19 problems

Let $\mathcal{B}_N$ be the family of Maass cusp forms of level $N$, let $h_T$ be the weight function defined in the source, let $\phi$ be an even Schwartz function with compactly s…

Let $q$ be a modulus, let $f$ be the test function used in the 1-level density, and let $D_{1;q}(\widehat{f})$ denote the corresponding density with scaling parameter $Q=q$. Write…

Consider a reasonable family of $L$-functions for which all functional equations have even sign and for which there is no corresponding family with odd functional equations. Folklo…

Katz–Sarnak conjecture. For every integer $j\geq 1$ and every compactly supported complex-valued function $h$ on ${\mathbf R}$,

Consider centered moments of one-level densities of zeros of Dirichlet $L$-functions, with identical test functions $\Phi_i=\Phi$ and initially restricted Fourier-transform support…

Lower-order universality conjecture. As long as the prime factors of $N$ tend to infinity sufficiently fast relative to the reciprocal of the error scale being considered, the lowe…

Let $\{L(s,f)\}_{f\in\mathcal{F}}$ be a family of $L$-functions with symmetry type $G=G(\mathcal{F})$, where…

Let $N_{\operatorname{eff}}$ be the effective matrix size chosen to match the leading lower-order term in the one-level density for the relevant quadratic-twist family. In the gene…

Low-lying zeros conjecture. For every $n\in\mathbb{Z}_{\geq1}$,

Let a homogeneous orthogonal family of $L$-functions be given, and consider the normalized low-lying zeros of the family. Sarnak–Shin–Templier conjecture. The low-lying zeros of a…

Central-value weighting conjecture. The density $W_w(\mathcal F)$ should differ from $W(\mathcal F)$ only when $w_\Pi$ essentially contains the central value $L(1/2,\Pi)$.

Let $\rho$ be an $L$-homomorphism, and let $\mathcal{P}$ be the characteristic function on $\hat{\mathbb{X}}_{\mathrm{gen}}$ of the representations $\pi$ for which…

Let $\rho$ be an $L$-homomorphism and let $C_{\rho,X}$ denote the average analytic conductor associated with the family indexed by representations with analytic conductor bounded b…

Let $\frak{F}$ be a family of $L$-functions, let $\frak{F}(X)=\{\pi\in\frak{F}:X<c_\pi<2X\}$, and let $D^{(n)}(\pi,\phi)$ denote the $n$-level statistic formed from the normalized…

Bias Conjecture. For any such family, the largest lower-order term in the second moment that does not average to $0$ is on average negative; equivalently, the first $\beta_r(p)$ te…

Folklore symmetry-sign conjecture. The symmetry of the family is determined by the sign of its functional equations.

Density conjecture. As $m$ tends to infinity, the family average converges to the unitary density:

Let $\mathfrak{F}$ be a rank $0$ essentially homogeneous family. Write the zeros of each completed $L$-function as $\frac12+i\gamma_j^{(\pi)}$ and normalize them by…

Let $f$ be the test function in the average 1-level density, and let $sigma$ denote the finite support parameter. Theorem 3/2(3) gives an expansion involving the terms of order…