Galois points for double-Frobenius nonclassical curves

We determine the distribution of Galois points for plane curves over a finite field of q elements, which are Frobenius nonclassical for different powers of q. This family is an important class of plane curves with many remarkable properties. It contains the Dickson–Guralnick–Zieve curve, which has been recently studied by Giulietti, Korchmáros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified.     

Mod 4 Galois representations and elliptic curves

Galois representations with cyclotomic determinant all arise from the -torsion of elliptic curves for . For , we show the existence of more than a million such representations which are surjective and do not arise from any elliptic curve.

     

Galois structure and de Rham invariants of elliptic curves

Let K be a number field with ring of integers OK. Suppose a finite group G acts numerically tamely on a regular scheme X over OK. One can then define a de Rham invariant class in the class group Cl(OK[G]), which is a refined Euler characteristic of the de Rham complex of X. Our results concern the classification of numerically tame actions and the de Rham invariant classes. We first describe how all Galois étale G-covers of a K-variety may be built up from finite Galois extensions of K and from geometric covers. When X is a curve of positive genus, we show that a given étale action of G on X extends to a numerically tame action on a regular model if and only if this is possible on the minimal model. Finally, we characterize the classes in Cl(OK[G]) which are realizable as the de Rham invariants for minimal models of elliptic curves when G has prime order.     

Abhyankar's conjecture on Galois groups over curves

Sans résuméOblatum 5-I-1993En hommage à Armand Borel avec notre admiration     

A birational embedding with two Galois points for quotient curves

A criterion for the existence of a birational embedding with two Galois points for quotient curves is presented. We apply our criterion to several curves, for example, some cyclic subcovers of the Giulietti–Korchmáros curve or of the curves constructed by Skabelund. New examples of plane curves with two Galois points are described, as plane models of such quotient curves.     

Reducible spectral curves and the hyperkahler geometry of adjoint orbits

We study the hyperkähler geometry of a regular semisimpleadjoint orbit of SL(k, ) via the algebraic geometry of the correspondingreducible spectral curve.     

Galois theory, discriminants and torsion subgroup of elliptic curves

We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve.     

Realization of modular Galois representations in the Jacobians of modular curves

The Ramanujan Journal - In Tian (Acta Arith. 164:399–412, 2014), the author improved the algorithm proposed by Edixhoven and Couveignes for computing mod $$ell $$ Galois representations...     

On the field of definition for modularity of CM elliptic curves

The purpose of this paper is to decide the conditions under which a CM elliptic curve is modular over its field of definition.     

A Lie theoretic Galois theory for the spectral curves of an integrable system. II

In the study of integrable systems of ODE's arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group and the Hecke algebra of double cosets of a parabolic subgroup of For each algebra a subtorus is identified that agrees with Kanev's Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.

     

On maximal curves which are not Galois subcovers of the Hermitian curve

We show that the generalized Giulietti-Korchmáros curve defined over $\mathbb{F}_{q^{2n} }$ , for n ?? 3 odd and q ?? 3, is not a Galois subcover of the Hermitian curve over $\mathbb{F}_{q^{2n} }$ . This answers a question raised by Garcia, Güneri and Stichtenoth.     

Galois Cohomology of Unipotent Algebraic Groups and Field Extensions

In this note we discuss, in the case of unipotent groups over nonperfect fields k of characteristic p, an analog of a theorem of Steinberg (formely a Serre's conjecture) for unipotent algebraic group schemes, which relates properties of Galois (or flat) cohomology of unipotent group schemes to finite extensions of k of degree divisible by p.