Arithmetic geometry is the study of number theoretic properties of geometric objects. A profitable way to study such objects is via their family of p-adic Galois representations which encode the essential arithmetic data. The local behavior of the p-adic Galois representation is especially subtle at the prime p itself. The investigator proposes to study various p-adic aspects of these p-adic Galois representations. Anticyclotomic Iwasawa theory is a subject which does by the studying the p-adic Galois representation attached to elliptic curves and modular forms not merely over the rational numbers but instead over certain p-towers of number fields. The main conjecture of Iwasawa theory gives a p-adic analytic interpretation of the arithmetic information extracted from the Galois representations. A primary goal of this project is to better understand the anticyclotomic main conjecture, especially its behavior under congruences. In addition, the investigator intends to study local properties of p-adic regulators, which are a method of constructing Galois cohomology classes for p-adic Galois representations. Finally, the investigator proposes to investigate a question of Ramakrishna on power residues of Frobenius traces of p-adic Galois representations.
Number theory, often considered the oldest mathematical discipline, has in recent times developed remarkable applications to cryptography. Many of these applications involve arithmetic geometric objects known as elliptic curves. Modular forms, the primary object of study in this proposal, are generalizations of elliptic curves which play a fundamental role in modern number theory. The questions investigated in this proposal deal with invariants which are closely related to those of interest in cryptography and may perhaps yield some insight into them.
