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Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations Arias De Reyna Dominguez, Sara ; ; Wiese, Gabor 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 ▲] Detailed reference viewed: 186 (23 UL)Classification of subgroups of symplectic groups over finite fields containing a transvection Arias De Reyna Dominguez, Sara ; ; Wiese, Gabor in Demonstratio Mathematica (2016), 49(2), 129-148 In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l ... [more ▼] In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving "big image" results for symplectic Galois representations. [less ▲] Detailed reference viewed: 143 (6 UL)Possible connection between a generalized Maeda's conjecture and local types ; Tsaknias, Panagiotis E-print/Working paper (2016) Detailed reference viewed: 41 (1 UL)Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image Arias De Reyna Dominguez, Sara ; ; Wiese, Gabor 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 ▲] Detailed reference viewed: 109 (4 UL)On modular forms and the inverse Galois problem ; Wiese, Gabor in Transactions of the American Mathematical Society (2011), 363(9), 4569--4584 In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the ... [more ▼] In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Particular care is taken in order to exclude nontrivial inner twists. [less ▲] Detailed reference viewed: 98 (2 UL) |
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