322 problems

Theta-lift nonvanishing conjecture. If $L(f,1)=0$, then

Rational Stark conjecture. There exists an element $u_\chi\in(k\otimes_{\mathbb{Z}}U_{v,S})^{\chi^{-1}}$ such that

Let $F$ be a quadratic extension of $\mathbb{Q}$ and let $K$ be a nontrivial finite abelian extension of $F$. If $F$ is real quadratic, assume that one infinite place of $F$ stays…

Generalized $\tau$-Li criterion. Non-vanishing of $F$ in the half-plane $\operatorname{Re}(s)>\tau/2$ is equivalent to growth of $\lambda_F(n,\tau)$ as $\mathcal{C}_F\tau n\log n$…

Let $p>2$ be a prime, let $E/{\mathbb Q}$ be an elliptic curve, and let $\Sigma_p$ denote the set of Dirichlet characters of order $p$. For $S\subset\Sigma_p$, say that $S$ has den…

Let $X$ be an affine spherical variety over $k$ with trivial arithmetic multiplicity. Let $\mathcal S(X(\mathbb A_k))$ be a Schwartz space in the sense described in the source, let…

Let $(L/K,S,T)$ be a basic triple, let $G=\operatorname{Gal}(L/K)$, and let $r\geq 0$ be an integer. Let $U_{S,T}$ be the group of $S$-units of $L$ that are congruent to $1$ at all…

Let $\overline{k}$ be a fixed algebraic closure of $k$, let $k^{\rm sep}$ be its separable closure, and let $G=\operatorname{Gal}(k^{\rm sep}/k)$. Let $\rho:G\to\mathbf C_\infty^*$…

Let $\Xi(\chi,t)=\Lambda(\chi,1/2+it)$ for an abelian character $\chi$ over a number field. The classical absolute-value conjecture. For every $e\geq0$, there are at most two zeroe…

Let $L(x,y)=\prod_i(1-\beta_i^{(y)}/x)$, where the nonzero $\beta_i^{(y)}$ are the zeroes of $L(x,y)$, and let $\lambda_i^{(y)}:=1/\beta_i^{(y)}$ be the reciprocal zeroes. The char…

Let $L(s)$, with $s=(x,y)\in S_\infty$, be a characteristic-$p$ $L$-series arising from arithmetic, and let $\mathbf K_L$ be the finite extension of $\mathbf K_{\mathbf V}$ generat…

Vanishing conjecture. $H_k^0 = 0$ for all positive integers $k$. Consequently, $M_k(T)$ is a polynomial for all positive integers $k$.

Let $p$ be a prime, let $f$ be the modular form in the construction, let $\widetilde{\mathscr{O}}$ and $\mathscr{O}'$ be the relevant orders, and write … Let…

Let $k$ and $n$ be natural numbers with $k-n$ even. Let $f\in S_{2k}(\Gamma_1)$ be a normalized Hecke eigenform, and let $F_{2n}\in S_{k+n}(\Gamma_{2n})$ be the Ikeda lift of $f$.…

Let $k$ be a global function field, let $w$ be any place of $k$, and let $N_w$ denote the non-classical set for the interpolation of $L(\chi,s)$ at $w$; when $w=\infty$, this inter…

Hecke–Fricke converse theorem. If the Dirichlet series satisfies the stated functional equation and Euler product, then it equals $L(s,f)$ for some

Let $N$ be a positive integer, let $k$ be a weight, and let $\chi_0$ be the trivial character modulo $N$. Let $\mathcal H=\{x+iy\in\mathbb C:y>0\}$, and let…

Let $E/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$, let $F_\infty/\mathbb{Q}$ be the false Tate curve extension, and let…

Let $E$ be an elliptic curve of conductor $Q$, let $q\nmid Q$ be prime, and let $S^{\pm}(X)$ be the relevant families of fundamental discriminants with sign $\pm$. For…

Let $M$ be a $K$-motive, let $\Delta_K(M)$ be its fundamental line, and let $\vartheta_\infty:K_{\mathbb R}\otimes_K\Delta_K(M)\cong\mathbf{1}_{K_{\mathbb R}}$ be the period-regula…

Deligne–Beilinson conjecture.

Let $\mathcal F$ be a family of $L$-functions, and for $F\in\mathcal F$ let $c(F)$ denote its conductor. Assume the family is ordered by conductor and has approximately $D$ members…

Let $\phi_i$ be a Hecke–Maass cusp form with spectral parameter $\lm_i$, and let $d_i$ denote its associated triple period coefficient. For every $\eps>0$, the notation…

Let $p$ be the prime and let $f$ be the weight-$2$ eigenform introduced in the construction above. Fix $\sigma\in\{+,-\}$, and let $c_f(d)$ be the coefficient of $q^d$ in … Let…

Let $\pi$ be a cuspidal automorphic representation of $GL(n)$ that is unramified at a prime $p$, and let $\alpha_{p,j}$ be the quantities appearing in its local Euler product. Gene…