Medvedovsky's NSF postdoctoral fellowship supported her work on modular forms, Hecke algebras, and Galois representations mod p in 2016-2021, first at the Max Planck Institute in Bonn, Germany, and then at Boston Univeristy in Boston, MA.

Classical modular forms are complex-valued arithmetic functions with a lot of symmetries, which beautifully capture a lot of number-theoretic phenomena. In particular, it is expected that a large class of 2-dimensional symmetries of the rational numbers (that is, representations of their absolute Galois group) come from modular forms in a way that is well understood, as described by a small part of the so-called Langlands program. Hecke algebras are an algebraic tool for studying spaces of modular forms. Medvedovsky's research concerned characteristic-p versions of all of these objects, a way of keeping track of their congruences modulo a prime p. 

During this time period of this fellowship, Medvedovsky worked on a number of projects, a few of which are described below, with emphasis on their intellectual merit. 

What has turned into the flagship project -- entirely conceived, researched, written, and published during the life of the fellowship -- concerns the size of the image of a two-dimenasional p-adic (that is, capturing all p-power congruences) representation of a profinite group (such as the Galois group of a number field, which captures all symmetries of all algebraic overfields). Under some mild conditions, Medvedovsky and her collaborators showed that these images are as big as they possibly can be given their own internal symmetries. These purely algebraic results, building on work of Pink and Bellaiche, recover and extend all p-adic cases of known big-image theorems about Galois representations coming from modular forms, which have been proved using arithmetic methods since the 1970s by Serre, Ribet, Momose, Nekovar, Hida, Lang, Conti, Iovita, and Tilouine; Lang and Conti are collaborators on this project. One can hope that these results point the way to the underlying algebraic nature to the big-image results of the future. For more on this project, please see https://arxiv.org/pdf/1904.10519.pdf. 

Another project, building on results from Medvedovsky's Ph.D. thesis about mod-p Hecke algebras, is about the mod-3 distribution of the coefficients of some modular forms. Modular forms, the functions with many symmetries, have a representation as a power series -- an infinite polynomial -- whose coefficients encode certain arithmetic properties. One can then ask how these coefficients are distributed modulo primes. Medvedovsky, inspired by work of Nicolas, Serre, and her Ph.D. advisor Bellaiche for the prime 2, has built a completely explicit universal Galois representation on the mod-3 Hecke algebra, has conjectured equidistribution for mod-3 modular forms outside of a thin set of exceptional forms, and has proved precise distribution formulas for many of those exceptional forms. Most recently, she has applied other work of Bellaiche to prove partial equidistribution results. This work is not yet published; a partial preprint is available on Medvedovsky's web site at BU (https://math.bu.edu/people/medved/Mathwriting/Mod3galoisrep.pdf). One has some modest hope that because of the relation of the generating function for the partition function to modular forms, mod-3 equidistribution of certain modular forms could point the way to something like mod-3 equidistribution of the partion function, though this pathway is not yet clear. 

The third project that we will mention here is a collaboration that began under the auspices of this fellowship and is now nearing its first publications. Medvedovsky and her collaborators managed to count modular forms separated by their mod-p Galois representation and the eigenvalue (+1 or -1) of an involution, the at-p Atkin-Lehner involution that helps organize modular forms into "old" and "new" forms, according to exactly how many symmetries they have. They have obtained very satisfying recursive formulas that generalize both earlier results that only keep track of the mod-p Galois representation (of Bergdall and Pollack) and earlier results that only keep track of the +1/-1 eigenvalue (of Fricke, Yamauchi, Wakatsuki, and Martin). To achieve this counting, Medvedovsky and her collaborators developed a new method for establishing isomorphisms between mod-p representations (or rather, counting mod-p eigenvalues) by finding deeper p-power congruences between traces of powers of elements. This last result is both a contribution to combinatorics and a new technique for working with mod-p representations. This work has not yet been published. 
For two additional projects of Medvedovsky supported by this fellowship, see https://www.ams.org/journals/btran/2019-06-08/S2330-0000-2019-00035-3/S2330-0000-2019-00035-3.pdf and https://msp.org/obs/2019/2-1/obs-v2-n1-p20-s.pdf. 

The broader impacts aspect of this work has been somewhat curtailed by the pandemic. Together with a collaborator, Medvedovsky begain running a math circle in a class of a local elementary school in the spring semester of 2020; sadly, the math circle had to be completely abandonded during remote schooling. For the last two years, no one except for students and staff have been allowed in school buildings.

Last Modified: 03/25/2022
Modified by: Anna Medvedovsky