Abstract: On average there are $q^r+O(q^{(r-1)/2+\epsilon})$ ${\bf F}_q$-rational points on a curve of genus $g$ defined over ${\bf F}_q$ if $r$ is odd or $r>2g$, but if $r$ is even and $r\le 2g$ there are $q^r+q^{r/2}+O(q^{(r-1)/2+\epsilon})$ ${\bf F}_q$-rational points on average. Of course these facts can be interpreted as statements about the moments of the roots of the Weil polynomial. Based on computations we conjecture precise formulas for the moments and product moments associated to the submoduli of hyperelliptic curves. In some cases we can prove our formulas. (The work is joint with Andrew Granville.)
Abstract: Given $x,y \in G$ an (additive) abelian group and $y \in \left< x \right>$ to compute $n \in {\bf{Z}}$ such that $y = [n]x$ is called the discrete logarithm problem. The supposed computational intractability of this problem in certain groups forms the core of many cryptographic systems. We provide efficient algorithms to compute discrete logarithms in certain formal groups and show how in particular this helps us to compute discrete logarithms in elliptic curves over local fields. Finally we provide some cryptographic applications.
Abstract: In this talk we will treat the classification of curves of genus 2 over arbitrary fields of any characteristic, including the case of characteristic 2. We will start by considering the moduli space M_2 that classifies curves of genus 2 up to isomorphisms defined over an algebraically closed field, in order to show the relationship between field of definition and field of moduli. We will give explicit representatives for moduli points corresponding to curves with non-trivial reduced group of automorphisms. Then, we will show how one can obtain the classification modulo isomorphisms over an arbitrary field via the study of certain cohomology sets. This will allow us to describe the isomorphism classes of curves of genus 2 in terms of field extensions. As an application, we will give formulas for the number of curves of genus 2 over arbitrary finite fields.
Abstract: The field of differential algebraic geometry results from the expansion of classical algebraic geometry to include algebraic differential equations and arithmetic analogs of algebraic differential equations. Specifically, given an algebraic variety $X/R$ and a differential operator $\delta :R \to R$, it is possible to construct a prolongation sequence of varieties $. . . \to \hat X^2 \to \hat X^1 \to \hat X^0 = \hat X$ compatible with the differential operator. We will discuss the explicit computation of some key objects connected with this construction.
Abstract: If an algebraic curve over Q is birationally equivalent over Q to the projective line, then such equivalence (a parametrization) can be found with existing computer implementations. However, the rational number coefficients in this parametrization often have much more digits than necessary. The problem we study is how to find a Moebius transformation that turns this parametrization into one with smaller coefficients.
Abstract: Let N denote the set of natural numbers. A subset S of N is said to be avoidable if there exists a partition of N into two (nonempty) disjoint sets A and B such that no element of S is the sum of two distinct elements of either A or B. While avoidable sets in N have been studied for some time, not many families of such sets are known. To date, the Fibonacci and Tribonacci sequences have been categorized.
Now let N(n) denote the set of the first n positive integers. We define the following function. Given positive integers k and n, each at least 3 and with n at least k, let U(k,n) denote the number of subsets S of N(n) of cardinality k that are unavoidable. We restrict k and n to be at least three because all singleton and size-two sets are avoidable. In this talk, we shall present a recursion algorithm to calculate U(3,n) efficiently, and we shall give a nontrivial lower bound for U(k,n) when k is at least 4.
We shall also discuss the following problem. Given an avoidable set S with maximum value u, what is the smallest number b greater than u such that S unioned with the singleton set consisting of the element b is also avoidable? Beyond the trivial observations that u+1 is at most b, which is in turn at most 2u-3, it is not clear what values b can take on in general. Parts of this work were done jointly with Patrick Mitchell of Midwestern State University.
Abstract: A smooth projective algebraic curve C is a Riemann surface, and hence can be represented as C=D/G where D is an open set in the finite complex plane, and G is a group of Möbius transformations acting on D. We will discuss algorithms and their implementations which allow one to approximate the domain D and the generators of G once C is given. We will also discuss the extensions of these algorithms to the case of stable curves (with double points).
Abstract: Let X be a genus g algebraic curve defined over an algebraically closed field k of characteristic zero. Determining the automorphism group Aut (X) of this curve, the field of moduli, and the minimal field of definition are some of the problems of classical algebraic geometry. We will give a survey of some of the techniques used in studying these problems. Further, we will discuss some of these problems from a computational viewpoint and show how GAP, MAGMA, MAPLE are used in the study of these problems.
Abstract: The existence of a non-trivial automorphism group on a compact Riemann surface leads to vanishings of Riemann's theta function. Accola showed that in some cases the converse is true. In particular, certain vanishings of $\theta$ at quarter-periods imply the existence of an involution in the automorphism group. This talk will discuss these vanishings, with special attention to the case when the existence of two or more involutions is known. It will show how to use the vanishings to determine the order of the dihedral group generated by two corresponding involutions; this is topological in nature. Then, group-theoretic arguments show how to use theta-vanishings to find equations for the locus of surfaces of genus three with {\sl given\/} automorphism group, in the case when the group is nonabelian and generated by its involutions.
Abstract: The talk will consider special loci in moduli space of curves of genus g with n marked points. Special loci in this space parameterize marked curves with extra automorphisms. A typical curve with marked points has no automorphisms; but some do, depending upon the choice of curves and position of marked points. This gives us certain subvarieties in the moduli space. For Riemann surfaces, these subvarieties are characterized by specifying a finite group of mapping-classes whose action on a curve is fixed topologically. Schneps considered the situation of genus 0 with n marked points, and genus 1 with n=1 or 2 marked points, corresponding to the curves having a cyclic group in its automorphism group, over the complex numbers. After reviewing that work, the talk will discuss more general cases in higher genus and in characteristic p.
Abstract: Let X be a proper, smooth, geometrically connected curve over any field k and J(X) be the jacobian variety of X. We say that J(X) is completely decomposable over k if J(X) is isogenous to the product of elliptic curves over k. In general, it seems to be difficult to find precise informations of factors of J(X) . Therefore, we restrict our attention to modular curves of type X_0(N). Then as a result, we give all positive integers N for which J_0(N) is completely decomposable over the rational number field. Let H is a subgroup of the group generated by Atkin-Lehner involutions with respect to N. We can determine all pairs (N,H) for which the jacobian variety of a quotient modular curve X_0(N)/H is completely decomposable. Furthermore we explore some open problems and conjectures in this area.