# Research

Please feel free to email me about preprints for any of the topics on this page, they are in various stages of completion but most of them are close to finished.

## Polysymmetric functions and topological invariants of the space of irreducible polynomials

The classical ring of symmetric functions in infinitely many variables, $\Lambda$, is a very intensely studied object and carries an amazing amount of structure (including a collection of important bases, an inner product and a lambda ring structure). In joint work with Andrew O’Desky, we generalize the classical ring of symmetric functions to the context of weighted variables and we call such functions polysymmetric functions and the ring $P\Lambda$.

The classical ring $\Lambda$ and its lambda ring structure is very relevant and useful when we deal with symmetric monoidal categories such as the category of representations of a group. Analogously, the graded lambda ring seems to be very useful when we are dealing with a “graded symmetric monoidal ring” such the category of representations of a group on a graded vector space. In fact, our construction of the ring of polysymmetric functions even clarifies certain aspects of the classical theory such as certain dualities.

We then apply this theory to computing topological and arithmetic invariants of the space of irreducible polynomials in $n$ variables of degree $d$. This space has been recently considered by a few different authors (including Vakil-Wood and Trevor Hyde). Our definition of $P\Lambda$ offers a clarifying framework for considering such questions. We construct a map $P\Lambda \to K_0(\mathrm{var})$ to the Grothendieck ring of varieties that sends special basis in $P\Lambda$ to relevant classes in the Grothendieck ring. This map preserves a lot of structure and helps us compute many invariants of the space of irreducible polynomials, generalizing and extending previous work.

## Unlikely intersections of general subvarieties

There has been a lot of recent work on proving that unlikely intersections on moduli spaces nevertheless happen infinitely often. For instance, any two distinct elliptic curves over a global field only have finitely many places at which their reductions are isomorphic. Nevertheless, there are infinitely many places at which the reductions are isogenous (proven by ChaiOort over positive characteristic and François Charles in the number field setting.

More recently, Ananth Shankar and Yunqing Tang have a series of papers (with some collabarators) where they tackle higher dimensional versions of this question (concerning isogenies between Abelian varieties and K3 surfaces). These results are generally restricted to studying the intersection of a special divisor with a curve on a moduli space (that is a Shimura variety).

In joint work with Qiao He and Ananth Shankar, we extend such results to the intersection (up to Hecke translates) of two arbitrary curves in a Hilbert Modular Surface over a finite field (with some restrictions on the characteristic). Since we work with arbitrary curves, we cannot use the same techniques as the earlier papers mentioned around this topic and we have to use quite different methods. Our methods also lets us compute the asymptotic behaviour of the change in height under an isogeny for Abelian surfaces associated to the Hilbert Modular Surfaces that we consider.

In a different paper, I answer a question of Shou-Wu Zhang about the intersection of arbitrary subvarieties inside the $n$-fold product of the modular curve. I prove that any two subvarieties of $X(1)^n$ of complementary dimension (under mild, necessary conditions) intersect infinitely often up to isogeny. I also show that there exist two curves in $X(1)^n$ (for $n$ arbitrarily large) with infinitely many common isogenous points. As far as I am aware, this is the first work proving that there are infinitely many unlikely intersections in very high codimension.

## Non abelian Iwasawa theory in the function field context

On the variation of the Frobenius in a non abelian Iwasawa tower:

Given a (smooth, proper) variety $X/\mathbb F_q$, the eigenvalues of the Frobenius on the (etale) cohomology carry a lot of precise arithmetic and geometric information. These eigenvalues are algebraic integers and the proof of the Weil Conjectures by Deligne affords us a good understanding of the absolute values of these eigenvalues (under any complex embedding). The precise value is a lot more mysterious and much less understood.

The etale cohomology of a curve is in some ways analogous to the class numbers of number fields and Iwasawa realized that it is easier to study the asymptotic behaviour of these class numbers in certain families than any individual class number. Taking a cue from Iwasawa theory, I study the variation of these eigenvalues in families of curves similar to $y^2 = f(x^{\ell^n} )$ for $\ell$ a prime different from the characteristic of the base field and $f$ a polynomial.

When $f$ is a linear polynomial, I show that the eigenvalues for all $n$ are determined by the eigenvalues for finitely many $n$ and I give precise formulas for what the eigenvalues can be. These results follow from old results of Coleman on the Gross-Kubota Gamma function because the eigenvalues turn out to be Gauss and Jacobi sums in this case.

For a general polynomial $f$, the eigenvalues in general do not seem to be determined by finitely many $n$ but nevertheless I show that there is a strong $\ell$-adic convergence of the eigenvalues. The proofs here involve interpreting the cohomology as a finitely generated module over a “non-commutative” or “twisted” Iwasawa algebra and proving that certain invariants of such modules converge asymptotically.