289 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 $L/K$ be as above, and let $S$ and $T$ be finite non-empty disjoint sets of places of $K$, with $S$ containing all places ramifying in $L^+_\infty/K$ and all infinite places. W…

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…

Throughout, let $f_\alpha$ be a $\theta$-critical normalized eigenform, and let $L_{\mathrm{S},\alpha}^{[0]}(f,\xi)$ and $L_{\mathrm{K},\alpha}(f,\xi)$ denote the corresponding cri…

Let $V$ be an $o_L$-linear representation of $G_L$, let $M_{\mathrm{LT}}=\mathbb{D}(V)$ be an étale $L$-analytic $(\varphi_L,\Gamma_L)$-module over $\mathbf{A}_L$, and let…

Let $L$ be a finite extension of $\mathbb{Q}_p$, let $K$ be a coefficient field containing $L$, and let $M_0$ be an $L$-analytic $(\varphi_L,\Gamma_L)$-module over $\mathcal{R}_L$.…

Let $L$ be a finite extension of $\mathbb{Q}_p$, let $K$ be a coefficient field containing $L$, and let $V$ be an $L$-analytic $o_L$-linear representation of $G_L$. Let $M$ be the…

Let $H$ be the number field occurring in the Artin representation $\rho$, let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, and let…

Let $p$ be a prime with $p\geqslant 3$, let $F^{cyc}$ be the cyclotomic extension of $F$, and let $F^S$ be the maximal extension of $F$ unramified outside the specified set $S$. Fo…

$C_{\mathrm{Iw}}(K_\infty/K,V)$. The canonical trivialization satisfies

$C_{\mathrm{EP}}(L/K,V)$. The map $\delta_{V,L/K}$ sends $\Delta_{\mathrm{EP}}(L/K,T)$ onto $\mathbf{Z}_p[G]_{V,L/K}$.

Let $\Lambda$ be the Iwasawa algebra of the relevant $p$-adic Lie group, let $\widetilde\Lambda$ be its unramified coefficient extension, and let $\mathbb T$ be the associated big…

Let $F/\mathbb Q$ be a $p$-adic Lie extension with Galois group $G$, let $\Lambda=\Lambda(G)$ be its Iwasawa algebra, let $T_p$ be a Galois-stable lattice, and put…

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.…

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

Maximal nondegeneracy conjecture. If $j$ is even and $0\leq j\leq 2\min(r^+,r^-)$, then

Perrin-Riou's integrality conjecture. One has

Zeta-distribution main conjecture. There exist $f,g\in\Lambda$ such that $(\operatorname{rn}(f_ng_n^{-1}))_n$ is the zeta distribution $\mathcal{L}_S(G_\infty,M^\lor,1-k)$ and the…

Non-abelian Main Conjecture. The element $\delta_p(G_\infty,M,k)$ is induced by a generator

Let $p$ be a prime, let $K$ be a number field, and let $\widetilde{K}$ be the compositum of all $\mathbb{Z}_p$-extensions of $K$. Let $X$ be the Galois group of the maximal abelian…

Mazur–Swinnerton-Dyer's conjecture. There is an equality of ideals in $\Lambda$

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