Congruences between modular forms, Galois representations, and arithmetic consequences

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A major theme in modern mathematics is that it is useful to study how objects can be changed or deformed. For instance, the value of a function at a point is just one number, but knowledge of how a (continuous) function behaves in a tiny neighborhood of that point gives qualitative information like whether the function is increasing or decreasing and how fast. This requires a notion of distance so that one can talk about perturbing an object “a little bit”. In number theory, one often replaces the usual notion of distance with a “p-adic” distance — one that measures how divisible a number is by a given prime number p. The PI will investigate the p-adic variation of “modular forms” — special functions whose graphs have many symmetries. Studying variation of modular forms is a topic in number theory that has led to many deep results such as the proof of Fermat’s Last Theorem. The PI and her collaborators have found new directions of p-adic variation that have yet to be explored and promise to yield new insights about arithmetic. This investigation will incorporate young scientists at many career stages as the PI mentors high school, undergraduate, and graduate students and organizes conferences at both a regional and international level. On a more technical level, there are three overarching topics to be studied. The first two concern congruences between modular forms, first in the “vexing” setting and then Eisenstein congruences. The final topic concerns images of Galois representations. Each topic contains multiple projects, both addressing the topic directly as well as arithmetic consequences. One new direction of (tame) p-adic variation referenced above occurs in the vexing setting, where the PI and her collaborators can prove a structure result on the relevant Hecke algebra. A deeper study of this phenomenon will allow her to explore consequences for tame analogues of Iwasawa theory and the Bloch-Kato Conjectures. Within the topic of Eisenstein congruences, the PI will explore the structure of pseudodeformation rings and Hecke algebras in weight 2 and prime-square level. This has natural applications to the p-rank of the class group of the field cut out by a p-th root of an integer. Finally, she will use her recent work with Conti and Medvedovsky on images of 2-dimensional Galois representations to understand images of finite products of such representations, doing this in a way that characterizes the availability of elements needed for Euler system machinery.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.