Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations

in Transactions of the American Mathematical Society (2017), 369

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine ... [more ▼]

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem. [less ▲]

Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image

in Pacific Journal of Mathematics (2016), 281(1), 1-16

This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part is concerned with ... [more ▼]

This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part is concerned with symplectic Galois representations having a huge residual image, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. We prove a classification result on those subgroups of a general symplectic group over a finite field that contain a nontrivial transvection. Translating this group theoretic result into the language of symplectic representations whose image contains a nontrivial transvection, these fall into three very simply describable classes: the reducible ones, the induced ones and those with huge image. Using the idea of an (n,p)-group of Khare, Larsen and Savin we give simple conditions under which a symplectic Galois representation with coefficients in a finite field has a huge image. Finally, we combine this classification result with the main result of the first part to obtain a strenghtened application to the inverse Galois problem. [less ▲]

Jacobian varieties of genus 3 and the inverse Galois problem

Presentation (2015, October 28)

The inverse Galois problem, first addressed by D. Hilbert in 1892, asks which finite groups occur as the Galois group of a finite Galois extension K/Q$. This question is encompassed in the general problem ... [more ▼]

The inverse Galois problem, first addressed by D. Hilbert in 1892, asks which finite groups occur as the Galois group of a finite Galois extension K/Q$. This question is encompassed in the general problem of understanding the structure of the absolute Galois group G_Q of the rational numbers. A deep fact in arithmetic geometry is that one can attach compatible systems of Galois representations of G_Q to certain arithmetic-geometric objects, (e.g. abelian varieties). These representations can be used to realise classical linear groups as Galois groups over Q. In this talk we will discuss the case of Galois representations attached to Jacobian varieties of genus n curves. For n=3, we provide an explicit construction of curves C defined over Q such that the action of G_Q on the group of l-torsion points of the Jacobian of C provides a Galois realisation of GSp(6, l) for a prefixed prime l. This construction is a joint work with Cécile Armana, Valentijn Karemaker, Marusia Rebolledo, Lara Thomas and Núria Vila, and was initiated as a working group in the Conference Women in Numbers Europe (CIRM, 2013). [less ▲]

Genus 3 curves and explicit realisations of symplectic groups as Galois groups over Q

Presentation (2015, April 06)

Abstract: Let n be a natural number and l a prime number. Given a genus n curve C defined over Q, the group of l-torsion points defined over an algebraic closure of Q of its Jacobian variety J_C is ... [more ▼]

Abstract: Let n be a natural number and l a prime number. Given a genus n curve C defined over Q, the group of l-torsion points defined over an algebraic closure of Q of its Jacobian variety J_C is endowed with an action of the absolute Galois group G_Q , giving rise to a Galois representation ρ: G_Q → GSp(2n, l). When ρ is surjective, it provides us with a realisation of GSp(2n, l) as a Galois group over Q. To study Galois realisations (over Q) with particular ramification properties at l, it is of great interest to have conditions at auxiliary primes different from l that ensure surjectivity, while allowing great flexibility in the behaviour at the prime l. In this talk we focus on the case n = 3, and provide an explicit construction of curves C defined over Q such that ρ is surjective for a prefixed prime l. This is joint work with Cécile Armana, Valentijn Karemaker, Marusia Rebolledo, Lara Thomas and Núria Vila, and was initiated as a working group in the Conference Women in Numbers Europe (CIRM, 2013). [less ▲]

Artin representations associated to modular forms of weight one: Deligne-Serre's theorem

Presentation (2015, January 27)

Automorphic Galois representations and the inverse Galois problem

in Chamizo, Fernando; Guàrdia, Jordi; Rojas-León, Antonio (Eds.) et al Trends in Number Theory (2015)

A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully ... [more ▼]

A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully chosen, they provide compatible systems of Galois representations satisfying some desired properties, e.g. properties that reflect on the image of the members of the system. In this article we survey some results obtained using this strategy. [less ▲]

Large Galois images for Jacobian varieties of genus 3 curves

E-print/Working paper (2015)

Given a prime number l greater than or equal to 5, we construct an infinite family of three-dimensional abelian varieties over Q such that, for any A/Q in the family, the Galois representation \rho_{A, l ... [more ▼]

Given a prime number l greater than or equal to 5, we construct an infinite family of three-dimensional abelian varieties over Q such that, for any A/Q in the family, the Galois representation \rho_{A, l}: Gal_Q -> GSp(6, l) attached to the l-torsion of A is surjective. Any such variety A will be the Jacobian of a genus 3 curve over Q whose respective reductions at two auxiliary primes we prescribe to provide us with generators of Sp(6, l). [less ▲]

Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties

in Mathematische Annalen (2015), 361(3), 909-925

This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the ... [more ▼]

This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer n and any positive integer d, PSp_n(F_{l^d}) or PGSp_n(F_{l^d}) occurs as a Galois group over the rational numbers for a positive density set of primes l. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of GL_n(A_Q) with local types chosen so as to obtain a compatible system of Galois representations to which the results from Part II of this series apply. [less ▲]

Compatible systems of symplectic Galois representations and the inverse Galois problem

Scientific Conference (2014, November 20)

Compatible systems of symplectic Galois representations and the inverse Galois problem

Presentation (2014, April 01)