Arithmetic Algebraic Geometry

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Abstract of DMS 0400877PI: Douglas UlmerDr. Ulmer proposes to work on three projects ultimately related to thearithmetic of abelian varieties over function fields. In the firstproject, he plans to prove that there exist abelian varieties ofarbitrary dimension whose L-functions vanish to arbitrarily high orderat the center of their critical strips. This generalizes his previouswork on elliptic curves. In the second and third projects, he plansto generalize his previous construction of local points on theuniversal elliptic curve over a modular curve to the context of anarbitrary non-isotrivial elliptic curve over a function field and tostudy the extent to which this construction may be used to constructglobal points. This research is intimately connected with theconjecture of Birch and Swinnerton-Dyer in the function field context.Dr. Ulmer works in the general area of Arithmetical AlgebraicGeometry, an area of fundamental mathematics that was previouslythought to be beyond application. However, it is now known to be ofcrucial importance to many important technologies, such as codingtheory and cryptography, which affect our everyday lives. CD and DVDplayers, mobile telephones, and secure internet communication all relyon mathematics originally created in the pursuit ofquestions in arithmetical algebraic geometry. The field also has deepand mysterious links to other areas of science such as quantum fieldtheory. Dr. Ulmer's work does not appear to have immediateapplication, but his knowledge of this area provides a broadperspective on modern mathematics and forms the basis for manyeducation, outreach, and training activities which he is engaged in.These include the Southwestern Center for Arithmetical AlgebraicGeometry, initiatives in training and mentoring of new graduatestudents, and numerous undergraduate research projects.