12 problems

Let $\mathcal F$ be a family in $\mathcal D$, meaning a class of $L$-functions in $\mathcal D$ whose thickened conductor tends to infinity. For $L(s)\in\mathcal D_m$ with archimede…

Let $\mathcal{C}$ be the family locus under consideration, let $\kappa_{{\mathbf{g}},{\mathbf h}}\vert_{\mathcal{C}}$ be the restriction of the Beilinson–Flach cohomology class, an…

Let $L_p^{(c)}(\pi,s)=\mathscr{L}_p^{(c)}(\pi,\langle-\rangle^s)$ for $s\in\mathbb{Z}_p$, where $\langle-\rangle:\mathbb{Z}_p^{\times}\to1+2p\mathbb{Z}_p$ is the natural map. Let…

Let $k(p^\infty)$ be the compositum of all $\mathbb Z_p$-extensions of $k$, let $\Gamma_\infty:=\operatorname{Gal}(k(p^\infty)/k)$, and let $\mathcal A_{\Gamma_\infty}$ be the augm…

Let $k_{\Gamma}/k$ be a $\mathbb{Z}_p$-extension with Galois group $\Gamma$, let $\Gamma_n:=\operatorname{Gal}(k_n/k)$, and put $L_\Gamma:=k_\Gamma L$. Let…

Exceptional-zero order conjecture. The constant term satisfies $g_{\chi,\varepsilon}(0)\ne0$. Thus the zero at the trivial character has order exactly $t$. The preceding argument g…

Exceptional zero formula. If $L(A/\mathbf{Q},1)\ne0$, then

Let $A/\mathbf{Q}$ be an elliptic curve with split multiplicative reduction at $p$. Let $L_p(A/\mathbf{Q},s)$ be its $p$-adic $L$-function, let $L(A/\mathbf{Q},s)$ be its Hasse–Wei…

Let $\mathbf{f}$ be the Hilbert modular form in the paper, let $F$ be its totally real coefficient field, and let $m$ be the parameter for which the trivial zero occurs at $s=m+1$.…

Let $G(T,V)$ and $\mathcal{E}(V)$ be as above, and let $g$ be the number of vanishing factors of $\mathcal{E}(V)$. Let $\mathcal{E}^*(V)$ denote the product of the nonzero factors,…

Let $\theta_F$ be the character associated with the setting of the paper, let $L_p^{\operatorname{Gross}}$ be Gross's $p$-adic $L$-function, let $R_p(\theta_F)$ be Gross's $p$-adic…

Let $M$ be a $p$-ordinary motive, let $L_p(s,M,\chi)$ be its analytic $p$-adic $L$-function, and let $a_0$ be a critical integer with character $\chi_0$ such that the interpolation…