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We determine the number of Del Pezzo surfaces of degree 2 over finite fields of odd characteristic with specified action of the Frobenius endomorphism, i.e., we solve the “quantitative inverse Galois problem”. As applications we determine the number of Del Pezzo surfaces of degree 2 with a given number of points and recover results of Banwait, Fité and Loughran and Loughran and Trepalin.

In this paper we study couples of finite separable extensions of the function field Fq(T) which are arithmetically equivalent, i.e. such that prime ideals of Fq[T] decompose with the same inertia degrees in the two fields, up to finitely many exceptions. In the first part of this work, we extend previous results by Cornelissen, Kontogeorgis and Van der Zalm to the case of non-geometric extensions of Fq(T), which are fields such that their field of constants may be bigger than Fq. In the second part, we explicitly produce examples of non-geometric extensions of F2(T) which are equivalent and non-isomorphic over F2(T) and non-equivalent over F4(T), solving a particular Inverse Galois Problem.

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. A key ingredient is a classification 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 strengthened application to the inverse Galois problem.

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. A key ingredient is a classification 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 strengthened application to the inverse Galois problem.

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. A key ingredient is a classification 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 strengthened application to the inverse Galois problem.

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. A key ingredient is a classification 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 strengthened application to the inverse Galois problem.

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. A key ingredient is a classification 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 strengthened application to the inverse Galois problem.

In this paper we will focus on a variant of the Inverse Galois Problem over the rationals, emphasizing the progress made through the analysis of the Galois representations arising from arithmetic-geometric objects.

In this paper we will focus on a variant of the Inverse Galois Problem over the rationals, emphasizing the progress made through the analysis of the Galois representations arising from arithmetic-geometric objects.