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Let $m > 2$ be an integer such that $S_m(m) = \sum_{n=1}^{m-1} n^m\equiv 1 \bmod{m}$. (Taking away $m^m$ does not harm the question, of course). Then $S_m(m)$ has the following expression in terms of …

The first question is answered in Serre's [Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15:4 (1972) 259--331], on page 304, section 5.2, exactly for this cur …

This is a long comment rather than an answer. Let $p$ be a prime. Consider the map $f(a)$ which assigns to $0<a<p$ the remainder of $p$ modulo $a$. The questions asks when the sequence $a$, $f(a)$, $ …

More generally let k be a local field of residual characteristic $\ell$ and $E/k$ an elliptic curve with good reduction. Then for any prime $p\neq\ell$, there is a perfect duality between $E(k)/p^n$ a …

$\newcommand{\FF}{\mathbb{F}}\DeclareMathOperator{\SL}{SL}$ OK, let me try. Write $M=E[p]$. If the order of $G$ is coprime to $p$, then $H^1(G,M)=0$. Assume that $p$ divides the order of $G$. Now by …

At what prime ? Well, let $p$ be an odd prime at which $E$ has good ordinary reduction. There are two ways: Use sage or magma to compute the analytic $p$-adic L-function via modular symbols. In sage …

Let $E$ be an elliptic curve over a $p$-adic field $k$. Let $\varphi: E \to E'$ be an isogeny defined over $k$. Write minimal Weierstrass equations with integer coefficients for both curves. Write $\ …

Both $\mu$ and $\lambda$ are about the $\Lambda$-torsion part of $M$. Neither of both invariants is visible in either $M/pM$ or $M[p]$ over $\Omega=\mathbb{F}_p[\![T]\!]$. The $\Lambda$-rank of $M$ …

Let $X$ be the Pontryagin dual of $M$. Then the dual of $M[p]$ is $X/pX$. Now it is easy to see from the decomposition theorem for modules over the Iwasawa algebra $\Lambda = \mathbb{Z}_p[\![T]\!]$ th …

Here is a rather long comment in which I try to justify why I think that the majority will have a surjective map on the Mordell-Weil group. I would not want to guess what the % is. Let $E$ be an elli …

sage: E = EllipticCurve([0,0,1,0,-1]) sage: E Elliptic Curve defined by y^2 + y = x^3 - 1 over Rational Field sage: E.integral_points() [(1 : 0 : 1), (7 : 18 : 1)]

Let me expand my comment above. While we believe that we expect this very very frequently, it is not always the case. As I commented, we have $$ E_d(\mathbb{Q})[n]\oplus E(\mathbb{Q})[n] = E(\mathbb …

I guess $X_{\infty}$ stands for the Galois group of the maximal abelian unramified $p$-extension of $K_{\infty}$, in other words the projective limit of the $p$-primary part of the class groups. Since …

Let $E$ be an elliptic curve over a number field $F$ and $p$ a prime such that $E$ has good ordinary reduction at all places above $p$. Suppose we know that the dual $X$ of the usual Selmer group over …

I have not read all the details of the article, but most of what I see is just descent by the isogeny $[2]$. The map is the Kummer map $$E(\mathbb{Q})\to \mathbb{Q}^{\times}/\square \times \mathbb{Q}^ …