307 problems

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

Mazur–Tate conjecture. The following claims are valid:

Poonen–Rains conjecture. As $E$ varies over all elliptic curves over $F$ ordered by height,

Let $\mathcal{R}_\infty$ be the two-variable Iwasawa algebra, let $\mathcal{L}_p(f/K)$ be the associated two-variable $p$-adic $L$-function, and let…

Assume that $\mathcal{R}$ is regular. Let $K_\infty/K$ be the $p$-anticyclotomic $\mathbb{Z}_p$-extension, let $\mathcal{R}_\infty$ be the associated Iwasawa algebra, and define ……

Let $K_\infty$ be the relevant infinite extension and let the signed Selmer groups constructed using Wach modules be the signed Selmer groups attached to a modular form that is non…

Selmer conjecture. The ranks $r(n)$ and $s(n)$ have the same parity.

Let $F$ be a number field, $E$ a number field, $M$ a polarized motive in the pseudo-Abelian category of Chow motives over $F$ with coefficients in $E$, and let $\lbd$ be a finite p…

Mazur–Rubin's conjecture. If $E_1$ and $E_2$ are $p$-Selmer near-companions over $K$, then there exists a $G_K$-module isomorphism

Let $K$ be an imaginary quadratic field in which the odd prime $p$ splits as $(p)=\mathfrak p\overline{\mathfrak p}$, let $\Lambda$ be the anticyclotomic Iwasawa algebra, let…

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

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 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…

Let $E$ be an elliptic curve with good ordinary reduction at an odd prime $p$, let $\theta_n(f)$ be the Mazur–Tate element attached to the modular form $f$ associated with $E$ at…

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

Let $F$ be an imaginary quadratic field, let $F_\infty$ be its unique $Z_p^2$-extension, and let $\Lambda(\Omega)$ denote the Iwasawa algebra associated with…

Euler system existence and nonvanishing conjecture. If $V$ is $r$-critical and satisfies the rank $r$ Panchishkin condition for $r\geq 0$, there exists a collection of cohomology c…

Let $n$ be a positive integer, and let $E$ range over elliptic curves, ordered by naive height. The $n$-Selmer group of $E$ is denoted by $\operatorname{Sel}_n(E)$. Bhargava–Kane–L…

Finiteness conjecture. The group

BDP characteristic-ideal conjecture. One has

Let $n$ be any positive integer. For elliptic curves $E$ over $\mathbb{Q}$ ordered by height, let $\operatorname{Sel}_n(E/\mathbb{Q})$ denote the $n$-Selmer group, and let…

Let $p$ be an odd prime, let $E$ be an elliptic curve over a number field $F$, and let $\Sigma^{ss}$ be the nonempty set of primes of $F$ above $p$ at which $E$ has good supersingu…

Let $E$ be an elliptic curve over $F$, let $F^{\mathrm{cyc}}$ be its cyclotomic extension, and write … For a choice of signs $\overrightarrow{s}$ at the relevant supersingular prim…

$p$-Parity conjecture. For every abelian variety $A$ over a number field $K$ and every prime number $p$,