number of nontrivial zeros of the associated D…

Beilinson's conjecture. One has

Let $f$ be a holomorphic cusp form with associated quadratic-twist family $\mathcal{F}_f^{+}(X)$, and let $R_f(\alpha,\gamma)$ denote the average of the ratio of shifted $L$-functi…

Let $L(s)$ be an $L$-function with nontrivial zeros written as $\rho=\frac12+i\gamma$, let $\tilde{\gamma}=\frac{\log c(L)}{2\pi}\gamma$, and for an even Schwartz function $\phi$ o…

Let $C$ be a curve over a number field $K$ with an action by a finite group $G$, let $\tau$ be an irreducible representation of $G$, and let $C^\tau$ be the corresponding motivic p…

Let $A$ be an abelian variety of dimension $g$ defined over $\mathbb{Q}$. For each prime $p$ of good reduction, write the normalized Euler factor as … Let…

Affine-linearity conjecture. Then

Let $f(\tau)=\sum_{n=1}^{\infty}a_f(n)q^n\in S_\kappa(N,\omega)$ be a normalized elliptic newform of weight $\kappa\geq 2$, and let $\chi$ be a Dirichlet character. For a critical…

Automorphic Goldfeld conjecture. Among quadratic characters $\chi'$ satisfying

Let $L/k$ be a finite algebraic extension of number fields, and let $\zeta_L(s)$ and $\zeta_k(s)$ denote their Dedekind zeta functions. Dedekind's conjecture. The quotient … is ent…

Brumer's conjecture.

Global Gan–Gross–Prasad conjecture. Let $\pi$ and $\pi^\prime$ be irreducible tempered cuspidal automorphic representations of $\operatorname{GSpin}(W)(\mathbb{A})$ and…

Gan–Gross–Prasad conjecture. If $n-m$ is odd, a nonzero Bessel period $\mathcal{B}_{k,\psi}^{\varepsilon}$ on $\pi_1 \boxtimes \pi_2$ implies that $(M,N)$ is a relevant pair. For a…

Let $\hat{\Delta}_{\mathrm{red}}=(\hat{L},\hat{H}',1,\hat{\rho}_{\hat{H}',\hat{\iota}'})$, where … Let $\pi_L$ be a tempered automorphic representation of $L(\mathbb A)$ and let…

Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathrm{GL}_n$ over $\mathbb Q$, with associated $L$-function … Let…

Assume the setup in which $G$ is simply connected and almost simple, $S$ is a set of places with at least two elements, and $T=\emptyset$. For each positive integer $m$, let…

Let $\pi=\bigotimes'\pi_v$ be an irreducible cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_\mathbb Q)$. At an unramified finite prime $p$, write the Satake parame…