Over the last decade, the Iwasawa theory of non-ordinary modular forms of weight 2 has seen a surge of results which has brought the theory more on par with the well-established ordinary case. However, in weights greater than 2, little is known in the non-ordinary case about even the most fundamental questions on Selmer groups and L-values. The most serious obstacles to progress along these lines are (1) the wild behavior of non-ordinary p-adic L-functions, and (2) the complicated nature of the local conditions at p which arise. To circumvent the first of these obstacles, Pollack seeks to make a systematic study of the Iwasawa theory of non-ordinary modular forms at each finite level of the cyclotomic Zp-extension. To deal with the local obstacle, he aims to use the p-adic local Langlands correspondence to control the local structures which arise, and to then formulate an algebraic theory of theta-elements which is the analogue of the analytically defined Mazur-Tate elements. Pollack aims to prove a series of theorems which show that these algebraically defined elements control the size and structure of Selmer groups, and to show that the main conjecture is equivalent to the equality of these elements with Mazur-Tate elements. One particular goal of this program is to combine results of Kato and the theory of algebraic theta-elements to prove a conjecture of Mazur and Tate which asserts that their analytic theta-element lies in the Fitting ideal of a certain dual Selmer group.
The motivation for the study of this project comes from the theory of elliptic curves which are certain mathematical objects whose points have the shape of a doughnut. Elliptic curves, once the focus of study of only pure mathematicians, have now become ubiquitous in both the theory and practice of cryptography. Further, as a result of the breakthrough proof of Fermat's Last Theorem by Andrew Wiles in the mid 90s, we now know that elliptic curves are intimately connected to modular forms which are functions of a complex variable with many many symmetries. Wiles' theorem essentially states that one can make a precise dictionary between elliptic curves over the rational numbers (which are geometric objects) and certain modular forms of weight 2 (which are calculus-type objects). This project pushes out beyond the case of weight 2 modular forms, and seeks to make a systematic study of certain properties of arbitrary weight modular forms. It remains to be seen if these higher weight modular forms (which can be thought of as generalized elliptic curves) are also highly important from a cryptographic viewpoint.
An elliptic curve is the set of solutions of an equation of the form y2 = x3+ax+b. For instance, consider y2 = x3-2, and one may ask what numbers x and y satisfy this equation? If one lets x and y run over all numbers (even complex numbers), the set of solutions looks like a torus (i.e. a doughnut). However, if we limit ourselves to x and y that are whole numbers or ratios of whole numbers, then it can be very difficult to get a handle on what the solutions are. In the case of y2 = x3-2, we have x=3 and y=5 is a solution since 33-2 = 25 = 52. But x= 164323/29241 and y=-66234835/5000211 is also a solution! And for this curve one can find fractional solutions with as many digits as you like in the numerators and denominators. A natural question to ask then is how can one find all such solutions? In the 1960s, Birch and Swinnerton-Dyer formulate an incredible conjecture (a guess) on how to get a handle on these solutions using certain complex analytic functions (L-series). That is, they proposed using a theory akin to calculus to solving algebraic questions. This was an extraordinarily novel idea, and had major ramifications across many fields in number field. In this NSF project, I examined an analgous question which also compares algebraic information to analytic information. In my context, I studied Selmer groups of modular forms (which can be viewed as generalizations of elliptic curves) and their associated Fitting ideals (an algebraic invariant). On the other hand, I also looked at Mazur-Tate elements of modular forms (an analytic invariant). Mazur and Tate conjectured a relationship between these two invariants (in the spirit of the Birch and Swinnerton-Dyer conjecture). One of the key outcomes of this project was establishing this conjecture for non-ordinary forms in certain cases using the (relatively new) p-adic local Langlands correspondence. This gives one example of a connection between p-adic local Langlands and Iwasawa theory, and I expect that there will be many more such connections discovered in the years to come. Other key outcomes of this project included me giving a number of lecture series (e.g. at the Arizona Winter School and at Heidelberg University) where hundreds of graduates students were exposed to many new mathematical ideas pitched at an accesible level. Further, I taught a class of high students at the PROMYS program a course on representation theory giving them a chance to be exposed to upper university mathematics while still in high school.
