24 problems

Let $L/\mathbb Q_p$ be finite, let $V$ be a finite-dimensional $L$-representation of $G_{\mathbb Q_p}$, and let $T$ be a Galois-stable $\mathcal O_L$-lattice. Let $G$ be the releva…

Let $p$ be an odd prime, let $F_\infty=\mathbb{Q}(\mu_{p^\infty},\sqrt[p^n]{m}:n\geq1)$, let $G=\operatorname{Gal}(F_\infty/\mathbb{Q})$, and let…

Let $E/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$, let $F_\infty/\mathbb{Q}$ be the false Tate curve extension, and let…

The main conjecture. The p-adic $L$-function ${\cal L}_E\in K_1(\Lambda(G)_{S^*})$ is a characteristic element of $X(E/F_{\infty})$.

The non-CM p-adic L-function conjecture. There exists ${\cal L}_E\in K_1(\Lambda(G)_{S^*})$ such that ${\cal L}_E(\rho)\ne\infty$ for every Artin representation $\rho$ of $G$, and

The noncommutative Selmer torsion conjecture. The Pontryagin dual of the Selmer group,

The four-case integrality conjecture. In each of the following four cases, the assertions (a), (b), (c), and (d) are equivalent for every $xiin K1(R3):

Equivariant Bloch–Kato conjecture. The Beilinson regulator induces an isomorphism $\Delta_f(G,M^\lor(1-k))\otimes {\Bbb{R}}\cong \1_{{\Bbb{R}}[G]}$, under which…

Let $p$ be an odd rational prime, let $K$ be a totally real number field, and let $\mathcal L/K$ be an admissible one-dimensional $p$-adic Lie extension. Write…

Let $\mathcal{K}_\infty/K$ and $S$ be as in the source, put $\Sigma=\Sigma_S(\mathcal{K}_\infty)$ and $r=|\Sigma|$, and let $\Lambda(\mathcal{G}_\infty)$ be the relevant Iwasawa al…

Let $L/K$, $\Pi$, and $\Pi'$ be the fixed data, let $\Sigma$ be a subset of $\Sigma(L)$ with $\Sigma e Pi$, and let $p$ be a prime such that $O^\times{L, Pi, Pi',p}$ is torsion-fre…

Let $\bar{A}$ be one of the coefficient objects considered in the source, with coefficient ring $\bar{R}$, and let $F_\infty/F$ be a $p$-adic Lie extension such that…

Let $F_\infty/F$ be a strongly admissible $p$-adic Lie extension, set $G=\operatorname{Gal}(F_\infty/F)$ and $H=\operatorname{Gal}(F_\infty/F^{\mathrm{cyc}})$, and let…

Let $F_\infty/F$ be a strongly admissible $p$-adic Lie extension: it is Galois, contains the cyclotomic $\mathbb{Z}_p$-extension, is unramified outside finitely many primes, and…

Let $F_{\infty}/F$ be the extension in the paper, let $G=\operatorname{Gal}(F_{\infty}/F)$, and let $H$ be the subgroup specified by the extension, with completed group ring…

Let $F_{\infty}/F$ be the $\mathbb{Z}_p^d$-extension considered in the paper, let $G=\operatorname{Gal}(F_{\infty}/F)$, and let $\Sigma_{\mathrm{ss}}$ be the set of primes where…

Adjoint Selmer-category conjecture. The dual Selmer group $\mathrm{Sel}^\ast_{E_\infty}(\mathrm{Ad}^0(\rho))$ is in the category…

Let $K_\infty/K$ be a strongly admissible $p$-adic Lie extension with Galois group $G$, let $E$ be an elliptic curve, and let $\Lambda(G)$ be the Iwasawa algebra. Let $R$ be the se…

Let $\pi$ be an irreducible infinite-dimensional smooth admissible representation of $\mathrm{GL}_n(\mathbf{Q}_p)$, and let $\pi^{\vee}=\operatorname{Hom}(\pi,\mathbf{F}_p)$. Let…

Let $K=\mathrm{SL}_n(\mathbf{Z}_p)$ and let $K(p)$ be its principal congruence subgroup. Write $\Lambda=\mathbf{Z}_p[[K(p)]]$. For a finitely generated $\Lambda$-module $M$, let it…

In the notation above, let $\mathcal{L}_E\ \epsilon\ K_1(\Lambda(\mathcal{G})_{S^*})$ be the conjectural $p$-adic $L$-function, let $X(E/F_\infty)$ be the Pontryagin dual of the Se…

Let $K\in\mathcal{F}^+$ be unramified outside a finite set of places $\Sigma$, and let $C_K$ be the associated compactly supported arithmetic complex. Generalized Tate-motive main…

Let $E/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$, let $K=\mathbb{Q}(E(p))$, and let $G$ be its relevant Galois group, assumed to have no element of order…

Let $K$ be a number field, let $p$ be a prime, let $E$ be an elliptic curve over $K$, and let $L/K$ be a Galois extension with $G=\operatorname{Gal}(L/K)$ containing a closed norma…