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By definition, $U_s$ is the only subgroup of the cyclic group $(\mathbb Z/p\mathbb Z)^{\times}$ of cardinality $s$. The cyclotomic character $\chi_p:\operatorname{Gal}(\mathbb Q(\zeta_p)/\mathbb Q)\lo …

I think that the answer to your questions depends in subtle ways on whether $r=0$ or $r=1$. In full generality, I believe you are right that none of the properties you state are known for all elliptic …

We don't know that there are infinitely many number fields with Class Number one, so a fortiori we don't know any explicit infinite family of such number fields.

Since an algebraic number is zero if and only if any of its conjugates is zero, $I_f J_1$ is stable under $W_N$ and so indeed $W_N$ descends to an automorphism of $A_f$. Now, the important thing to re …

Recently, I stumbled coincidentally on the paper Computation of invariants in the theory of cyclotomic fields K. Iwasawa and C. Sims J. Math. Soc. Japan Vol.18 (1966) This explains in full details how …

The difference is the generality of the setting: Hida families (first introduced by Hida in the early 80s) apply only to eigencuspforms which are so-called ordinary at $p$ (roughly speaking, the $p$-a …

Well, the answer of the question in the title in certainly Yes, many things in fact, but let me be more precise. In 1958, Serre gave a Bourbaki talk on the recent works of Iwasawa on class groups in t …

Going in the other direction, the Néron-Ogg-Shafarevich criterion and Weil pairing imply that the Tate module $T_{\ell}E$ is a Galois representation which is ramified at $p$. So if $n$ is large enough …

If by $f_1\equiv f_2$ modulo $p$, you mean that $a_n(f_1)\equiv a_{n}(f_2)$ modulo $p$ for all $n\in\mathbb N$ or maybe for all except finitely many, then this theorem cannot be true. Let's start with …

I don't see any contradiction: the Selmer group also has a contribution of rational points. Indeed, the group of 2-torsion rational points on this elliptic curve is isomorphic to $\mathbb Z/2\mathbb Z …

When he introduced $p$-adic numbers, Kurt Hensel produced an incorrect local/global proof of the fact that $e$ is transcendental. Apparently, the intended proof goes along the following lines: studyin …

A Galois representation $\rho_\ell:\operatorname{Gal}(\bar{\mathbb Q}_{\ell}/\mathbb Q_{\ell})\longrightarrow\operatorname{GL}_2(\mathbb Q_{\ell})$ can be semistable (technically $B_{st}$-admissible i …

This semester, I teach a graduate course in epistemology of mathematics and as a case study, I assigned students a discussion on the epistemological status of Fermat's Last Theorem according to differ …

How did we end up with the such complicated constructions of $B$? To add to Laurent's answer remark that "these rings did not, however, come out of nowhere", I believe that in the early 80s, Fontain …

Denote by $\Phi$ the quotient of $\mathcal A^\vee$, the special fiber of the smooth (but not necessarily proper) model of the dual abelian variety $A^\vee$, by the connected component of $0$ of $\math …