Research Interests:

My research interest lies at the intersection of number theory and arithmetic geometry. On the one hand, on the number theory side, I work on local-global representation theory, automorphic forms, and classical modular forms ; on the other hand, I pursue various projects on elliptic curves over global and local fields on the arithmetic geometry side.
Another part of my research focuses on exploring the connection between number theory and arithmetic geometry using L-functions and Galois representations, which broadly comes under the Langlands Program. Recently, I am also working on a few projects in computational number theory. My recent and ongoing research are in the following three categories:

• Classical modular forms and automorphic forms. I am interested in the following questions in this area:


• Can we find the automorphic representations associated with classical modular forms explicitly?
• How to obtain the nice properties (e.g., meromorphic continuation, functional equations, special values) for L-functions of automorphic representations?
• How to count a certain set of cuspidal automorphic representations using their relationship with classical modular forms?
• How to compute the dimension of spaces of classical modular forms and Siegel modular forms?

In the papers 1,4, 5, 8, 9 listed below, I deal with some of the above questions for special cases.


• Elliptic curves over number fields and local fields. In the context of the arithmetic of elliptic curves, I am currently working on the following questions:


• How does the local data (Kodaira-Néron types, local Tamagawa numbers, conductor exponent) at each prime change for rational elliptic curves with non-trivial torsion (or non-trivial isogeny)?
• Can we classify all rational elliptic curves with a specific global Tamagawa number?
• Can we classify all the elliptic curves over number fields with the same discriminants and conductors?
• What is the density of rational elliptic curves with non-trivial torsion with specified local data in the set of rational elliptic curves with non-trivial torsion?

The papers 2, 7 listed below addresses some of the above questions.


• Connection between number theory and arithmetic geometry. The Langlands program predicts that to each arithmetic geometry object one can associate an automorphic representation of an appropriate group such that their L-functions are the same. My research focuses on better understanding this connection for some specific cases; specifically, I am considering the following questions:


• Can we classify automorphic representations attached to elliptic curves over number fields in terms of their Weierstrass coefficients?
• What kind of automorphic objects do we get for a specific lifting from abelian varieties?

For some of my work in this direction, see the articles 6, 10, 12 below.

Publications and Preprints:

(The versions below might differ slightly from their published counterparts.)

1. Dimension formulas for Siegel modular forms of level 4 (with Ralf Schmidt and Shaoyun Yi, and with an appendix "Modular forms of Klingen level 4 and small weight" by Cris Poor and David Yuen).
   Preprint, 2022, (Submitted).


2. Prime isogenous discriminant twins over number fields (with Alexander J. Barrios, Alyson Deines, Maila Hallare, and Piper H)
   Preprint (available on request), 2022.


3. Generalized Ramanujan-Sato Series Arising from Modular Forms (with Angelica Babei , Lea Beneish , Holly Swisher, Bella Tobin, and Fang-Ting Tu)
   Preprint, 2022, (Submitted).


4. Classical and adelic Eisenstein series (with Ralf Schmidt and Shaoyun Yi).
   Preprint, 2021, (Submitted).


5. The completed standard L-function of modular forms on G_2 (with Fatma Çiçek, Giuliana Davidoff, Sarah Dijols, Trajan Hammonds, and Aaron Pollack).
   Mathematische Zeitschrift 302 (2022), 483–517, DOI.


6. Representations attached to elliptic curves with a non-trivial odd torsion point (with Alexander J. Barrios).
   Bulletin of the London Mathematical Society, 2022, (published online), DOI.


7. Local data of rational elliptic curves with non-trivial torsion (with Alexander J. Barrios).
   Pacific Journal of Mathematics 318 (2022), No.1,1-42, DOI.


8. Congruences for dimensions of spaces of Siegel cusp forms and 4-core partitions (with Chiranjit Ray and Shaoyun Yi).
   The Ramanujan Journal 58 (2022), 1011-1023, DOI.


9. On counting cuspidal automorphic representations for GSp(4) (with Ralf Schmidt and Shaoyun Yi).
   Forum Mathematicum 33 (2021), no. 3, 821-843, DOI.


10. Paramodular forms coming from elliptic curves.
    Journal Number Theory 233 (2022), 126-157, DOI.


11. Elliptic curves and paramodular forms.
    Doctoral dissertation, University of Oklahoma, 2019.


12. Level of Siegel modular forms constructed via sym^3 lifting.
    Automorphic forms and related topics, Contemp. Math.,732 (2019), 225-227, DOI.