Milnor K-Groups and Zero-Cycles on Products of Curves over p-Adic Fields

We investigate the Chow groups of zero cycles of products of curves over a p-adic field by means of the Milnor K-groups of their Jacobians as introduced by Somekawa. We prove some finiteness results for CH ₀(X)/m for X a product of curves over a p-adic field.     

The statistics of the number of rational points in the hyperelliptic ensemble over a finite field

We study the distribution of the numbers of \({F_{{q^r}}}\)-rational points of hyperelliptic curves over a finite field F_q in odd characteristic. This extends the result of Kurlberg and Rudnick [4]. We also study the distribution of the number of \({F_{{q^r}}}\)-rational points and the trace of high powers of the Frobenius class of real hyperelliptic curves over a finite field F_q in even characteristic.     

Some remarks on the Picard curves over a finite field

In this paper we study the Newton polygon of the L ‐polynomial L (t) associate to the Picard curves y³ = x⁴ – 1, y³ = x⁴ – x defined over a finite field ??_p . In the former case we get a complete classification. In the latter case we obtained a partial result. As a consequence of our result we obtain a criterion to find a supersingular Picard curves for the above two cases. Our main results are stated in Theorems 3.1 and 4.1. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Union curvature of a vector field in a subspace

The differential equations of union curves on a hypersurfaceV _n immersed in a RiemannianV _n+1 have been obtained by Springer [1]. These results were generalized later for a subspace in a Riemannian space by Mishra [2]. The author [3] has defined the union curvature of a vector field with respect to a curve on a hypersurfaceV _n of a RiemannianV _n+1. The purpose of this paper is to consider union curvature of a vector field with respect to a curve in a subspaceV _n of a RiemannianV _m. The author is indebted to the referee for helpful suggestions.     

n-Dimensional algebras over a field with a cyclic extension of degree n

We give a geometric method of classifying algebras A _n,K , n-dimensional over a field K, with a cyclic extension of degree n. Algebras A _n,K without zero divisors satisfying some conditions are classified. In particular, we determine all n-dimensional division algebras over a finite field F _q when n is prime and q is large enough.This research was supported in part by a grant from the M U R S T (40 % funds).     

On the number of curves of genus 2 over a finite field

In this paper we compute the number of curves of genus 2 defined over a finite field k of odd characteristic up to isomorphisms defined over k; the even characteristic case is treated in an ongoing work (G. Cardona, E. Nart, J. Pujolàs, Curves of genus 2 over field of even characteristic, 2003, submitted for publication). To this end, we first give a parametrization of all points in , the moduli variety that classifies genus 2 curves up to isomorphism, defined over an arbitrary perfect field (of zero or odd characteristic) and corresponding to curves with non-trivial reduced group of automorphisms; we also give an explicit representative defined over that field for each of these points. Then, we use cohomological methods to compute the number of k-isomorphism classes for each point in .     

Reduction theory for a rational function field

LetG be a split reductive group over a finite field F_q. LetF = F_q(t) and let A denote the adèles ofF. We show that every double coset inG(F)/G(A)/K has a representative in a maximal split torus ofG. HereK is the set of integral adèlic points ofG. WhenG ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.     

Rational Surfaces with Many Nodes

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain n b ₂ –2 disjoint smooth rational curves with self-intersection –2, where b ₂ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with p _g = 0 and K² = 8, 9 which admit an automorphism of order 2.     

Algebras with involution that become hyperbolic over the function field of a conic

We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra Q. We classify these algebras in degree 4 and give an example of such a division algebra with orthogonal involution of degree 8 that does not contain (Q,⁻), even though it contains Q and is totally decomposable into a tensor product of quaternion algebras.     

Torsion points on elliptic curves over a global field

Let C be an elliptic curve defined over a global field K and denote by C_K the group of rational points of C over K. The classical Nagell-Lutz-Cassels theorem states, in the case of an algebraic number field K as groud field, a necessary condition for a point in C_K to be a torsion point, i.e. a point of finite order. We shall prove here two generalized and strongthened versions of this classical result, one in the case where K is an algebraic number field and another one in the case where K is an algebraic function field. The theorem in the number field case turns out to be particularly useful for actually computing torsion points on given families of elliptic curves.     

Systems of two iterated functions over skew field of quaternions

Systems of linear iterated functions f ₀(z) = qz + a, f ₁(z) = qz + b over the field of complex numbers have been investigated since 1985 (Barnsley and Harrington). Much attention is paid to the question of the connection of their attractors. We consider systems of iterated functions f ₀(z) = qzp + a, f ₁(z) = qzp + b over the skew field of quaternions. We simplify the form of such systems and study the structure of their attractors.     

Class field theory for open curves over p-adic fields

We introduce the idèle class group for quasi-projective curves over p-adic fields and show that the kernel of the reciprocity map is divisible. This extends Saito’s class field theory for projective curves (Saito in J Number Theory 21:44–80, 1985).     

The dressed nonrelativistic electron in a magnetic field

We consider a nonrelativistic electron interacting with a classical magnetic field pointing along the x₃‐axis and with a quantized electromagnetic field. When the interaction between the electron and photons is turned off, the electronic system is assumed to have a ground state of finite multiplicity. Because of the translation invariance along the x₃‐axis, we consider the reduced Hamiltonian associated with the total momentum along the x₃‐axis and, after introducing an ultraviolet cutoff and an infrared regularization, we prove that the reduced Hamiltonian has a ground state if the coupling constant and the total momentum along the x₃‐axis are sufficiently small. We determine the absolutely continuous spectrum of the reduced Hamiltonian and, when the ground state is simple, we prove that the renormalized mass of the dressed electron is greater than or equal to its bare one. We then deduce that the anomalous magnetic moment of the dressed electron is nonnegative. Copyright © 2006 John Wiley & Sons, Ltd.     

Every finitely generated regular field extension has a stable transcendence base

A fieldK is called stable if every finitely generaed regular field extensionF/K has a transcendence basex ₁, …,x _n with the following properties: The field extensionF/K(x ₁,…,x _n ) is separable and the Galois hull ofF/K(x ₁,…,x _n ) remains regular overK, i.e.K is algebraically closed in . We prove in this paper thatevery field is stable. This generalizes results from [FJ1] and [GJ] which prove that fields of characteristic 0 and infinite perfect fields are stable, respectively. [G] showed that finite fields are stable in dimension 1, i.e. every finitely generated regular field extension of transcendence degree 1 over a finite field has a stable transcendence base. In the last section of this paper we apply the theorem to the construction of PAC fields with additional properties. A fieldK is called PAC if every absolutely irreducible variety overK has at least oneK-rational point.     

Linear groups over a skew field of quaternions containing the group T_{n}(A, \Phi) over a subsfield

We study the subgroups of GL_n(D) (n \geqq 3) GL_{n}(D) (n \geqq 3) over a skew field of quaternions D that comprise the subgroup of the unitary group U_n(A, F) U_{n}(A, \Phi) over a subsfield A \subseteqq D A \subseteqq D generated by all transvections in U_n(A, F) U_{n}(A, \Phi) .     

Varieties of solvable index-two alternative algebras over a field of characteristic three

The subvarieties of the variety Alt₂ of solvable index-two alternative algebras over an arbitrary field of characteristic 3 are studied. The main types of such varieties are singled out in the language of identities, and inclusions between these types are established. The main results is the following.Theorem.The topological rank of the variety Alt₂ of solvable index-two alternative algebras over an arbitrary field of characteristic 3 is equal to five. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 556–566, October, 1999.     

Converse theorems assuming a partial euler product

Associated to a newform f(z) is a Dirichlet series L _f (s) with functional equation and Euler product. Hecke showed that if the Dirichlet series F(s) has a functional equation of a particular form, then F(s)=L _f (s) for some holomorphic newform f(z) on Γ(1). Weil extended this result to Γ₀(N) under an assumption on the twists of F(s) by Dirichlet characters. Conrey and Farmer extended Hecke’s result for certain small N, assuming that the local factors in the Euler product of F(s) were of a special form. We make the same assumption on the Euler product and describe an approach to the converse theorem using certain additional assumptions. Some of the assumptions may be related to second order modular forms. This work resulted from an REU at Bucknell University and the American Institute of Mathematics. Research supported by the American Institute of Mathematics and the National Science Foundation.     

Singular values of some modular functions and their applications to class fields

Since the modular curve has genus zero, we have a field isomorphism where X ₂(z) is a product of Klein forms. We apply it to construct explicit class fields over an imaginary quadratic field K from the modular function j _Δ,25(z):=X ₂(5z). And, for every integer N≥7 we further generate ray class fields K _(N) over K with modulus N just from the two generators X ₂(z) and X ₃(z) of the function field , which are also the product of Klein forms without using torsion points of elliptic curves. J.K. Koo was supported by Korea Research Foundation Grant (KRF-2002-070-C00003).     

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form Y ²=f ₃ X ⁶+f ₂ X ⁴+f ₁ X ²+f ₀, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field ℚα. If each of these elliptic curves has rank less than the degree of ℚα : ℚ, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over ℚα) on the elliptic curves, for which x∈ℚ. This in turn allows us to find all ℚ-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over ℚ), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over ℚ. Received: 27 November 1998 / Revised version: 4 June 1999