11 problems

Let $\phi$ be a suitable test function, and let Theorem denote the density theorem established in the paper for the family of elliptic curves. Unrestricted-support conjecture. Theor…

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…

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

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…

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…

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 $\mathcal{B}_N$ 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 $f$ 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…

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…