34 problems

Nonvanishing conjecture. The first Fourier coefficient satisfies

A smooth projective variety is a variety that is smooth and projective. A line bundle is nef if its degree on every curve is nonnegative, and it is adjoint to an ample line bundle…

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…

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

Let $(X,Delta)$ be a minimal compact Kähler klt pair, and let $K_X+Delta$ be its adjoint $\mathbb{Q}$-line bundle. Nonvanishing Conjecture. The Kodaira dimension satisfies … This i…

Let $S_{2\ell}^{0,D}(1)$ denote the subspace of cusp forms associated with the discriminant $D$ and satisfying the relevant nonvanishing condition, and let $S_{2\ell}(1)$ denote th…

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…

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…

A primitive Dirichlet character is a Dirichlet character that is not induced by a character of smaller conductor. For such a character, let $L(s,\chi)$ denote its Dirichlet $L$-fun…

Let $L(s,\chi)$ be a Dirichlet $L$-function, with $\chi$ a Dirichlet character. Central nonvanishing conjecture. The central value … is never zero. This is a classical conjecture a…

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 $\mathbb{K}$ be an imaginary-quadratic number field, and let $G^{(\mathbb{K})}_k$ denote the Hermitian Eisenstein series appearing above. Nonvanishing conjecture. For every eve…

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…

Let $\chi$ be a Dirichlet character, and let $L(s,\chi)$ denote its Dirichlet $L$-function. Dirichlet central-value nonvanishing conjecture. One should have … for all Dirichlet $L$…

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.

Let $\lambda_1$ and $\lambda_2$ be two distinct integers. Carnevale–Voll conjecture. … This conjecture concerns the absence of unitary factors in polynomials arising in the study o…

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 $\chi$ be a primitive Dirichlet character of prime modulus. Its Dirichlet $L$-function is denoted by $L(s,\chi)$. Central nonvanishing conjecture. The central value should neve…

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

Let $\chi$ be a primitive Dirichlet character. Folklore nonvanishing conjecture. … This folklore conjecture extends Chowla's assertion from primitive quadratic characters to all pr…