165 problems

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…

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 $naturalin{sharp,flat}$ 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…

Iwasawa–Greenberg main conjecture. The ${\mathbb{I}}_{\mathcal F,\mathcal G}[[\Gamma_{\mathrm{cyc}}]]$-module…

Let $E$ be the Anderson $t$-module under consideration over $L/K$, let $P$ be a prime of $A$, and let $\Omega_E$ be the period lattices associated with $E$. Write $r_{\Omega_E}$ fo…

Let $E$ be the Anderson $t$-module under consideration, let $P$ vary over primes of $A$, and define the order of vanishing of the $P$-adic $L$-series at $z=1$ by … to be the greate…

Let $E$ be an Anderson $t$-module, let $P$ be a prime of $A$, and define … and … Also write $\mathscr{U}(E;\mathscr{O}_L)=\exp_E(U(E;\mathscr{O}_L))$. Leopoldt conjecture. The…

Injectivity conjecture. The $P$-adic $L$-series is non-zero if and only if the exponential map $\exp_E:L_\infty^d\rightarrow L_\infty^d$ is injective.

The p-adic Artin formalism conjecture. In full generalities, there should be a factorization

Let $E/\mathbb{Q}$ have supersingular reduction at $p$, let $L_p^\pm(E,T)$ be Pollack's signed $p$-adic $L$-functions, and let $f_E^\pm(T)$ be the characteristic series of the corr…