Relating Special Values of L-Functions with Orders of Tate-Shafarevich Groups

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Finding rational numbers that solve a polynomial equation is harder than finding solutions that are real numbers. This is because the real numbers form a number system called a completion of the rational numbers, and the solutions in a completion can usually be found more easily. One may hope to find the hard rational number solutions by scrutinizing the easier solutions living in the completions, but there may be a discrepancy between these two types of solutions. One central object in number theory designed to measure such a discrepancy is the Tate-Shafarevich group, and this group is only understood in some special cases. Two central conjectures (the Birch and Swinnerton-Dyer conjecture and the Bloch-Kato conjecture) relate sizes of Tate-Shafarevich groups to special values of an appropriate function. This project outlines some progress on these conjectures via a theory called Iwasawa theory. Another goal is to understand the interplay between the special values from the perspective of a modern form of analysis, called p-adic analysis, and use this interplay to give an easy explanation of complicated phenomena of certain crystalline representations.The Iwasawa main conjecture for elliptic curves at supersingular primes was proved by Wan in the case in which the trace of Frobenius vanishes, and the PI in the general supersingular case. The plan is to extend this work to more general modular forms. One central idea in the supersingular case is the construction of two appropriate p-adic power series, which is known explicitly by work of Pollack when the Frobenius trace is zero. The PI will work with Otsuki to explicitly construct the appropriate pair of p-adic power series in more general cases by developing the analytic aspect of supersingular Iwasawa theory further. One goal of such an explicit construction is determining reductions of crystalline Galois representations, shedding light on some conjectures of Breuil which describe these reductions purely in terms of the Frobenius trace and its Hodge-Tate weight. Another goal is to establish asymptotic formulas for the size of the p-primary components of Tate-Shafarevich groups as defined by Bloch and Kato, generalizing work of Lei, Loeffler, and Zerbes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.