Computational Arithmetic Geometry

Recent Developments in Computational Arithmetic Geometry

From 16.05.04 to 21.05.04, the Dagstuhl Seminar 04211 Algorithms and Number Theory was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and i...

Computational Arithmetic Geometry I. Sentences Nearly in the Polynomial Hierarchy

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: I. Given a polynomial f ∈Z[v, x, y], decide the sentence ∃v ∀x ∃y f(v, x, y) ? =0, with all three quantifiers ranging over N (or Z). II. Given polynomials f1, . . . , fm∈Z[x1, . . . , xn] with m≥n, decide if there is a rational solution to f1= · · · =fn = 0...

Arithmetic in Geometry

I thank Dean Martin for his very flattering introduction. I thank the College for bestowing a chair to me and the Mathematics Department for providing a wonderful atmosphere for my work. I thank Don Lewis and my current and former colleagues for their support. I take this opportunity to also thank my family, friends, and collaborators. My mathematical horizon has widened considerably after join...

Arithmetic Algebraic Geometry

[3] , Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Finiteness results for modular curves of genus at least 2, Amer.

Arithmetic and Diophantine Geometry