Algebra, Algebraic Geometry, Number Theory

• Hirotachi Abo
A central topic in algebraic geometry is the study of the common zero locus of a system of polynomial equations. Such systems arise very naturally throughout mathematics, statistics, computer science and engineering. These objects are thought of geometrically and are called algebraic varieties.

One of the guiding problems in algebraic geometry is to classify all projective varieties (i.e., common zero loci of systems of homogeneous polynomial equations) up to isomorphism. The problem in its full generality is an impossible task so attention is focused on particular open and challenging sub-problems. One such sub-problem is the classification of non-singular subvarieties of ``low" codimension in projective n-space. A basic question in the classification problem is whether for a given set of invariants the corresponding family of varieties is non-empty. Explicit constructions provide one avenue for establishing that given families are non-empty. It appears, however, that non-singular subvarieties of low codimension in projective n-space are exceedingly rare. The research will focus on the refinement of existing methods and will focus on the development of new theoretical, algorithmic, computational and experimental methods in the context of the search for special varieties and towards the solution of the classification problem of non-singular subvarieties of low codimension.

Applications have emerged in many areas in mathematics, science and engineering as a result of advances arising from computational methods in algebraic geometry. It is the expectation that techniques developed through this research will help to continue this trend.

• Monte Boisen
• Arie Bialostocki
• Ralph Neuhaus \
• Brooks Roberts
Classical modular forms are holomorphic functions on the upper half plane that transform under the action of an arithmetic subgroup of SL(2,R), such as SL(2,Z), according to a so-called factor of automorphy. Thus, they satisfy many internal symmetries, and are rather rare. Classical modular forms admit Fourier expansions, and their Fourier coefficients contain much number-theoretic information. One manifestation of this connection to number theory is the duality between modular forms and elliptic curves: there is a correspondence between classical modular newforms of level N with rational Fourier coefficients and elliptic curves defined over the rational numbers of conductor N that preserves zeta functions. This duality lies behind the solution of Fermat's Last Theorem.

Other problems in number theory involve modular forms of more than one variable. The next case in complexity after classical modular forms is that of Siegel modular forms of degree two. Siegel modular forms of degree two are holomorphic functions of three complex variables defined on the Siegel upper space of degree two. For these Siegel modular forms one again expects applications to number theoretic topics such as abelian surfaces, which are the two dimensional analogues of elliptic curves, and four dimensional Galois representations. In contrast to the case of classical modular forms, research on Siegel modular forms of degree two and their applications to number theory is at an early stage.

My work investigates Siegel modular forms of degree two from a representation-theoretic point of view. Siegel modular forms of degree two are closely related to automorphic representations of the Lie group GSp(4) over the rational numbers, while classical modular forms are connected to automorphic representations of the group GL(2) = GSp(2). In a series of papers completed in 2001, I proved a case of Arthur's conjecture about the structure of the discrete automorphic spectrum of GSp(4) over a number field. Since 2001, my work has centered on formulating and investigating the concept of newforms and level for Siegel modular forms of degree two. In joint work with Ralf Schmidt, we proved that a good theory of newforms and level, analogous to the theory for classical modular forms, does exist for Siegel modular forms of degree two with respect to the paramodular congruence subgroups. This theory is novel, and was not predicted by any conjectures. This project is described in a recently completed monograph. We are currently investigating newforms and level for other congruence subgroups.