VT Algebra Seminar — Fall 2025

(Joint work in progress with Ki Fung Chan, Kwokwai Chan and Chin Hang Eddie Lam) I will discuss a module action of the trigonometric degeneration of DAHA on the equivariant quantum cohomology of $T^*G/P$. It is constructed using Gromov-Witten invariants and can be described explicitly in terms of the stable envelopes. I will also discuss its relation to the “quantum equals affine” phenomenon.

Certain isogeny-based cryptographic protocols rely on a class group action. We study a large family of generalized class groups of imaginary quadratic orders . We prove that they act freely and (essentially) transitively on the set of primitively O-oriented elliptic curves over a field k equipped with appropriate level structure. This extends, in several ways, a recent observation due to Galbraith, Perrin, and Voloch for the ray class group. We show that this leads to a reinterpretation of the action of the class group of a suborder O’ of O on the set of O’-oriented elliptic curves and discuss several other examples. This is joint work with W. Castryck, J. Komada Eriksen, G. Lorenzon, and F. Vercauteren, with ongoing follow-up work joint with J. Macula and E. Orvis.

In [1] it was shown that the minimal genus of irreducible curves lying on an abelian surface influences the bound on the minimum distance of algebraic geometry codes constructed from it: the highest the minimal genus, the better the bound. In this talk, we will study abelian surfaces defined over finite fields containing no irreducible curves of genus less than or equal to 3. Our starting point will be the characterisation of isogeny classes of abelian surfaces without curves of genus less than or equal to 2, initiated in [1] and completed in [2]. We will then show that, for simple abelian surfaces, containing a curve of genus 3 is equivalent to admitting a polarisation of degree 4. Hence, we will characterise the isogeny classes of abelian surfaces without curves of genus ≤ 2 containing no abelian surface with a polarisation of degree 4, using the tools developed in [3,4]. Finally, if time permits, we will describe absolutely irreducible curves of genus 3 lying on abelian surfaces containing no curves of genus less than or equal to 2, and show that they have few rational points.

Bibliography:

[1] Y. Aubry, E. Berardini, F. Herbaut, and M. Perret, Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus, Finite Fields and Their Applications, 70 (2021), p. 101791.

[2] E. Berardini, A. J. Giangreco-Maidana, S. Marseglia, Abelian surfaces over finite fields containing no curves of genus 3 or less, arXiv preprint 2408.02493v2

[3] E. W. Howe, Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc., 347 (1995), pp. 2361--2401.

[4] E. W. Howe, Kernels of polarizations of abelian varieties over finite fields, J. Algebraic Geom., 5 (1996), pp.583--608.

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