294 problems

For every totally real number field $k$ and every prime number $p$, let $k_\infty^{\mathrm{c}}/k$ be the cyclotomic $\mathbb{Z}_p$-extension, and let $M(k_\infty^{\mathrm{c}})$ be…

For every prime $p\ge 11$ and every nonsplit très ramifiée representation $\bar\rho\cong\begin{pmatrix}\omega&*\\0&1\end{pmatrix}$, let $M$ be a primitive projective-augmented modu…

Let $E$ be an elliptic curve with good reduction at $p$, let $K_\infty/K$ be the anticyclotomic $\mathbb{Z}_p$-extension, and let $K_n$ be a finite layer. Under the hypotheses…

For every constant $\mathbb{Z}_p$-tower of Cayley graphs $(X_n)_{n\geq 0}$, the defect $\delta(X_n)$ is eventually constant; that is, there exist $N\geq 0$ and $d$ such that…

Let $K$ be a number field, let $p$ be a prime, and let $R(E/L)$ denote the fine Selmer group of an elliptic curve $E/K$ over an algebraic extension $L/K$. Write $K^{\mathrm{cyc}}$…

Mazur's main conjecture for modular forms.

Let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, let $\Lambda$ be its Iwasawa algebra, and assume that $\rho_f$ has large image. For a height-one…

Greenberg's conjecture. The module $X(k^\mathrm{c}_\infty)$ is finite.

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

Let $k_\infty/k$ be a $\mathds Z_p$-power extension with no finite place splitting completely, let $K_\infty=Kk_\infty$, and use the source's decomposition and character $\chi$ to…

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…

Greenberg's generalized conjecture. The module $X(\widetilde{k})$ is pseudo-null as a $\mathbb{Z}_p\llbracket\operatorname{Gal}(\widetilde{k}/k)\rrbracket$-module.

Coates–Sujatha's pseudo-nullity conjecture. The module

Let $K$ be an imaginary quadratic field, let $K_\infty$ be the relevant anticyclotomic extension, and let $\Lambda$ be its Iwasawa algebra. Let $\mathfrak{X}$ denote the relevant P…

Let $K$ be a number field and let $K^{\text{cyc}}$ be the cyclotomic $\mathbf{Z}_p$-extension of $K$. The Iwasawa $\mu$-invariant is the coefficient governing the exponential term…

Let $F$ be a number field, let $\widetilde{F}$ be the compositum of all $\mathbb{Z}_{p}$-extensions of $F$, and let $K(\widetilde{F})$ be the maximal unramified abelian pro-$p$ ext…

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…

The $p$-adic Beilinson conjecture. For each odd integer $j>1$,

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.