# Melissa Emory: Oklahoma State University

## About me

I am an Assistant Professor at Oklahoma State University in the Mathematics Department. Previously I was a NSF Postdoctoral Fellow at the University of Toronto with mentor James Arthur . I earned my PhD from the University of Missouri in 2018 with advisor Shuichiro Takeda who is now at Osaka University .

My VitaI am a co-organizer of the 36th Automorphic Forms Workshop to be held at OSU May 20 - 24, 2024

and a co-organizer of the Texas-Oklahoma Autmorphic Representations and Automorphic Forms (TORA) Conference

I am also a faculty adviser for the Oklahoma State University Chapter of The Association for Women in Mathematics

Home Page of the Math Department## Contact info

Office : MSCS 526

Office hours: Office hours for Spring 2024: T 12:30-2:00 and Th 12:30-2:00 PM in MSCS 526, or by appointment

Email: melissa.emory@okstate.edu

## Teaching

Fall 2023: MATH 2153 Calculus II and MATH 4713 Number Theory and MATH 2890 Honors Topics in Calculus II (1 credit hour add-on)

Spring 2024: Math 4453/5453 Mathematical Interest Theory - see the bottom of the page for more informatoin on being an actuary!

## Research

My research interests include automorphic forms and representations, number theory, and representation theory.

One area of research that I work in deals with a strategy of Langlands known as *Beyond Endoscopy* which utilizes the trace formula to study the analytic behavior of automorphic L-functions. Its goal is to establish the principle of functoriality which is the fundamental pillar of the Langlands program, with enormous implications for number theory, arithmetic geometry, and representation theory.

Another area of research that I am currently working on and have made some progress towards (see [6] below) is on the restriction of representations of a non-classical group, the general spin group. In general, the question is how a representation of a group decomposes when restricted to a subgroup. The goal of my research is to formulate and prove a local Gan-Gross-Prasad conjecture for general spin groups. The representation theory of general spin groups is important as it subsumes the representation theory of the special orthogonal groups; and having a non-trivial center (as occurs with general spin groups) is preferred for application purposes. The first step was to prove a multiplicity one theorem for GSpin groups as in which was completed in [6] below. Another important portion of this project is to complete the endoscopic classification of representations for general spin groups. There is also a global Gan-Gross-Prasad conjecture which forms a relationship between a certain period integral and special values of L-functions (see [2], and [3]).

If instead of algebraic solutions to curves, one is interested in the distributions of solutions mod p, we are interested in the Sato-Tate conjecture. This area of my research aims to determine the full Sato-Tate group for curves of certain forms. Roughly speaking, we are interested in studying how the coefficients of the (local) normalized L-polynomials of a curve vary as the prime p goes to infinity. This research area is important because the coefficients of the L-polynomial encode arithmetic information about the curve. The generalized Sato-Tate conjecture predicts that as p goes to infinity this distribution converges to the distribution of traces in the Sato-Tate group, a compact subgroup of USp(2g) associated to the Jacobian of the curve.

**In Preperation**

1. *Beyond Endoscopy via Poisson Summation for GL(2,K)* (joint with Malors Espinosa-Lara, Debanjana Kundu, and Tian An Wong).

Beyond Endoscopy is a strategy proposed by Langlands to prove the principle of functoriality. A first step in the strategy was achieved by Ali Altug who worked over the rationals. This work generalizes Altug's work to an arbitrary number field.

** Publications and Preprints **

7. Nondegeneracy and Sato-Tate Distributions of Two Families of Jacobian Varieties with H. Goodson, submitted. Preprint arXiv:2401.06208

6. Contragredients and a multiplicity one theorem for general Spin groups with S. Takeda, Math Ziet (2023). Preprint arXiv:2104.04814

5. Sato-Tate distributions of y^2=x^p-1 and y^2=x^{2p}-1 with H. Goodson, J. Algebra (2022).Preprint arXiv:2004.10583

4. On Sato-Tate groups of trinomial hyperelliptic curves with H. Goodson and A. Peyrot , International Journal of Number Theory (April 2021), 2175-2206. Preprint arXiv:1812.00242

3. On the global Gan-Gross-Prasad conjecture for general spin groups, Pacific Journal of Mathematics 306-1 (2020), 115--151. Preprint arXiv:1901.01746

1. The Diophantine equation x^4+y^4=D^2z^4 in quadratic fields, Integers 12 (2012), article A65

## Actuary Information

An app that many have found useful is Coaching Actuaries to study for the exams. Ideally, students will take MATH 4453, Mathematical Theory of Interest, in the spring of their junior year, take the FM exam the summer after their junior year, so that they will have 2 exams passsed prior to applying for positions their senior year. Students should also attempt to complete an internship in the field prior to graduating.

**Some helpful websites**

This is for taking a practice FM exam online

This is for taking a practice P exam online

Other helpful links, Be an Actuary

Society of Actuaries

SOA Internships