Related papers: Galois Groups Via Atkin-Lehner Twists

We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour,…

We compute the rational points on the Atkin-Lehner quotient $X^+_0(125)$ using the quadratic Chabauty method. Our work completes the study of exceptional rational points on the curves $X^+_0(N)$ of genus between 2 and 6. Together with the…

For an elliptic curve E over Q, the Galois action on the l-power torsion points defines representations whose images are subgroups of GL_2(Z/l^n Z). There are three exceptional prime powers l^n=2,3,4 when surjectivity of the mod l^n…

For each of the groups PSL2(F25), PSL2(F32), PSL2(F49), PGL2(F25), and PGL2(F27), we display the first explicitly known polynomials over Q having that group as Galois group. Each polynomial is related to a Galois representation associated…

Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the \'etale fundamental group of $X$ onto the absolute Galois group of $k$ has a…

We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a G-cover of curves in characteristic 0. We prove that all…

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve $E/\mathbb{Q}$, a positive proportion of its quadratic twists $E^{(d)}$ have rank 1. Using tools from Galois cohomology, we give criteria on E and d which…

We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the…

Let p>3 be a prime, f a positive integer and Q_{p^f} the unramified extension of Q_p of degree f. After Breuil and Paskunas, to a generic semi-simple continue modulo p representation of the absolute Galois group of Q_{p^f}, we can associate…

Let E be an elliptic curve over a number field F, A the abelian surface E x E, and T_F(A) the F-rational albanese kernel of A, which is a subgroup of the degree zero part of Chow group of zero cycles on A modulo rational equivalence. The…

In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of $\ell$-adic Galois representations valued in a semisimple…

Consider a rational point on an elliptic curve under an isogeny. Suppose that the action of Galois partitions the set of its pre-images into n orbits. It is shown that all such points above a certain height have their denominator divisible…

If P is an algebraic point on a commutative group scheme A/K, then P is _almost_rational_ if no two non-trivial Galois conjugates sigma(P), tau(P), have sum equal to 2P. In this paper, we classify almost rational torsion points on…

Let $C \subset \mathbb{P}^2$ be a plane curve of degree at least three. A point $P$ in projective plane is said to be Galois if the function field extension induced by the projection $\pi_P: C \dashrightarrow \mathbb P^1$ from $P$ is…

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve E over an arbitrary number field K. Under the assumption that Gal(K(E[2])/K) = S_3 we show that the density (counted in a non-standard way)…

For \ell \neq p odd primes, we examine PSL_2(\ell)-covers of the projective line branched at one point over an algebraically closed field of characteristic p, where PSL_2(\ell) has order divisible by p. We show that such covers can be…

A (projective, geometrically irreducible, non-singular) curve $\mathcal{X}$ defined over a finite field $\mathbb{F}_{q^2}$ is maximal if the number $N_{q^2}$ of its $\mathbb{F}_{q^2}$-rational points attains the Hasse-Weil upper bound, that…

In 1923 Schur considered the following problem. Let f(X) be a polynomial with integer coefficients that induces a bijection on the residue fields Z/pZ for infinitely many primes p. His conjecture, that such polynomials are compositions of…

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…

In this article we consider outer Galois actions on a free profinite group of rank two, induced by the \'etale fundamental group of a projective line minus three points or of a pointed elliptic curve over a number field. Under mild…