Topics in p-adic analysis

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Johns Hopkins University
Since their introduction by Hensel more than 100 years ago, the p-adic numbers have been playing a fundamental role in mathematics, number theory in particular, and nowadays it is hardly an exaggeration to contend that they are no less important than real and complex counterparts, due to its wide range of connections with other branches like rigid analytic geometry, p-adic Hodge theory, Iwasawa theory, and so on. In this dissertation, we present a total of 3 topics that crucially employ ideas and techniques from p-adic analysis. More precisely, these topics are: - Sum expressions for totally real p-adic L-functions: In Chapter 2 and 3, we establish the infinite sum expressions of totally real p-adic L-functions, generalizing the earlier works of Delbourgo. This uses crucially the computational tool provided by the Amice-Mazur p-adic Fourier transform. As applications, we present a generalization of the Ferrero-Greenberg formula in the totally real setting, as well as a numerical criterion of the vanishing of the mu-invariant à la Iwasawa--Ferrero. - On the BDP Iwasawa main conjecture for modular forms: Under the Heegner hypothesis, Kobayashi--Ota showed that one inclusion of the Iwasawa main conjecture involving the Bertolini--Darmon--Prasanna p-adic L-function holds after tensoring by Q_p. In Chapter 4, under certain hypotheses, we present the joint work with Antonio Lei, in which we show that the same inclusion holds integrally. This uses several modern developments in p-adic analysis, including Perrin-Riou's exponential/logarithm map, Pollack-Sprung's factorization into Coleman maps and its formulation in terms of p-adic Hodge theory by Lei and collaborators. - Note on p-adic local functional equation: Given distinct primes l, p, in the first part of Chapter 5 we record a p-adic valued Fourier theory on a local field K over Q_l, which is developed under the perspective of group schemes. The second part studies the Haar measure of K and befitting Schwartz class functions, and their zeta integrals in a rigid analytic setting. Eventually, following Tate’s original argument, it proves a p-adic local functional equation over K that uncannily resembles the complex one found in Tate thesis.
p-adic analysis, p-adic L-functions, Iwasawa theory