192 problems

Mazur's main conjecture for modular forms.

Let $M$ be a motive with $L$-function $L(M,s)$ and $p$-adic Selmer group $\mathrm{Sel}_{p^\infty}(M)$. Suppose its functional equation is … with $s=\frac{1}{2}$ a critical value. B…

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 $f$ 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…

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

Kolyvagin's conjecture. The set of Kolyvagin classes is nontrivial:

Mazur–Tate conjecture. The following claims are valid:

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

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 $E$ be an elliptic curve over a number field $F$, let $p$ be a prime at which $E$ has good reduction, and let $F_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $F$. Deno…

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.…

Suppose that $a_p(f)=0$ and $\psi=\mathbf{1}$. Let $\mathrm{Sel}^{\pm}(\mathbb{Q}_\infty,A_f(1))^\vee$ be the Pontryagin duals of the signed Selmer groups, let…

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

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 $E/\mathbf{Q}$ be an elliptic curve, and let $p$ be an odd prime of good ordinary reduction. The representation $E[p]$ is irreducible as a Galois module when it has no nontrivi…

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

Let $E$ be the elliptic curve and let $F_\infty$ be the cyclotomic extension considered in the paper. For each supersingular prime $w$, choose $s_w\in\{\flat,\sharp\}$, and write…

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…

Mazur's growth number conjecture. For all sufficiently large $n$,

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 $E/{\mathbb Q}$ be an elliptic curve of conductor $N$, let $p>2$ be a prime at which $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field in which…

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…

Let $X$ be as above, let $p$ be a prime of good reduction, and let $m<2r-1$. Write $H^m_{\operatorname{\acute{e}t}}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_p(r))$ for the $p$-adic éta…

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 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…