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Luis Dieulefait

Publications and source records attributed to Luis Dieulefait.

62 records · Page 4Linked to original sources
arXiv2003

Galois actions on Q-curves and Winding Quotients

We prove two "large images" results for the Galois representations attached to a degree $d$ Q-curve $E$ over a quadratic field $K$: if $K$ is arbitrary, we prove maximality of the image for every prime $p >13$ not dividing $d$, provided that $d$ is divisible by $q$ (but $d \neq q$) with $q=2$ or 3 or 5 or 7 or 13. If $K$ is real we prove maximality of the image for every odd prime $p$ not dividing $d D$, where $D = \disc(K)$, provided that $E$ is a semistable Q-curve. In both cases we make the (standard) assumptions that $E$ does not have potentially good reduction at all primes $p \nmid 6$ and that $d$ is square-free.

math.NT↗
arXiv2003

Solving diophantine equations x^4 + y^4 = q z^p

We give a method to solve generalized Fermat equations of type $x^4 + y^4 = q z^p$, for some prime values of $q$ and every prime $p$ bigger than 13. We illustrate the method by proving that there are no solutions for $q= 73, 89$ and 113.

math.NT↗
arXiv2003

Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture

In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensional representation. We apply this result to establish new cases of the Fontaine-Mazur conjecture, namely, an irreducible Barsotti-Tate $λ$-adic 2-dimensional Galois representation unramified at 3 and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p^2 \}) = \Q $ always comes from an abelian variety. We also show the non-existence of irreducible Barsotti-Tate 2-dimensional Galois representations of conductor 1 and apply this to the irreducibility of Galois representations on level 1 genus 2 Siegel cusp forms.

math.NT↗
arXiv2001

On the images of modular and geometric three-dimensional Galois represetations

We address the problem of the determination of the images of three-dimensional geometric and modular Galois representations. In the modular case the existence of these representations is only conjectural. We give conditions to ensure that the images are as large as possible for almost every prime. We compute some examples and stress the applications to inverse Galois theory.

math.NT↗