CAREER: Cohomological Methods in Algebraic Geometry and Number Theory

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The investigator studies the use of p-adic analytic techniques in several aspects of arithmetic geometry. One focus is on the p-adic cohomology of algebraic varieties over finite fields, including theoretical questions like the stability of coefficient objects under cohomological operations, and computational problems like the determination of zeta functions of specific curves and surfaces. Another focus is the classification of Galois representations over local fields (p-adic Hodge theory); goals of this include gaining insight into proposed Langlands-style correspondences between Galois representations and representations of p-adic Lie groups, as well as unifying the treatment of Hodge theory in the complex and p-adic settings. These foci share common (and recently developed) technical elements from the theory of p-adic differential equations, which again have strong complex-analytic analogues.The use of p-adic analytic methods in arithmetic geometry, as in the investigator's work, has great significance within mathematics; for instance, it is currently being used to pursue generalizations of the techniques underlying the proof of Fermat's Last Theorem. But it also has surprising practical significance due to the appearance of systems of polynomial equations over finite fields (such as the integers modulo a prime number) in combinatorics and computer science. For instance, elliptic curve-based cryptography has been adopted by NIST as a standard for secure communications thanks to its balance of efficiency versus security; p-adic methods can be used to select curves suitable for this construction. Geometry over finite fields also appears in the construction of many error-correcting codes; p-adic methods can be deployed to search for codes which correct transmission errors without carrying too much overhead. So far these applications rely on proven statements in the theory of p-adic cohomology, but the theory is somewhat unfinished and it seems likely that future applications will rely for their provable correctness on statements not yet proved. It is thus important to pursue theoretical and practical aspects in parallel.