相关论文: On a parity result for the symmetric square of mod…

We study generalisations to totally real fields of methods originating with Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction.…

Let $p$ be an odd prime number, and $E$ an elliptic curve defined over a number field with good reduction at every prime of $F$ above $p$. In this short note, we compute the Euler characteristics of the signed Selmer groups of $E$ over the…

We introduce the notions of symmetric and symmetrizable representations of $\text{SL}_2(\mathbb{Z})$. The linear representations of $\text{SL}_2(\mathbb{Z})$ arising from modular tensor categories are symmetric and have congruence kernel.…

This paper aims at studying the Iwasawa $\lambda$-invariant of the $p$-primary Selmer group. We study the growth behaviour of $p$-primary Selmer groups in $p$-power degree extensions over non-cyclotomic $\mathbb{Z}_p$-extensions of a number…

We give a short proof of the anticyclotomic analogue of the "strong" main conjecture of Kurihara on Fitting ideals of Selmer groups for elliptic curves with good ordinary reduction under mild hypotheses. More precisely, we completely…

A result due to R. Greenberg gives a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic $\mathbb Z_p$-extensions…

Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in E(K)$ be a point of infinite order, and let $N^{-1}\alpha$ be the set of $N$-division points of $\alpha$ in $E(\bar{K})$. We prove strong effective and uniform…

We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides $A$ and $B$. As an example, let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $p$ be a…

We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees…

The main result of this paper is an application of the topology of the space $Q(X)$ to obtain results for the cohomology of the symmetric group on $d$ letters, $\Sigma_d$, with `twisted' coefficients in various choices of Young modules and…

We prove modularity of certain residually reducible ordinary 2-dimensional $p$-adic Galois representations with determinant a finite order odd character $\chi$. For certain non-quadratic $\chi$ we prove an $R=T$ result for $T$ the weight 1…

Let p be an odd prime and let E be an elliptic curve defined over a quadratic imaginary field where p splits completely. Suppose E has supersingular reduction at primes above p. We define and study the fine double-signed residual Selmer…

Let p be an odd prime. Suppose that E is a modular elliptic curve/Q with good ordinary reduction at p. Let Q_{oo} denote the cyclotomic Z_p-extension of Q. It is conjectured that Sel_E(Q_{oo}) is a cotorsion Lambda-module and that its…

Let $\ell \geq 5$ be a prime, and let $\nu_\eta$ denote the Dedekind eta multiplier. For an odd integer $r$, and a real Dirichlet character $\psi$, recent work of Ahlgren, Andersen, and the author showed that quadratic congruences modulo…

For given linear action of a finite group on a lattice and a positive integer q, we prove that the mod q permutation representation is a quasi-polynomial in q. Additionally, we establish several results that can be considered as mod…

This is an expository article, based on a lecture course given at CRM Barcelona in December 2009. The purpose of these notes is to prove, in a reasonably self-contained way, that finiteness of the Tate-Shafarevich group implies the parity…

Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group…

We show that if $E/\mathbb{Q}$ is an elliptic curve without complex multiplication and for which there is a prime $q$ such that the image of $\bar{\rho}_{E,q}$ is contained in the normaliser of a split Cartan subgroup of…

We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular…

Our main result in this article is a proof (under mild technical assumptions) of an analogue for $p$-adic Galois representations attached to a newform $f$ of even weight $k\geq4$ of Kolyvagin's conjecture on the $p$-indivisibility of…