Defining Integrality at Prime Sets of High Density over Function Fields

 Let F be a function field of characteristic over a finite field . Then for any finite set of F-primes S and any , there exists a set of F-primes W of density greater than such that and has a Diophantine definition over . (Here and is defined analogously.)     

On some questions related to the Gauss conjecture for function fields

We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.     

Going up of the <Emphasis Type="Italic">u</Emphasis>-invariant over formally real fields

Let F be a field of characteristic not 2, and assume that F has finite reduced stability. Let K/F be any finite extension. We prove that if the general u-invariant u(F) is finite, then u(K) is finite. This article is based on part of the author’s Ph.D. thesis, written under the supervision of Richard Elman.     

Higher relative class number formulae

Let E be a totally complex abelian number field with maximal real subfield F, and let denote the non-trivial character of . Similar to the classical case n=1 the value of the Artin L-function at for odd is given by a relative class number formula of the form Here is a higher Q-index, which is equal to 1 or 2 and is a higher relative class number. Here for any number field L the higher class number is the order of the finite group closely related to the order of the higher K-theory group of the ring of integers in L. Received: 4 June 1999 / Revised version: 27 September 2001 / Published online: 26 April 2002     

Curves with a prescribed number of rational points

We show that for any finite field Fq, any N?0 and all sufficiently large integers g there exist curves over Fq of genus g having exactly N rational points.     

The local root number of elliptic curves with wild ramification

Let E be an elliptic curve over a number field F. The root number is conjecturally the sign of the functional equation of L-function of E/F. It is defined as the product of local signs over all places of F. The purpose of this paper is to describe this local sign by the coefficients of a Weierstra? equation of E. Received: 31 March 2000 / Revised version: 22 November 2001 / Published online: 23 May 2002     

Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

Let F be a finite extension of ℚ _p . For each integer n≥1, we construct a bijection from the set ?_F ⁰ (n) of isomorphism classes of irreducible degree n representations of the (absolute) Weil group of F, onto the set ? _F ⁰ (n) of isomorphism classes of smooth irreducible supercuspidal representations of GL _n (F). Those bijections preserve epsilon factors for pairs and hence we obtain a proof of the Langlands conjectures for GL _n over F, which is more direct than Harris and Taylor’s. Our approach is global, and analogous to the derivation of local class field theory from global class field theory. We start with a result of Kottwitz and Clozel on the good reduction of some Shimura varieties and we use a trick of Harris, who constructs non-Galois automorphic induction in certain cases. Oblatum 1-III-1999 & 21-VII-1999 / Published online: 29 November 1999     

The distribution of class groups of function fields

For any sufficiently general family of curves over a finite field F_q and any elementary abelian ?-group H with ? relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[?](F_q)≅H. In doing so, we prove a conjecture of Friedman and Washington.     

Torsion points on elliptic curves over function fields and a theorem of Igusa

If F   is a global function field of characteristic p>3$p>3$, we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a fixed finite set of places of F. We end the paper with an application to torsion points rational over abelian extensions of F.     

Gauss‐Green theorem for weakly differentiable vector fields,sets of finite perimeter,and balance laws

We analyze a class of weakly differentiable vector fields F : ?ⁿ → ?ⁿ with the property that F ∈ L^∞ and div F is a (signed) Radon measure. These fields are called bounded divergence‐measure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence‐measure field F over the boundary of an arbitrary set of finite perimeter that ensures the validity of the Gauss‐Green theorem. To achieve this, we first establish a fundamental approximation theorem which states that, given a (signed) Radon measure μ that is absolutely continuous with respect to ??^{N ? 1} on ?^N, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure‐theoretic interior of the set with respect to the measure ||μ||, the total variation measure. We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter E as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary so that the Gauss‐Green theorem for F holds on E. With these results, we analyze the Cauchy flux that is bounded by a nonnegative Radon measure over any oriented surface (i.e., an (N ? 1)‐dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of the balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure‐valued source terms from the formulation of the balance law. This framework also allows the recovery of Cauchy entropy flux through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. © 2008 Wiley Periodicals, Inc.     

The minimal polynomial of a matrix ∗

Let Rbe a finite dimensional central simple algebra over a field F A be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A (λ) of A over F. By using q_A (λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

Conditions on decomposable symmetric tensors as an algebraic variety

Let V be a finite dimensional vector space of dimension at least 2 over an infinite field F. We show that the set of all decomposable elements in the rth symmetric product space over i:V(r≥ 2) is an algebraic set if F is algebraically closed and only if every polynomial of degree at most r splits completcly over F.     

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.     

Maximal class numbers of CM number fields

Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis and Artin's conjecture on the entirety of Artin L-functions, we derive an upper bound (in terms of the discriminant) on the class number of any CM number field with maximal real subfield F. This bound is a refinement of a bound established by Duke in 2001. Under the same hypotheses, we go on to prove that there exist infinitely many CM-extensions of F whose class numbers essentially meet this improved bound and whose Galois groups are as large as possible.     

Determination of conductors from Galois module structure

Let E denote an unramified extension of , and set for an odd prime p and . We determine the conductors of the Kummer extensions of F by those elements such that is Galois. This follows from a comparison of the Galois module structure of with the unit filtration of F. Received: 28 August 2000; in final form: 11 October 2001 / Published online: 4 April 2002     

Writing representations over minimal fields

The chief aim of this paper is to describe a procedure which, given a d-dimensional absolutely irreducible matrix representation of a finite group over a finite field E, produces an equivalent representation such that all matrix entries lie in a subfield F of E which is as small as possible. The algorithm relies on a matrix version of Hilbert's Theorem 90, and is probabilistic with expected running time O(|E:F|d³) when |F| is bounded. Using similar methods we then describe an algorithm which takes as input a prime number and a power-conjugate presentation for a finite soluble group, and as output produces a full set of absolutely irreducible representations of the group over fields whose characteristic is the specified prime, each representation being written over its minimal field.     

Identities of Group Algebras

We consider polynomial identities of group algebras over a field F of characteristic zero. We prove that any PI group algebra satisfies the same identities as a matrix algebra M _n (F ), where n is the maximal degree of finite dimensional representations of the group over algebraic extensions of F.     

On the tensor rank of the multiplication in the finite fields

First, we prove the existence of certain types of non-special divisors of degree g−1 in the algebraic function fields of genus g defined over Fq. Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields Fq by using Shimura and modular curves defined over Fq. From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields Fq, notably in the case of F₂. These upper bounds attain the best asymptotic upper bounds of Shparlinski-Tsfasman-Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in: Lecture Notes in Math., vol. 1518, Springer-Verlag, Berlin, 1992, pp. 145-169].     

Torsion Elements in Incidence Algebras

Abstract

Let I(X, F) denote the incidence algebra of the locally finite partially ordered set, X, defined over the field, F. This paper considers when the set of torsion elements of I(X, F) forms a group. If F is of finite characteristic, it is shown that the torsion elements of I(X, F) form a group if and only if X is bounded, while if F is of characteristic 0, the torsion elements of the incidence algebra form a group if and only if X is an antichain.