Number Theory Problems Over Local Fields and Function Fields

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DMS-0600919Brian ConradThe PI proposes to work on some geometric questions that arise in number-theoretic questions over global function fields and p-adic fields. In the theory of overconvergent classical modular forms, the canonical subgroup of Lubin and Katz (for a p-adic analytic family of elliptic curves) has been an important tool. The recent interest in p-adic modular forms beyond the classical case has motivated several different constructions of canonical subgroups for abelian varieties,and recent work of the PI has led to another higher-dimensional theory that, at least in its geometric aspects, has a wider range of applicability than the other approaches. The PI proposes to use deformation-theoretic methods to make explicit certain abstract estimates in the theory, and to draw arithmetic consequences for p-adic families of automorphic forms.In another direction, the PI and co-workers have used methods from deformation theory and rigid-analytic geometry to develop a theory of a new global parity obstruction to randomness properties of prime specialization of inseparable irreducible polynomials overcertain coordinate rings of curves over finite fields. This has given rise to families of elliptic curves with unexpected rank behaviorover global function fields (under some standard conjectures) and seems likely to have further Diophantine applicationsto Mordell--Weil ranks for families of abelian varieties over global function fields. The PI proposes to develop a betterunderstanding of this basic arithmetic phenomenon and to work out some further consequences for ranks in families.Number theory is among the oldest subjects in mathematics, since it is ultimately concerned with the relatively concreteproblem of studying properties of whole numbers. Such problems can take on the form of finding whole number solutionsto systems of equations in many variables, or problems relating to properties of primes, and so on. Typically one has to bring in some deeper structural information (provided by geometric, algebraic, or analytic ideas) to make progress on such questions, and in the last several decades a very wide range of sophisticated geometric techniques have been developed to attack these problems. Moreover, though the initial motivation comes from within pure mathematics, security ofelectronic telecommunications has come to be related in an essential way to many of these number-theoretic problems (such asprime factorization and studying points on curves over finite fields) and the geometric concepts used to attack them. This proposal focuseson both geometric and prime factorization questions that arise in several number-theoretic settings, aiming to develop somenew theoretical methods and to apply them to specific problems. In addition, the PI proposes to continue his long-standing tradition of supervising high-level number theory research by talented high school students and giving talks to larger groups of students at the high school, undergraduate, and graduate levels. He will also complete two books that, together with assorted freely available notes already posted on his web site, will provide useful references for graduate students who wish to learn about some fundamental topics in arithmetic geometry.