8 problems

Effective non-vanishing conjecture. One has

Chai's conjecture. A complex number $s=s_0$ is an exceptional pole of type $1$ with level $m$ for the pair $(\pi_1,\pi_2)$ if and only if it is an exceptional pole of type $2$ with…

Conjecture on archimedean zeta integrals. There exists a number field $E(\varepsilon^{(n+1)}_v,\varepsilon^{(n)}_v)$ such that, for every $m\in\mathrm{Crit}(\pi^{(n+1)},\pi^{(n)})$…

Let $F$ be a totally real number field, let $n'=n$ or $n'=n-1$, and let $\pi=\otimes'_v\pi_v$ and $\pi'=\otimes'_v\pi'_v$ be irreducible cuspidal automorphic representations with t…

Let $F$ be a totally real number field, let $\psi$ be a non-trivial unitary character of $\mathbb{A}/F$, and fix a totally positive $\xi\in F^\times$. Let $\pi$ be an irreducible r…

Let $K$ be the quadratic imaginary field under consideration. Let $\Pi$ and $\Pi'$ be regular algebraic cuspidal representations of $GL_n(\mathbb{A}_K)$ and $GL_{n'}(\mathbb{A}_K)$…

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…

Yuan–Zhang–Zhang-type conjecture. There exists $b\in\widehat{B}^{\times}$ with $k(b,\omega_{\varphi})\neq0$ such that