Algebraic Geometry

We will talk about Drinfeld modules, and how they compare to elliptic curves for algorithms and computations.

Drinfeld modules can be seen as function field analogues of elliptic curves. They were introduced in the 1970's by Vladimir Drinfeld, to create an explicit class field theory of function fields. They were instrumental to prove the Langlands program for GL2 of a function field, or the function field analogue of the Riemann hypothesis.

Elliptic curves, to the surprise of many theoretical number theorists, became a fundamental computational tool, especially in the context of cryptography (elliptic curve Diffie-Hellman, isogeny-based post-quantum cryptography) and computer algebra (ECM method).

Despite a rather abstract definition, Drinfeld modules offer a lot of computational advantages over elliptic curves: one can benefit from function field arithmetics, and from objects called Ore polynomials and Anderson motives.

We will use two examples to highlight the practicality of Drinfeld modules computations, and mention some applications.

We show that the reduction of a projective endomorphism modulo a discrete valuation naturally takes the form of a set-theoretic correspondence. This raises the possibility of classifying "reduction types" of such dynamical systems, reminiscent of the additive/multiplicative dichotomy for elliptic curves. These correspondences facilitate the exact evaluation of certain integrals of dynamical Green's functions, which arise as local factors in the context of counting rational points ordered by the Call-Silverman canonical height. No prior knowledge of arithmetic dynamics will be assumed.

The Polya group P o ( K ) of a Galois number field K coincides with the subgroup of the ideal class group C l ( K ) of K consisting of all strongly ambiguous ideal classes. We prove that there are only finitely many imaginary abelian number fields K whose "Polya index" [ C l ( K ) : P o ( K ) ] is a fixed integer. Accordingly, under GRH, we completely classify all imaginary quadratic fields with the Polya indices 1 and 2. Also, we unconditionally classify all imaginary biquadratic and imaginary tri-quadratic fields with the Polya index 1. In another direction, we classify all real quadratic fields K of extended R-D type (with possibly only one more field K ) for which P o ( K ) = C l ( K ) . Our result generalizes Kazuhiro's classification of all real quadratic fields of narrow R-D type whose narrow genus numbers are equal to their narrow class numbers.

This is a joint work with Amir Akbary (University of Lethbridge).

Over the years, there have been several open problems involving polynomials that I would love to tell others about. This opportunity to speak at my “home ground” seems the perfect time to do so. More specifically, I will discuss the following:

- A conjecture of Ruzsa for integers and a related problem in a joint work with Bell for polynomials over finite fields.
- A conjectural lower bound for the degree of irreducible factors of certain polynomials from a joint work with DeMarco, Ghioca, Krieger, Tucker, and Ye.
- The irreducibility of certain Gleason polynomials.

Given two elliptic curves, the path finding problem asks to find an isogeny (i.e. a group homomorphism) between them, subject to certain degree restrictions. Path finding has uses in number theory as well as applications to cryptography. For supersingular curves, this problem is known to be easy when one small endomorphism or the entire endomorphism ring are known. Unfortunately, computing the endomorphism ring, or even just finding one small endomorphism, is hard. How difficult is path finding in the presence of one (not necessarily small) endomorphism? We use the volcano structure of the oriented supersingular isogeny graph to answer this question. We give a classical algorithm for path finding that is subexponential in the degree of the endomorphism and linear in a certain class number, and a quantum algorithm for finding a smooth isogeny (and hence also a path) that is subexponential in the discriminant of the endomorphism. A crucial tool for navigating supersingular oriented isogeny volcanoes is a certain class group action on oriented elliptic curves which generalizes the well-known class group action in the setting of ordinary elliptic curves.