Jennifer Johnson-Leung

Research

My research centers on automorphic forms and the arithmetic of L-functions, with ongoing work in paramodular theory and representations of GSp(4). I am particularly interested in structural questions in representation theory and their arithmetic manifestations, especially in settings where explicit computation plays a role. I also collaborate on modeling projects in cybersecurity and public health, and am developing mathematically grounded approaches to trustworthy and interpretable AI systems, including tools designed to better support mathematical research and reasoning.

Number Theory

• 2024 Johnson-Leung, J., McGlade, F., Negrini, I., Pollack, A., & Roy, M. The quaternionic Maass Spezialschar on split SO(8). arXiv:2401.15277. Submitted.
  Abstract

  The classical Maass Spezialschar is a Hecke-stable subspace of the level one holomorphic Siegel modular forms of genus two, i.e., on Sp(4), cut out by certain linear relations between the Fourier coefficients. It is a theorem of Andrianov, Maass, and Zagier, that the classical Maass Spezialschar is exactly equal to the space of Saito-Kurokawa lifts. We study an analogous space of quaternionic modular forms on split SO(8), and prove the analogue of the Andrianov-Maass-Zagier theorem. Our main tool for proving this theorem is the development of a theory of a Fourier-Jacobi coefficient of quaternionic modular forms on orthogonal groups.
• 2024 Johnson-Leung, J. & Rupert, N. An explicit theta lift to Siegel Modular Forms. To appear in Women in Numbers 6.
  Abstract

  Let $E/L$ be a real quadratic extension of number fields. We construct an explicit map from an irreducible cuspidal automorphic representation of $\mathrm{GL}(2,E)$ which contains a Hilbert modular form with $\Gamma_0$ level to an irreducible automorphic representation of $\mathrm{GL}(4,L)$ which contains a Siegel paramodular form and exhibit local data which produces a paramodular invariant vector for the local theta lift at every finite place, except when the local extension has wild ramification.
• 2024 Johnson-Leung, J., Parker, J., & Roberts, B. The paramodular Hecke algebra. Research in Number Theory 10, article 91 (2024). arXiv:2310.13179.
  Abstract

  We give a presentation via generators and relations of the local graded paramodular Hecke algebra of prime level. In particular, we prove that the paramodular Hecke algebra is isomorphic to the quotient of the free $\mathbb{Z}$-algebra generated by four non-commuting variables by an ideal generated by seven relations. Using this description, we derive rationality results at the level of characters and give a characterization of the center of the Hecke algebra. Underlying our results are explicit formulas for the product of any generator with any double coset.
• 2023 Johnson-Leung, J., Roberts, B., & Schmidt, R. Stable Klingen Vectors and Paramodular Newforms. Book Springer Lecture Notes in Mathematics, Volume 2342.

  Abstract

  We introduce the family of stable Klingen congruence subgroups of $\mathrm{GSp}(4)$. We use these subgroups to study both local paramodular vectors and Siegel modular forms of degree 2 with paramodular level. In the first part, when $F$ is a nonarchimedean local field of characteristic zero and $(\pi,V)$ is an irreducible, admissible representation of $\mathrm{GSp}(4,F)$ with trivial central character, we establish a basic connection between the subspaces $V_s(n)$ of $V$ fixed by the stable Klingen congruence subgroups and the spaces of paramodular vectors in $V$ and derive a fundamental partition of the set of paramodular representations into two classes. We determine the spaces $V_s(n)$ for all $(\pi,V)$ and $n$. We relate the stable Klingen vectors in $V$ to the two paramodular Hecke eigenvalues of $\pi$ by introducing two stable Klingen Hecke operators and one level lowering operator. In contrast to the paramodular case, these three new operators are given by simple upper block formulas. We prove further results about stable Klingen vectors in $V$ especially when $\pi$ is generic. In the second part we apply these local results to a Siegel modular newform $F$ of degree 2 with paramodular level $N$ that is an eigenform of the two paramodular Hecke operators at all primes $p$. We present new formulas relating the Hecke eigenvalues of $F$ at $p$ to the Fourier coefficients $a(S)$ of $F$ for $p^2\mid N$. Finally, for $p^2\mid N$ we express the formal power series in $p^{-s}$ with coefficients given by the radial Fourier coefficients $a(p^tS)$, $t\geq0$, as an explicit rational function in $p^{-s}$ with denominator $L_p(s,F)^{-1}$, where $L_p(s,F)$ is the spin $L$-factor of $F$ at $p$.

  
• 2017 Johnson-Leung, J. & Roberts, B. Fourier Coefficients for Twists of Siegel Paramodular Forms. J. Ramanujan Math. Soc. 32(2), 101–119. Expanded version on arXiv.
  Abstract

  In this paper, we calculate the Fourier coefficients of the paramodular twist of a Siegel modular form of paramodular level $N$ by a nontrivial quadratic Dirichlet character mod $p$ for $p$ a prime not dividing $N$. As an application, these formulas can be used to verify the nonvanishing of the twist for particular examples. We also deduce that the twists of Maass forms are identically zero.
• 2017 Johnson-Leung, J. & Roberts, B. Twisting of Siegel Paramodular Forms. Int. J. Number Theory 13(7), 1755–1854.
  Abstract

  Let $S_k(\Gamma^{\mathrm{para}}(N))$ be the space of Siegel paramodular forms of level $N$ and weight $k$. Let $p\nmid N$ and let $\chi$ be a nontrivial quadratic Dirichlet character mod $p$. Based on our previous work, we define a linear twisting map $\mathcal{T}_\chi:S_k(\Gamma^{\mathrm{para}}(N))\rightarrow S_k(\Gamma^{\mathrm{para}}(Np^4))$. We calculate an explicit expression for this twist and give the commutation relations of this map with the Hecke operators and Atkin-Lehner involution for primes $\ell\neq p$.
• 2014 Johnson-Leung, J. & Roberts, B. Twisting of paramodular vectors. Int. J. Number Theory 10, 1043–1065.
  Abstract

  Let $F$ be a non-archimedean local field of characteristic zero, let $(\pi,V)$ be an irreducible, admissible representation of $\mathrm{GSp}(4,F)$ with trivial central character, and let $\chi$ be a quadratic character of $F^\times$ with conductor $c(\chi)>1$. We define a twisting operator $T_\chi$ from paramodular vectors for $\pi$ of level $n$ to paramodular vectors for $\chi \otimes \pi$ of level $\max(n+2c(\chi),4c(\chi))$, and prove that this operator has properties analogous to the well-known $\mathrm{GL}(2)$ twisting operator.
• 2013 Johnson-Leung, J. The local equivariant Tamagawa number conjecture for almost abelian extensions. In Women in Numbers 2: Research Directions in Number Theory, Contemp. Math. 606, 1–27.
  Abstract

  We prove the local equivariant Tamagawa number conjecture for the motive of an abelian extension of an imaginary quadratic field with the action of the Galois group ring for all split primes $p\neq 2, 3$ at all integer values $s < 0$.
• 2012 Johnson-Leung, J. & Roberts, B. Siegel modular forms of degree two attached to Hilbert modular forms. Journal of Number Theory 132, 543–564.
  Abstract

  Let $E/\mathbb{Q}$ be a real quadratic field and $\pi_0$ a cuspidal, irreducible, automorphic representation of $\mathrm{GL}(2,\mathbb{A}_E)$ with trivial central character and infinity type $(2,2n+2)$ for some non-negative integer $n$. We show that there exists a Siegel paramodular newform $F: \mathfrak{H}_2 \to \mathbb{C}$ with weight, level, Hecke eigenvalues, epsilon factor and $L$-function determined explicitly by $\pi_0$. We tabulate these invariants in terms of those of $\pi_0$ for every prime $p$ of $\mathbb{Q}$.
• 2011 Grundman, H., Johnson-Leung, J., Lauter, K., Salerno, A., Viray, B., & Wittenborn, E. Embeddings of Quartic CM Fields and Intersection Theory on the Hilbert Modular Surface. In WIN—Women in Numbers, Fields Inst. Commun. 60, 35–60.
  Abstract

  Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, $CM(K).T_m$, where $CM(K)$ is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field $K$, and $T_m$ is the Hirzebruch-Zagier divisors parameterizing products of elliptic curves with an $m$-isogeny between them. In this paper, we examine fields not covered by Yang's proof of the conjecture. We give numerical evidence to support the conjecture and point to some interesting anomalies. We compare the conjecture to both the denominators of Igusa class polynomials and the number of solutions to the embedding problem stated by Goren and Lauter.
• 2011 Johnson-Leung, J. & Kings, G. On the equivariant main conjecture for imaginary quadratic fields. J. reine angew. Math. 653, 75–114.
  Abstract

  In this paper we first prove the main conjecture for imaginary quadratic fields for all prime numbers $p$, improving earlier results by Rubin. From this we deduce the equivariant main conjecture in the case that a certain $\mu$-invariant vanishes. For prime numbers $p\nmid 6$ which split in $K$, this is a theorem by a result of Gillard.
• 2005 Johnson-Leung, J. Artin $L$-functions for abelian extensions of imaginary quadratic fields. PhD Thesis, California Institute of Technology, 1–69.
  Abstract

  Let $F$ be an abelian extension of an imaginary quadratic field $K$ with Galois group $G$. We form the Galois-equivariant $L$-function of the motive $M=h^0(\mathrm{Spec}(F))(j)$ where the Tate twists $j$ are negative integers. The leading term in Taylor expansion at $s=0$ decomposes over the group algebra $\mathbb{Q}[G]$ into a product of Artin $L$-functions indexed by the characters of $G$. We construct a motivic element $\xi$ via the Eisenstein symbol and relate the $L$-value to periods of $\xi$ via regulator maps. Working toward the equivariant Tamagawa number conjecture, we prove that the $L$-value gives a basis in étale cohomology which coincides with the basis given by the $p$-adic $L$-function according to the main conjecture of Iwasawa theory.

Interdisciplinary Collaborations

• 2025 Mankotia, S., Conte de Leon, D., & Johnson-Leung, J. Hierarchical Firmware-level Security Policy for Industrial Control Systems. IEEE, 2025.
  Abstract

  Current digital systems may be vulnerable to a variety of low-level attacks, including out-of-bounds operations and unsafe deserialization. We present BHPol, a binary implementation of a hierarchical security policy framework that enables fast firmware-level declaration and enforcement of security policies on-chip through associative memory that validates each instruction against predetermined security policies, blocking unauthorized requests.
• 2024 Sarathchandra, D. & Johnson-Leung, J. Influence of Political Ideology and Media on Vaccination Intention in the Early Stages of the COVID-19 Pandemic in the United States. COVID 4(5), 658–671.
  Abstract

  As a pharmaceutical intervention, vaccines remain a major public health strategy for mitigating the effects of COVID-19. Yet, vaccine uptake has been affected by various cognitive and cultural barriers. We examine how a selected set of barriers (i.e., knowledge, concern, media, peer influence, and demographics) shaped COVID-19 vaccination intention in the early phase of the pandemic (Fall 2020). Using a survey conducted in three US states (Idaho, Texas, and Vermont) just prior to the roll out of the first vaccines against COVID-19, we find that COVID-19 concern was the primary driver of vaccination intention. Concern was shaped mainly by two factors: political ideology and media sources. Yet, ideology and media were much more important in affecting concern for those who leaned politically conservative, as opposed to those who leaned liberal or remained moderate. The results from our Structural Equation Models affirm that the information politically conservative respondents were receiving reinforced the effects of their ideology, leading to a greater reduction in their concern. We discuss the potential implications of these findings for future pandemic preparedness.
• 2024 Moxley, T.A., Johnson-Leung, J., Seamon, E., Williams, C., & Ridenhour, B.J. Application of Elastic Net Regression for Modeling COVID-19 Sociodemographic Risk Factors. PLoS ONE 19(1): e0297065.
  Abstract

  COVID-19 has been at the forefront of global concern since its emergence in December 2019. Determining the social factors that drive case incidence is paramount to mitigating disease spread. We gathered data from the Social Vulnerability Index (SVI) along with Democratic voting percentage to understand which county-level sociodemographic metrics had a significant correlation with case rate for COVID-19. We used elastic net regression due to issues with variable collinearity and model overfitting. Statistically, elastic net improved prediction when compared to multiple regression, as almost every HHS region model consistently had a lower RMSE and satisfactory R² coefficients. These analyses show that the percentage of minorities, disabled individuals, individuals living in group quarters, and individuals who voted Democratic correlated significantly with COVID-19 attack rate. Our findings can assist policymakers in distributing resources to more vulnerable counties in future pandemics.
• 2023 Seamon, E., Ridenhour, B.J., Miller, C.R., & Johnson-Leung, J. Spatial Modeling of Sociodemographic Risk for COVID-19 Mortality. Preprint, submitted.
  Abstract

  In early 2020, COVID-19 rapidly spread across the United States, exhibiting significant geographic variability. The objective of this analysis is to examine spatiotemporal variation of COVID-19 deaths in association with socioeconomic, health, demographic, and political factors, using regionalized multivariate regression as well as nationwide county-level geographically weighted random forest (GWRF) models. Analyses were performed on data from three separate timeframes: pandemic onset until May 2021, May 2021 through November 2021, and December 2021 until April 2022. GWRF results indicate a more nuanced modeling strategy is useful for capturing the diverse spatial and temporal nature of the COVID-19 pandemic.
• 2022 Ridenhour, B.J., Sarathchandra, D., Seamon, E., Brown, H., Leung, F-Y., Johnson-Leon, M., Megheib, M., Miller, C.R., & Johnson-Leung, J. Effects of trust, risk perception, and health behavior on COVID-19 disease burden: Evidence from a multi-state US survey. PLoS ONE 17(5): e0268302.
  Abstract

  Early public health strategies to prevent the spread of COVID-19 in the United States relied on non-pharmaceutical interventions (NPIs) as vaccines and therapeutic treatments were not yet available. Implementation of NPIs, primarily social distancing and mask wearing, varied widely between communities within the US due to variable government mandates, as well as differences in attitudes and opinions. To understand the interplay of trust, risk perception, behavioral intention, and disease burden, we developed a survey instrument to study attitudes concerning COVID-19 and pandemic behavioral change in three states: Idaho, Texas, and Vermont. The best fitting structural equation models show that trust indirectly affects protective pandemic behaviors via health and economic risk perception. Notably, political ideology is the only exogenous variable which significantly affects all aspects of the social cognitive model.
• 2022 Fox, S.J., Johnson, K., Owirodu, B., Johnson-Leung, J., Elizondo, M., Walkes, D., & Ancel Meyers, L. COVID-19 Risk Assessment for Public Events. (pdf) University of Texas COVID-19 Modeling Consortium.
  Abstract

  We describe a risk assessment framework to support event planning during COVID-19 waves. The method was developed in partnership with public health officials in Austin, Texas.
• 2015 Yopp, D., Ely, R., & Johnson-Leung, J. Generic Example Proving Criteria for All. For the Learning of Mathematics 35(3), 8–13.
  Abstract

  Our goal is to provide criteria for determining if an example in an argument is being used as a generic example. We write in response to Leron and Zaslavsky's (2013) discussion of generic examples, which we agree with in some ways and disagree with in others.

Other Work

• 2018 Bouw, I., Ozman, E., Johnson-Leung, J., & Newton, R. (Eds.) Women in Numbers Europe II: Contributions to Number Theory and Arithmetic Geometry. Edited Volume Association for Women in Mathematics Series (AWMS, vol. 11), Springer Cham.

  Abstract

  Inspired by the September 2016 conference of the same name, this second volume highlights recent research in a wide range of topics in contemporary number theory and arithmetic geometry. Research reports from projects started at the conference, expository papers describing ongoing research, and contributed papers from women number theorists outside the conference make up this diverse volume. Topics cover arithmetic dynamics, failure of local-global principles, geometry in positive characteristics, and heights of algebraic integers.

  
• 2017 Johnson-Leung, J. Hyperelliptic Threshold Noise: A Mathematician's Perspective. Essay accompanying the Visualizing Science Exhibit, Pritchard Art Gallery, University of Idaho.

Funding

• Co-PI: Fundamental Principles of Cybersecurity (2023 - )
  Schweitzer Engineering Laboratory — $865,000

Mentoring

I supervise doctoral students in automorphic forms, paramodular theory, and arithmetic aspects of representation theory. I am also interested in projects that connect rigorous mathematics with computation and trustworthy AI systems. Prospective students are welcome to contact me by email; availability varies by year.

PhD Students

• Jordan Hardy PhD 2023
  Abelian Surfaces with Complex Multiplication Admitting Nonprincipal Polarizations
• Joshua Parker PhD 2022
  Prime Level Paramodular Hecke Algebras
• Daniel Reiss PhD 2019
  Arithmetic Relations Between Fourier Coefficients of Siegel Paramodular Forms
• Nina Rupert PhD 2017
  An Explicit Theta Lift from Hilbert to Siegel Paramodular Forms

Undergraduate Research

I have mentored over a dozen undergraduate students in research and independent study. The students listed below received Undergraduate Research Fellowships.

• Ryan Burke (current, Hill)
• Katie Thiessen (BS 2025, Hill, co-advised with James Nagler)
• Trevor Griffin (BS 2021, Hill)
• Kirk Bonney (BS 2020, Hill)
• Beau Horenberger (BS 2019, SURF)

Bio