Number Theory and Algebraic Geometry

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Abstract Noam Elkies The investigator studies problems in number theory, notablythe arithmetic of varieties such as elliptic curves and surfacesand their relations and applications to other areas of mathematicssuch as error-correcting codes and Euclidean lattices. Specificprojects include: proof and computational exploitation of newrelations between p-adic modular forms of weight 3/2 and the arithmeticof quadratic twists of odd analytic rank; refinement of the Shioda-Usuitheory relating Mordell-Weil lattices with Weyl groups, andgeneralization to certain complex reflection groups; continued studyof optimal recursive towers and the investigator's modularity conjecturefor such towers; extended study of the connection between bilinear(Somos) recurrences, theta sequences, and explicit moduli spaces;and novel applications of lattice reduction to study and efficientlycompute solutions of Diophantine equations and inequalities.The investigator continues his study of problems in number theoryconcerning the mathematical structure of algebraic solutionsof equations. This should lead both to better understandingof those equations and their solutions (elliptic curves, etc.),and to further connections with other topics such as efficientcomputation, error-correcting codes, and sphere packing.