26 problems

Nonvanishing conjecture. The first Fourier coefficient satisfies

Let $\Gamma_n=\operatorname{Sp}_n(\mathbf Z)$, let $T$ be a positive-definite index for the Siegel Poincaré series $P_{T,\Gamma_n}$, and assume the usual parity condition for scala…

Let $(X,\Delta)$ be a klt pair on a normal projective variety $X$ such that $K_X+\Delta$ is pseudo-effective. Let $L$ be a nef $\mathbb{Q}$-divisor on $X$. Generalized Nonvanishing…

Fix a prime $p$ and an elliptic curve $E/\mathbb{Q}$ of conductor $N$, with $L$-function coefficients $a_E(n)$, and assume that $p$ is prime to the Manin constant $c_E$ and that…

Let $X$ be a projective manifold, let $q$ satisfy $1 e q$ and $q e \dim X$ only if excluded?

Chowla–Chowla conjecture. $L(2,f)\ne0$, except when

Let $N$ be a positive integer, and let $f$ be an arithmetic function with period $N$ such that $f(n)\in\{-1,1\}$ for $n=1,2,\dots,N-1$ and $f(n)=0$ whenever $n\equiv 0\pmod{N}$. Er…

Let $X$ be a smooth projective variety of dimension $n$ over $\a0\mathbb{C}$, and let $K_X$ be its canonical divisor. A nef divisor is 1-big when $L^{n-1}\cdot H>0$ for some ample…

Let $X$ be a smooth projective variety of dimension $n$ over $\a0\mathbb{C}$, and let $K_X$ be its canonical divisor. Ambro's conjecture. If $D$ is an ample divisor on $X$ and…

Brent's cusp-coefficient nonvanishing conjecture. There are corresponding polynomials $T_n\in\mathbb{Q}[x]$ satisfying $\tau_m(n)=T_n(m)$ for $m=3,4,\ldots$, together with the fact…

Brent's nonvanishing conjecture. For each fixed integer $n\geq -1$ and every integer $m\geq 3$, one has

Carnevale–Voll conjecture.

Assume that $\mathbb{I}=\mathfrak{O}_L[[T]]$, and for each $k\in\mathbf{Z}_{\geqslant 2}$ let $\boldsymbol{f}_k$ be the $p$-stabilized newform obtained by setting $T=(1+p)^{k-2}-1$…

Let $\tau(n)$ be Ramanujan's tau function, the Fourier coefficients of the discriminant modular form $\Delta$, and let $p$ range over primes. Lehmer's conjecture. If $n\geq 1$, the…

Nonvanishing conjecture. For every prime $q$ satisfying $q\equiv3\pmod 8$,

Weak Generalised Nonvanishing Conjecture. Under these hypotheses, the numerical class of $K_X+\Delta+tL$ belongs to the effective cone for every $t\geq0$.

Generalised Nonvanishing Conjecture. The numerical class of $K_X+\Delta+tL$ belongs to the effective cone for every $t\geq0$.

Let $\Pi$ and $\Pi'$ be the automorphic representations specified in the surrounding hypotheses, and let $\Pi_\chi$ and $\Pi_{\chi'}$ be the associated twists by conjugate self-dua…

Let $(X,L)$ be a projective variety of dimension $n$ with at most terminal $\mathbb{Q}$-factorial singularities, where $L$ is a nef and big Cartier divisor; such a pair $(X,L)$ is…

Let $X$ be a complex projective manifold, and let $K_X$ denote its canonical bundle. A line bundle is pseudoeffective when its numerical class lies in the closure of the cone of ef…

For every odd integer $k\geq 3$, define the element $(B_{p-k}\pmod p)_p\in\mathcal{A}_{\mathbb{Q}}$, where $B_m$ denotes the $m$-th Bernoulli number. Bernoulli finite zeta nonvanis…

Let $F \in S_k^{(2)}(N)$ be a Hecke eigenform, let $K$ be an imaginary quadratic field, and let $\Lambda$ be an ideal class group character of $K$. Suppose that the quantity…

Let $f$ be the eigenform, $K$ the quadratic field, and let $\mathfrak{X}$ denote the relevant set of characters. Write a ring class character as $\rho=\rho_0\rho_w$, of any given…

Let ${\bf{f}} \in \mathcal{S}_2(\mathfrak{N})$ be a cuspidal Hilbert modular eigenform, and let $K/F$ be a totally imaginary quadratic extension such that the root number of…

Let $\chi$ be a Dirichlet character and let $L(s,\chi)$ denote its Dirichlet $L$-function. The central-point nonvanishing conjecture. … for every Dirichlet character $\chi$. The so…