494 problems

Let $F$ be a number field, let $r_2$ denote the number of pairs of complex embeddings of $F$, and let $F_\infty$ be the compositum of all $\mathbb{Z}_p$-extensions of $F$. Write ……

Iwasawa's conjecture. For every cyclotomic $\mathbb{Z}_p$-extension of a number field, the $\mu$-invariant is $0$.

Iwasawa main conjecture. The $p$-adic $L$-function is integral and generates the characteristic ideal:

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

Coates–Sujatha's conjecture B. The module $Y(E/\mathds{K}_{\infty})$ is pseudo-null as a $3bLambda_2$-module.

Let $k$ be totally real, let $K/k$ be an abelian CM extension with group $G$, let $p$ be a rational prime, and let $\chi$ be a one-dimensional character of $G$. Define … Gross's hi…

Let $S$ consist of the infinite place and the primes dividing $Np$, let $\mathbb{Z}_S=\mathbb{Z}[1/Np]$, let $\Lambda$ be the cyclotomic Iwasawa algebra, and let $\mathbf{T}$ be th…

Let $K$ be a number field, let $K_\infty^{\mathrm{cycl}}$ be its cyclotomic $\mathbb{Z}_p$-extension, and let $T_p^{\mathrm{cycl}}(K)$ be the associated Kuz'min–Tate module. Write…

Perrin-Riou's conjecture. Equality should hold in this inclusion:

Let $k_\infty$ be the cyclotomic $\mathds{Z}_p$-extension of $k$, let $K$ be a finite extension in the setup, and let $\mathcal Q_K$ and $\Lambda_K$ be the total quotient ring and…

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…

Constancy conjecture. One has

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 $K_infty/K$ be a $\mathbb{Z}_p$-extension, let $\mathbf{A}(rho)$ be the $p$-divisible module attached to a $p$-adic Galois representation, and let $Y(\mathbf{A}(rho)/K_infty)$…

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…

Let $E/\mathbb{Q}$ be an elliptic curve with supersingular reduction at $p$, where $p$ is an odd prime and $a_p(E)=0$. Let $\mathbb{Q}_\mathrm{cyc}$ be the cyclotomic…

Mazur's growth number conjecture. The Mordell–Weil rank of $E$ stays bounded along any ${\mathbb Z}_p$-extension of $K$, unless the extension is anticyclotomic and the root number…

Let $E/\mathbb{Q}$ have good ordinary reduction at $p$, let $K_\infty/K$ be the anticyclotomic $\mathbb{Z}_p$-extension, let $\Lambda$ be its Iwasawa algebra, and let…

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…

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.…

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…

Fine Selmer torsion conjecture. The module