14 problems

Mazur–Tate–Teitelbaum conjecture. One has $L_p(f,1)=0$ and

Let $E/F$ be a solvable extension, let $\mathfrak{q}$ be a prime of $E$ above $\mathfrak{p}$, and let $\pi_E$ be the base-change lift of $\pi$ to $PGL_{n,E}$. Let…

Let $\pi$ be the automorphic representation and let $c\in\{1,\dots,n-1\}$, $0\le r\le\ell$, and $\varepsilon\in\mathcal{E}_\pi$. Let $\mathcal{L}_c^{r,\varepsilon}(\pi)$ be the sub…

Bergdall–Pollack equidistribution conjecture. As $T$ tends to infinity, the set $Y_T\subseteq(-\infty,+\infty)$ is evenly distributed on $[0,1]$.

Distributional conjecture. For each sign $\pm$, the sets $\mathbf y_T^{\pm}(\overline{\rho})$ become equidistributed for Lebesgue measure on $[-1,0]$ as $T\to\infty$. This conjectu…

Let $\psi$ be the character in the source, let $L_p(\psi,s)$ be its $p$-adic $L$-function, let $L(\psi,0)$ be the corresponding complex $L$-value, and let $\mathscrbf{L}(\psi)$ be…

Let $\rho_{\pi,\mathfrak{p}}\colon\operatorname{Gal}(\overline{\mathbb{Q}}_p/F_{\mathfrak{p}})\to\widehat{G'}(\Omega)$ be the local Galois representation attached to $\pi$, and let…

Let $G$ be a reductive group, let $\a3$ be a sign character, let $d$ be an integer with $0\leq d\leq\delta$, and let $i\in\Delta$ be a root. The automorphic invariant…

Let $f$ be the Bianchi cuspidal eigenform and $V_p(f)$ its attached $p$-adic Galois representation. For each embedding $\sigma:F_{\mathfrak p}\hookrightarrow L$, let…

Let $F$ be a number field, let $b\mathfrak{p}d$ be a fixed prime above $p$, let $\pi$ be the automorphic representation and let $i$ and $s$ be the cohomological indices occurring i…

Let $E/F$ be the finite extension and let $S_p(cpi)$ denote the relevant set of primes above $p$. For a prime $\tilde{\mathfrak{p}}\in S_p(\pi)$, let $e_{\tilde{\mathfrak{p}}}$ and…

Fix a prime number $p$. Let $F$ be a totally real field, set $g=[F:{\mathbb Q}]$, and let ${\mathfrak p}$ be a prime ideal of $F$ above $p$. Let $f_\infty$ be a Hilbert eigen newfo…

Let $F$ be a totally real field, let ${\mathfrak p}$ be a prime of $F$ above $p$, and let $f_\infty$ be a Hilbert eigen newform of even weight whose level is divisible exactly by…

Let $V_p$ be the $p$-adic representation attached to the space of $p$-new modular forms, let $\mathbf{D}^{FM}=D_{st}(V_p)$ be its associated two-dimensional monodromy module, and l…