166 problems

For every elliptic curve $E/\mathbb{Q}$ with complex multiplication and $\operatorname{rank}E(\mathbb{Q})=2$, and for every good ordinary prime $p$ of $E$, is…

Let $f$ be the modular form under consideration, let $G_ fty$ be the relevant Galois group, and let $\mathcal{O}$ be the coefficient ring. For $\natural\in\{\sharp,\flat\}$ and…

Let ${\mathcal{K}}$ be a CM field with maximal totally real subfield ${\mathcal{F}}$, let $p$ be an odd prime unramified in ${\mathcal{F}}$ such that every prime above $p$ splits i…

Mazur–Tate–Teitelbaum's p-adic BSD conjecture. One should have

Let $K$ be an imaginary quadratic field satisfying the Heegner, discriminant, and splitting hypotheses, let $K_\infty$ be its anticyclotomic $Z_p$-extension, let $\Lambda$ be the a…

Let $V\cong\mathcal K^{2d}$ be an even-dimensional $p$-adic representation of $G_{\mathbb Q}$ satisfying condition $\mathrm{(Geom)}$, with no Artin representation as a subquotient.…

Let $F$ be a number field, let $H/F$ be a finite cyclic CM extension, let $p$ be a prime, and let $R$, $R_0$, and $R_1$ be the subsets of primes of $F$ above $p$ that split, ramify…

$p$-adic Birch–Swinnerton-Dyer conjecture.

Greenberg–Benois conjecture. If $\mathcal{L}_p(S,V)$ has $e_0+t$ trivial zeros, then $S^{e_0+t+r}$ divides it exactly and

Let $V\in {\mathscr{V}}^{\circ}$ be an incoherent pair, and let $\pi$ be a distinguished, stable, ordinary, cuspidal automorphic representation of $\mathrm{G}^{V}(\mathbf{A})$, tri…

Let $L/K$ be as above, and let $S$ and $T$ be finite non-empty disjoint sets of places of $K$, with $S$ containing all places ramifying in $L^+_\infty/K$ and all infinite places. W…

Throughout, let $f_\alpha$ be a $\theta$-critical normalized eigenform, and let $L_{\mathrm{S},\alpha}^{[0]}(f,\xi)$ and $L_{\mathrm{K},\alpha}(f,\xi)$ denote the corresponding cri…

Let $H$ be the number field occurring in the Artin representation $\rho$, let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, and let…

Fix an odd prime $p$, a field isomorphism $\iota:\mathbb{C}\simeq\mathbb{C}_p$, a character $\psi\in\operatorname{Irr}_{\mathbb{C}_p}(\mathcal{G})$, and a finite set of places $S$…

Let $E/\mathbb Q$ be the number field and $\bar G$ its relevant Galois group, let $E_0=\{x\in E:\operatorname{Tr}_{E/\mathbb Q}(x)=0\}$, and let $\mu_\infty$ and $\mu_p$ be the arc…

The main conjecture. The p-adic $L$-function ${\cal L}_E\in K_1(\Lambda(G)_{S^*})$ is a characteristic element of $X(E/F_{\infty})$.

The non-CM p-adic L-function conjecture. There exists ${\cal L}_E\in K_1(\Lambda(G)_{S^*})$ such that ${\cal L}_E(\rho)\ne\infty$ for every Artin representation $\rho$ of $G$, and

Two-variable $p$-adic Birch–Swinnerton-Dyer conjecture. The element $\mathbf{L}$ lies in $\mathbf{I}^r$ and satisfies

Mazur–Swinnerton-Dyer's conjecture. There is an equality of ideals in $\Lambda$

Let $F$ be a totally real number field, let $K$ be a quadratic imaginary extension of $F$, let $F'$ be a finite totally real extension of $F$, and let $f_{F'}$ be a Hilbert modular…

Let $U$ be the affinoid neighbourhood of a finite-slope classical point $x_0$ on the Rankin–Selberg eigenvariety, with $U$ smooth, the weight map finite étale, and classical cuspid…

Let $\underline{\Pi}$ be a family on $\operatorname{GSp}_4$ with root number $-1$ and no endoscopic specialisations, let $\underline{\sigma}$ be the accompanying $\mathrm{GL}_2$ fa…

Let $\underline{\Pi}$ be a family on $\operatorname{GSp}_4$ whose specialisations have root number $-1$, let $\underline{\sigma}$ be the accompanying $\mathrm{GL}_2$ family, and le…

Let $K/F$ be a $p$-ordinary CM quadratic extension for an odd prime $p$, and let $h_K^-$ be its relative class number. Fix embeddings…

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…