8 problems

Let $\mathcal{H}_N(\tau)$ be the integrated correlator under consideration at finite $N$, let $B_N(t)$ be its kernel, and let $\Gamma_{2,2}(\tau;t)$ be the genus-two Narain lattice…

Non-isotropic $(2,2)$ meromorphicity conjecture. The image of $\frac{i^{m}}{2}\Phi_{L,m,m,0}$ under $\frac{1}{-4\pi^{2}}R_{m}^{(2)}$, namely…

$S_{m}^{(b_{-})}$ operator conjecture. For any dimension $b_{-}$ and any weight $m$, there exists an operator $S_{m}^{(b_{-})}$ satisfying properties (i)--(vii) stated in the sourc…

Non-isotropic $(2,2)$ meromorphicity conjecture. Applying $R_{m}^{(2)}$ to the theta lift yields a meromorphic Hilbert modular form also when $L$ is non-isotropic.

Odd-dimensional operator conjecture. Appropriate operators should exist for other odd values of $b_{-}$ as well.

Vanishing conjecture. If $\xi(f)=0$, then

Factorization conjecture. The arithmetic theta lift factors through $\xi$, so that there is a commutative diagram

Let $F$ be a totally real number field, let $\boldsymbol{c}$ be an odd square-free level, let $g\in S_{\mathbf{2}}(\boldsymbol{c})$ be a newform, and let $v_g$ be a Hecke eigenvect…