Number Theory

Number Theory Seminar

Schedule 2026/27

Abstracts 2026/27

Ilan Ehrlich

Classical modular polynomials classify pairs of \(\ell\)-isogenous elliptic curves via their \(j\)-invariant. Following work of Atkin, Morain, Enge and others, it has been observed experimentally that using invariants corresponding to richer level structures, such as the Weber functions, gives rise to modular polynomials of significantly smaller height. We provide a proof of this phenomenon: given an invariant on a genus 0 modular curve which is a degree-\(d\) cover of \(X(1)\), we show that the height of the associated \(\ell\)-modular polynomial is bounded above by \(\frac{6}{d} \ell \log \ell + O(\ell \sqrt{\log \ell})\). Along the way, we recast the construction of such modular polynomials within the modern framework of modular curves.

Emanuele Pavia

Supergeometry is the natural mathematical framework that accommodates the notion of supersymmetry in theoretical physics and string theory. In the last dozen years, this field of research has found renewed popularity after the foundational work of Donagi and Witten, who proved that super moduli spaces of super Riemann surfaces cannot be straightforwardly recovered and studied starting from their bosonic ( = classical) reductions.
The goal of this talk is twofold. First, I shall briefly revisit the theory and the technical machinery underlying the field of supergeometry (that is, \(\mathbb Z/2 \mathbb Z\)-graded commutative algebra with a Koszul twist in the commutativity constraint), describing how notions from classical algebraic geometry can be lifted and generalized to the super-setting. Secondly, I will introduce the moduli spaces ( = superstack) of (pre-)stable supercurves and of stable supermaps, which provide superenhancements of the usual moduli spaces of (pre-)stable Riemann surfaces and stable maps with spin structures, and explain what we know about them and why we care.
This is based on joint work with Ugo Bruzzo, Daniel Hernández Ruipérez, and Andrea Ricolf.