37 problems

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 $E/\mathbb Q$ be an elliptic curve, let $M=h^1(E)(1)$, and let $F_\infty=\mathbb Q(E(p))$ be the extension generated by the $p$-power torsion points. Let $\mathbb T$ be the ass…

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 $K=\mathbb{Q}(\mu_3)$ and let $F$ be the corresponding false Tate layer. Let $\rho$ and $\sigma$ be the two congruent Artin representations, and let $\mathcal L_E(\rho)$ and…

Let $K=\mathbb{Q}(\mu_p)$ and let $F$ be the first false Tate layer, and write $\rho$ and $\sigma$ for the two congruent Artin representations used in the paper. 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…

Let $E/bbQ$ be an elliptic curve with good ordinary reduction at an odd prime $p$, and let $F_\infty=\bigcup_n\mathbb{Q}(\mu_{p^n},\sqrt[p^n]{m})$. For a self-dual Artin representa…

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):

Noncommutative main conjecture. There exist $f,g\ \epsilon\ \mathcal T$ such that, for every irreducible Artin representation $\rho$ of $G$ for which $L(E,\rho,1)$ is defined, one…

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 $E$ be a $p$-ordinary elliptic curve, and let $\mathcal{K}_\infty/\mathcal{K}_0$ be a $p$-adic Lie extension of a number field $\mathcal{K}_0$ containing the cyclotomic…

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

Main conjecture over $\mathbb I[[{\mathcal G}]]$. There exists an element $\widetilde\xi\in K'_1(\mathbb I[[{\mathcal G}]]_{\mathscr S})$ such that

Noncommutative main conjecture. There exists an element

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

Coates–Fukaya–Kato–Sujatha–Venjakob conjecture. The Selmer groups defined over $F_\infty$ attached to $p$-ordinary Galois representations are in the category…