Related papers: A Finitary Hasse Principle for Diagonal Curves

It is known that the Cauchy's argument principle, applied to an holomorphic function $f$, requires that $f$ has no zeros on the curve of integration. In this short note, we give a generalization of such a principle which covers the case…

The inverse problem of Galois Theory was developed in the early 1800 s as an approach to understand polynomials and their roots. The inverse Galois problem states whether any finite group can be realized as a Galois group over Q (field of…

We present an algorithm which, given a connected smooth projective curve $X$ over an algebraically closed field of characteristic $p>0$ and its Hasse--Witt matrix, as well as a positive integer $n$, computes all \'etale Galois covers of $X$…

Using Mazur's theorem on torsions of elliptic curves, an upper bound 24 for the order of the finite Galois group $\mathcal{H}$ associated with weighted walks in the quarter plane $\mathbb{Z}^2_+$ is obtained. The explicit criterion for…

We classify Galois objects for the dual of a group algebra of a finite group over an arbitrary field.

Lichtenbaum proved that index and period coincide for a curve of genus one over a $p$-adic field. Salberger proved that the Hasse principle holds for a smooth complete intersection of two quadrics $X \subset P^n$ over a number field, if it…

We prove that there are infinitely many finite simple groups of symplectic Lie type, of any specified characteristic and rank, which appear as Galois groups over the field of rational numbers. This generalizes a result of Wiese, which…

Fixed an algebraic scheme $Y$. We suggest a definition for the conjugate of an algebraic scheme $X$ over $Y$ in an evident manner; then $X$ is said to be Galois closed over $Y$ if $X$ has a unique conjugate over $Y$. Now let $X$ and $Y$…

We give explicit formulas for the number of distinct elliptic curves over a finite field, up to isomorphism, in the families of Legendre, Jacobi, Hessian and generalized Hessian curves.

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we…

We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the etale Brauer-Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then…

We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does…

Consider a third order linear differential equation $L(f)=0$, where $L\in\mathbb{Q}(z)[\partial_z]$. We design an algorithm computing the Liouvillian solutions of $L(f)=0$. The reducible cases devolve to the classical case of second order…

We give sufficient conditions for a linear differential equation to have a given semisimple group as its Galois group. For any linear algebraic group G given as a semidirect product of a finite subgroup and a normal subgroup that is a…

A fundamental theorem of Katz \cite{Katz87} determines the differential Galois groups of rank $n$ connections on algebraic curves with slope $r/n$ at a singularity, where $\gcd(r,n)=1$. We extend this result to $G$-connections, where $G$ is…

An elementary approach is shown which derives the value of the Gauss sum of a cubic character over a finite field $\mathbb F_{2^s}$ without using Davenport-Hasse's theorem (namely, if $s$ is odd the Gauss sum is -1, and if $s$ is even its…

We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each…

A Fano problem consists of enumerating linear spaces of a fixed dimension on a variety, generalizing the classical problem of 27 lines on a cubic surface. Those Fano problems with finitely many linear spaces have an associated Galois group…

Over a global field (number field or function field of a curve over a finite field), theorems for the Galois cohomology of algebraic groups have long been known. For $F$ the function field of a curve over the formal series field…

We give a quantitative version of a result due to N. Katz about L-functions of elliptic curves over function fields over finite fields. Roughly speaking, Katz's Theorem states that, on average over a suitably chosen algebraic family, the…