DIAMANT Symposium Autumn 2020

Algebra over FI

FI is the category of finite sets with injections. An FI-something F is a functor from FI to the category of somethings. In particular, for S in FI, the symmetric group Sym(S) acts on F(S) by something-automorphisms of F(S).

FI-somethings arise in several areas of mathematics; for instance, FI-modules arise as cohomology groups of configuration spaces of n labelled points on a given manifold (with n varying), and FI-algebras arise as coordinate rings of certain algebraic-statistical models, where the number of observed random variables varies.

Sometimes, FI-somethings inherit good properties of the category of somethings (and indeed the same applies when FI is replaced by other suitable base categories). I will present several, by now, classical examples of this phenomenon, primarily due to Church, Ellenberg, Farb, Daniel Cohen, Aschenbrenner, Hillar, Sullivant, Sam, and Snowden. For instance, a finitely generated FI-module M over a Noetherian ring is Noetherian, and if the ring is a field, then the dimension of M(S) is eventually a polynomial in |S|.

After this overview, I will zoom in on new joint work with Rob Eggermont and Azhar Farooq that says that certain contravariant functors X from FI to schemes over a Noetherian ring have the beautiful property that the number of orbits of Sym(S) on irreducible components of X(S) is a quasipolynomial in |S| for all sufficiently large S.

Factoring RSA of 240 decimal digits and computing discrete logarithms in
a 240-decimal-digit prime field with the same software and hardware

In December 2019 were announced two new record computations: the
factorization of RSA-240 (240 digits, 795 bits) and discrete logarithm
computation in a prime field of the same size, with the same software,
running on the same platforms. This is the first time that integer
factorization (IF) and discrete logarithm (DL) of the same size are
computed together. The previous RSA factorization record was in Dec.
2009 by Kleinjung et al., who factorized RSA-768 (bits, 232 decimal
digits). The previous DL record computation was in June 2016 by
Kleinjung et al., for a prime field of 768 bits: there were seven years
between RSA factorization and DL computation records of the same size,
and ten years between the two RSA factorization records.

The best known algorithm to address challenges of this size is the
Number Field Sieve, designed in the 90’s, first for integer
factorization, then adapted to discrete logarithm computation. The free
software Cado-NFS implements the NFS
algorithm, and has been developed for ten years. The same software
modules were used, with different parameters, on four different
computing resources in EU and US, to achieve the two records. Thanks to
algorithmic variants well-suited for large sizes, and fine tuning of the
parameters, the DL record was actually three times faster than expected
compared to the previous DL record, when comparing on the same hardware.
Moreover our work shows that computing a discrete logarithm is not much
harder than a factorization of the same size.

In this talk, I will present the Number Field Sieve and its variants for
IF and DL. Then I will present our algorithmic improvements, some of the
software properties, and parameter options chosen for the records.
Finally I will discuss on expectations on how the computations would
scale for larger records.

Published paper:
Boudot, Gaudry, Guillevic, Heninger, Thomé, Zimmermann, Comparing the
difficulty of factorization and discrete logarithm: a 240-digit
experiment, CRYPTO’2020,
https://crypto.iacr.org/2020/program.php#session-26 ePrint 2020/697,
https://eprint.iacr.org/2020/309 DOI 10.1007/978-3-030-56880-1_3,
https://doi.org/10.1007/978-3-030-56877-1_13

Folded Alcove Walks and Moduli of Curves

I will explain the combinatorial tool of folded alcove walks, which enjoy a wide range of applications in combinatorics, representation theory, and algebraic geometry.  We will survey the construction of both finite and affine flag varieties through this lens, focusing on the problem of understanding the intersections of different kinds of Schubert cells.  We will conclude with one sample application, discussing joint work with Arun Ram which uses these folded alcove walks to label the points of the moduli space of genus zero curves in the finite flag variety.

Campana points, a new arithmetic challenge

This talk introduces Campana points, an arithmetic notion, first studied by Campana and Abramovich, that interpolates between the notions of rational and integral points. Campana points are expected to satisfy suitable analogs of Lang’s conjecture, Vojta’s conjecture and Manin’s conjecture, and their study introduces new number theoretic challenges of a computational nature. I will report on the latest results for varieties of Fano type.

Singular moduli for real quadratic fields

The theory of complex multiplication occupies an important place in number theory, an early manifestation of which was the use of special values of the j-function in explicit class field theory of imaginary quadratic fields, and the works of Eisenstein, Kronecker, Weber, Hilbert, and many others. In the early 20th century, Hecke studied the diagonal restrictions of Eisenstein series over real quadratic fields, which later lead to highly influential developments in the theory of complex multiplication initiated by Gross and Zagier in their famous work on Heegner points on elliptic curves. In this talk, we will explore what happens when we replace the imaginary quadratic fields in CM theory with real quadratic fields, and propose a framework for a tentative ‘RM theory’, based on the notion of rigid meromorphic cocycles, introduced in joint work with Henri Darmon. I will discuss some recent progress obtained in various joint works with Henri Darmon, Alice Pozzi, and Yingkun Li.

Topological Noetherianity for polynomial functors over rings

In the last decade of the 19th century, Hilbert established finiteness properties in finite dimensional algebraic geometry. These results, like Hilbert’s basis theorem, do not carry over to the infinite dimensional framework.
Recently, finiteness results for infinite dimensional varieties have been found by exploiting symmetries with respect to the action of a group such as the infinite symmetric group Sym(N) or the infinite general linear group GL.
GL-equivariant infinite dimensional algebraic geometry can be expressed with the language of polynomial functors. These objects were defined by Schur in the study of representations of GL_n. In geometry, they describe an infinite dimensional variety with an action of GL.
In 2017, J. Draisma proved that polynomial functors over infinite fields are topologically Noetherian implying that closed subsets can be described set-theoretically as the zero loci of finitely many GL-orbits of polynomial equations.
I present, after introducing the necessary background, a generalization of Draisma’s theorem to polynomial functors defined over base rings whose spectrum is Noetherian. Finally, I illustrate some applications.
Joint work with A. Bik and J. Draisma.

Generalised Symmetric Chabauty for cubic points on modular curves with infinitely many quadratic points

Josha Box described all quadratic points on certain modular curves with the positive rank of its Jacobian. Some of them have infinitely many points. We will describe a way to find all cubic points despite the infinitude of quadratic points. We will focus on X0(65). X0(65) has finitely many truly cubic points. The original question by David Zureick-Brown at the Arizona Winter School 2020 was about that curve because it plays a role in classifying the potential torsion subgroups of Mordell-Weil groups of elliptic curves over cubic number fields. This is joint work with Josha Box and Pip Goodman.

A Kaneko-Zagier equation for Jacobi forms

The Kaneko-Zagier equation is a second order differential equation depending on a parameter k which gives rise to an infinite family of modular forms as solutions. These solutions can be expressed as a residue in terms of the Weierstrass p function, corresponding to the fact that this differential equation originates from the study of supersingular elliptic curves. In this talk, we study an analogue of the Kaneko-Zagier differential equation for Jacobi forms, originating from the Gromov-Witten theory of K3 surfaces. The solutions of this differential equation can be expressed as a residue in terms of a ratio of Jacobi theta functions.
Joint work with Georg Oberdieck and Aaron Pixton.

Explicit Vologodsky Integration for Hyperelliptic Curves

Let X be a curve over a p-adic field with semi-stable reduction and let ω be a meromorphic 1-form
on X. There are two notions of p-adic integration one may associate to this data: the Berkovich–Coleman
integral which can be performed locally; and the Vologodsky integral with desirable number-theoretic properties.
In this talk, we present a theorem comparing the two, and describe an algorithm for computing Vologodsky
integrals in the case that X is a hyperelliptic curve. We also illustrate our algorithm with a numerical example
computed in Sage. This talk is partly based on joint work with Eric Katz.

Computing Virtual Classes of Representation Varieties using Topological Quantum Field Theories

We study a geometric technique for computing invariants of the G-representation variety of closed orientable surfaces of arbitrary genus, using Topological Quantum Field Theories. Explicitly, we compute their virtual classes in the Grothendieck ring of varieties for G the groups of complex upper triangular matrices of rank 2, 3 and 4. Moreover, we show this method can be extended to non-orientable surfaces and to character stacks.