On the Modularity of Q-curves
نویسندگان
چکیده
A Q-curve is an elliptic curve over a number field K which is geometrically isogenous to each of its Galois conjugates. K. Ribet [17] asked whether every Q-curve is modular, and he showed that a positive answer would follow from J.-P. Serre’s conjecture on mod p Galois representations. We answer Ribet’s question in the affirmative, subject to certain local conditions at 3.
منابع مشابه
On the rank of certain parametrized elliptic curves
In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.
متن کامل2 9 A ug 2 00 7 On the modularity of supersingular elliptic curves over certain totally real number fields
We study generalisations to totally real fields of the methods originating with Wiles and Taylor-Wiles ([32], [31]). In view of the results of Skinner-Wiles [26] on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some app...
متن کاملGeneralized Jacobian and Discrete Logarithm Problem on Elliptic Curves
Let E be an elliptic curve over the finite field F_{q}, P a point in E(F_{q}) of order n, and Q a point in the group generated by P. The discrete logarithm problem on E is to find the number k such that Q = kP. In this paper we reduce the discrete logarithm problem on E[n] to the discrete logarithm on the group F*_{q} , the multiplicative group of nonzero elements of Fq, in the case where n | q...
متن کاملOn Modular Forms and Elliptic Curves over Q(ζ5)
We survey our joint work with Paul Gunnells and Farshid Hajir on a computational investigation of the modularity of elliptic curves over the cyclotomic field Q(ζ5), including the techniques we developed for describing the action of the Hecke operators using Voronoï theory.
متن کاملMining Overlapping Communities in Real-world Networks Based on Extended Modularity Gain
Detecting communities plays a vital role in studying group level patterns of a social network and it can be helpful in developing several recommendation systems such as movie recommendation, book recommendation, friend recommendation and so on. Most of the community detection algorithms can detect disjoint communities only, but in the real time scenario, a node can be a member of more than one ...
متن کاملOn Silverman's conjecture for a family of elliptic curves
Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...
متن کامل