Algebra, Geometry, and Number Theory Seminar

Abstracts

Edray Goins - Monodromy Groups of Belyi Lattes Maps

An elliptic curve $E: y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_1x + a_6$ is a cubic equation which has two curious properties: (1) the curve is nonsingular, so that we can draw tangent lines to every point $P = (x,y)$ on the curve; and (2) the collection of complex points, namely $E(C)$, forms an abelian group under a certain binary operation $\bigoplus: E(C) \times E(C) \to E(C)$. In particular, for every positive integer $N$, the map $P \overset{[N]}{\to} P$ which adds a point $P \in E(C)$ to itself $N$ times is a group homomorphism. A rational map $\gamma: \mathbb{P}^1(C) \to \mathbb{P}^1(C)$ from the Riemann Sphere to itself is said to be a Lattes Map if there are ``well-behaved'' maps $\phi: E(C) \to \mathbb{P}^1(C)$ and $\psi: E(C) \to E(C)$ such that $\gamma \phi = \phi \psi$. We are interested in those Lattes Maps $\gamma$ which are also Belyi Maps, that is, the only critical values are 0, 1, and infinity. Work of Zeytin classifies all such maps: For example, if $E: y^2 = x^3 + 1$ then $\phi: (x,y) \mapsto (y+1)/2$ while $\psi = [N]$ for some positive integer $N$. We would like to know more about Belyi Lattes Maps $\gamma$. What can we say about such maps? What are their Dessin d'Enfants? In some cases, this is a bipartite graph with $3 N^2$ vertices. What are their monodromy groups? Sometimes this is a group of size $3 N^2$. In this talk, we explain the complete answers to these questions, exploiting the relationship between fundamental groups of Riemann surfaces and Galois groups of function fields. This work is conducted as part of the Pomona Research in Mathematics Experience (DMS-2113782). (See also his Solomon Lecture at 6:00 in WMBB 231!)

Fall Break - --

Eun Hye Lee - Subconvexity of Shintani Zeta Functions

Subconvexity problem has been a central interest in analytic number theory for over a century. The strongest possible form of the subconvexity problem is the Lindelof hypothesis, which is a consequence of the RH, in the Riemann zeta function case. There have been many attempts to break convexity for diverse zeta and L functions, usually using the moments method. In this talk, I will introduce the Shintani zeta functions, and present another way to prove a subconvex bound. The pretalk will give an introduction to prehomogeneous vector spaces.

Artane Siad - Explaining Anomalous Class Group Statistics in Certain Thin Families of Number Fields

The Cohen--Lenstra heuristics describe the distribution of class groups of number fields. After reviewing the philosophy leading to this description, we describe observed anomalies in class group statistics in certain thin families of number fields and outline an explanation in terms of extra structure on the class group. This structure is analogous to a structure enjoyed by spin 2- and 3-manifolds: a choice of a spin structure determines a quadratic refinement of, respectively, the intersection pairing mod 2 and the linking pairing.

Thanksgiving Break - --

Daniel Keliher - Dominant Galois Groups for Large Degree Number Fields

Via Hilbert Irreducibility, 100% of the time, the Galois group of a “random" degree n polynomial is S_n. One might then ask if a "random’’ degree n extension of the rationals is likely to have Galois group S_n. While the answer can be extracted for small n from various number field counts, this talk will address this question for large n. We prove, assuming a conjecture of Bhargava, that the density of degree n S_n-extensions among all degree n extensions is 0 as n, supported on a finite set of primes, goes to infinity. Under other standard conjectures, we also prove that extensions with Galois group S_t wreath G with t << log(n)^2 has density 1 among all degree n extensions as n goes to infinity. This is joint work-in-progress with Jiuya Wang.

Last semester's seminar.