192 problems

Let $K$ be the imaginary quadratic field, let $(\tilde{f},\chi_\varepsilon)$ be a Heegner pair, and let $\Lambda$ be the relevant Iwasawa algebra. Let $\bT_\varepsilon$ and…

Let $p$ be an odd prime and $E$ an elliptic curve with good ordinary reduction at $p$. Let $… be the cyclotomic Iwasawa algebra, and let$L_p(E)\in\Lambda$be the$p$-adic$L…

Let $K$ be a number field, let $A/K$ be an abelian variety, and for each prime $p$ let $\selrk{p}(A/K)$ denote the $p$-Selmer rank. p-independence conjecture. The parity of…

$p$-parity conjecture. The $p^\infty$-Selmer rank $\operatorname{rk}_p E/K$ is even if and only if $w(E/K)=1$.

Let $E$ be an elliptic curve over a number field $K$. Let $w(E/K)\in\{\pm 1\}$ be the root number, defined as the conjectural sign in the functional equation for $L(E/K,s)$ under…

Let $E$ be the number field in the deformation problem, let $V$ be an absolutely irreducible crystalline $p$-adic representation with the stated self-duality and distinct local dat…

Vanishing conjecture for high tensor powers. One has $Sel^0_T(V_n)=0$ for $n\gg0$. This conjecture concerns the disappearance of globally unramified Selmer classes in sufficiently…

Let $p$ be an odd prime, let $F_\infty=\mathbb{Q}(\mu_{p^\infty},\sqrt[p^n]{m}:n\geq1)$, let $G=\operatorname{Gal}(F_\infty/\mathbb{Q})$, and let…

Let $F$ be a number field, let $F_{\infty}$ be its cyclotomic $\mathbf{Z}_{p}$-extension, let $V$ be a critical nearly ordinary Galois representation, and let $A=V/T$ be a torsion…

Let $\bar{\rho}:G_{\mathbf{Q}}\to\operatorname{GL}_{2}(k)$ be the residual representation, let $\mathcal H(\bar{\rho})$ be its Hida family, and let $Y$ be a branch of this family.…

The noncommutative Selmer torsion conjecture. The Pontryagin dual of the Selmer group,

Let $A$ be an abelian variety over a finite extension $F$ of $\mathbb Q$, let $p>2\dim(A)+1$ be prime, let $F_\infty=F(A_{p^\infty})$, and put…

Cyclotomic and anticyclotomic main conjectures. Both assertions hold:

Two-variable main conjecture. The two-variable $p$-adic $L$-function $\mathbf{L}$ generates the ideal

Selmer cotangent-space conjecture. The inclusion is an isomorphism:

Mazur's conjecture. The $\Lambda$-module ${\rm Sel}_{E}(K_\infty)[p^\infty]^{\lor}$ is a finitely generated torsion $\Lambda$-module.

Let $A$ be as in the cyclotomic Iwasawa setting, let $X(A/F_\mathrm{cyc})$ have a pseudo-isomorphism with elementary factors including…

Selmer criterion for plectic Heegner classes. If $\chi$ is primitive, then

Let $\Lambda^{\rm ac}$ be the anticyclotomic Iwasawa algebra, let $\mathbf{T}$ be the big Galois representation, and suppose ${\rm Sel}_{\rm Gr}(\mathbf{T})$ is torsion-free of…

Let $\boldsymbol{\delta}=\{\delta_n\}_{n\in\mathcal{N}}$, and let $\mathscr{M}_\infty(\boldsymbol{\delta})$ be the limiting divisibility invariant defined from these quantities. Le…

Let $E/\mathbb{Q}$ be the elliptic curve in the setup, let $p$ be an odd prime satisfying the irreducibility and Manin-constant hypotheses, and let…

Let $M$ be a critical object of $\operatorname{{\bf PM}}_K$, let $b(M)$ be the basis supplied by Deligne's period conjecture, let $\lambda$ be a place of $K$ for which the stated v…

Let $M$ be an $L$-admissible premotivic structure over $K$, let $\tau:K\to\mathbb{C}$ be an embedding, and let $\lambda$ be any finite prime of $K$. Write…

Let $M$ be the residual-representation lattice and let $A=(M\otimes_{\mathbb{Z}_p}\mathbb{Q}_p)/M$. For each choice of signed local condition indexed by…

Pessimistic conjecture. The image of this map is closed in the Banach-space topology of $H^1(\mathcal{K}_\infty,V)$. This weaker assertion is stated as sufficient for the paper's p…