Jennifer JohnsonLeung
Assistant Professor
University of Idaho
jenfns@uidaho.edu
Jennifer JohnsonLeung
Department of Mathematics
PO Box 441103
University of Idaho
Moscow ID 838441103
USA
Office Hours:
303 Brink Hall
Mon. 10:30am, Wed. 1:30pm, and by appointment.
My primary research area is number theory, including
arithmetic geometry and automorphic representation
theory.
(Show/Hide
Research Summary).
I study Siegel modular forms, automorphic representations of $\GSp(4)$, and abelian surfaces. My work is motivated by several theorems in the case of elliptic curves and elliptic modular forms that are conjectured to generalize to genus two. In particular, I am interested in the paramodular conjecture and the equivariant Tamagawa number conjecture. The paramodular conjecture states that every simple abelian surface which is not of $\GL(2)$type is paramodular in the sense that if $N$ is the conductor of the abelian surface, then there is a Siegel modular form of paramodular level $N$. The equivariant Tamagawa number conjecture on special values of $L$functions unifies and generalizes the Dedekind class number formula, the generalized Stark's conjectures, and the Birch and SwinnertonDyer conjecture.
In addition, I spend some time thinking about Math Education
(Show/Hide
MathEd Summary).
As a CoPI on the Making Mathematical Reasoning Explicit MSP grant, I had the opportunity to work with teachers of 4th through 12th grade to develop methods and activities that encourage active learning while giving access to learners at various levels. I am working to apply what I have learned through working with teachers to my classrooms at the graduate and undergraduate level.
This fall, I will be giving a presentation at the
UI Advising Symposium on
September 11, 2015 on the topic of "Perserverance in
Problem Solving."
If you are interested in doing research
with me or need a letter of recommendation, please
come to my office hours or send me an email to set up
an appointment.
Mathematics Publications
 JohnsonLeung, J., & Roberts,
B. (2014). Fourier Coefficients for Twists of
Siegel Paramodular Forms. 59 pages,
expanded version (preprint). 14 pages,
submitted version.
(Show/Hide 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.
 JohnsonLeung, J., & Roberts,
B. (2014). Twisting of Siegel Paramodular
Forms. 29 pages, submitted.
(Show/Hide 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 AtkinLehner involution for primes $\ell\neq p$.

JohnsonLeung, J., & Roberts, B. (2014). Appendix to Twisting of Siegel
Paramodular Forms. 60 pages, preprint.
(Show/Hide Abstract)
In this appendix we present an expanded version of Section 4 of our paper arXiv:1404.4596, including the proofs of all of the technical lemmas.

JohnsonLeung, J., & Roberts, B. (2014). Twisting of paramodular vectors.
International
Journal of Number Theory,
10 1043–1065 (doi:
10.1142/S1793042114500146).
(Show/Hide Abstract)
Let $F$ be a nonarchimedean local field of characteristic zero, let
$(\pi,V)$ be an irreducible, admissible representation of $\GSp(4,F)$
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 wellknown
$\GL(2)$ twisting operator.

JohnsonLeung, J. (2013). The local equivariant Tamagawa number conjecture for almost abelian extensions.
In Women in Numbers 2: Research Directions
in Number Theory, Contemporary Mathematics, 606 1–27.
(Show/Hide 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 $s<0$.

JohnsonLeung, J., & Roberts, B. (2012). Siegel modular forms of degree two attached to Hilbert modular forms.
Journal of Number Theory, 132 ,543–564.
(Show/Hide Abstract)
Let $E/\mathbb{Q}$ be a real quadratic field and $\pi_0$ a cuspidal, irreducible, automorphic representation of $\GL(2,\mathbb{A}_E)$ with trivial central character and infinity type $(2,2n+2)$ for some nonnegative 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 $\mathbb{Q}$.
 H. Grundman, J. JohnsonLeung, K. Lauter, A. Salerno,
B. Viray, and E. Wittenborn. (2011). Embeddings of
Quartic CM fields and Intersection theory on the
Hilbert Modular Surface, in WINWomen in Numbers:
Research Directions in Number Theory, Fields Institute
Communications Series, Volume 60, 35–60. (Show/Hide 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 zerocycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field $K$, and $T_m$ is the HirzebruchZagier divisors parameterizing products of elliptic curves with an misogeny 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.
 JohnsonLeung, J., & Kings, G. (2011). On the equivariant
main conjecture for imaginary quadratic fields,
J. reine angew. Math. 653
75–114. (Show/Hide 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.
 JohnsonLeung, J. (2005) Artin $L$functions for abelian
extensions of imaginary quadratic fields. PhD Thesis,
California Institute of
Technology. 1–69.
(Show/Hide Abstract)
Let $F$ be an abelian extension of an imaginary quadratic field $K$ with Galois group $G$. We form the Galoisequivariant $L$function of the motive $M=h^0(\Spec(F))(j)$ where the Tate twists $j$ are negative integers. The leading term in Taylor expansion at $s=0$ decomposes over the group algebra $\Q[G]$ 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.
Math Education Publications
 Yopp, D., Ely, R., and JohnsonLeung, J. (2015) Generic
Example Proving Criteria for All. To appear in
For the Learning of Mathematics.
 Adams, A. and JohnsonLeung,
J. (2014) Justification as Exploration in Understanding
Divisibility Rules. preprint.
 JohnsonLeung, J. and Ely, R. (2015)
Learning Trajectories in Linear Algebra. in preparation.
 Algebraic Geometry (558) syllabus
 Linear Algebra (330) syllabus
 Course in Arithmetic Reading Group (504)
I use
bblearn for my
course web pages.
I have been an Assistant Professor in the
Department of
Mathematics at the
University of Idaho since 2007. Prior to joining the
faculty at Idaho, I was an instructor at
Brandeis
University for 2 years. I received my PhD in 2005
from
Caltech under the
direction of Matthias Flach. I am also an alumna of
The College of William and
Mary (1998). I enjoy working and playing in Moscow, Idaho with
my husband and four children.
Grant Activity
Conferences Organized
Selected Presentations
Graduate Students
 Malcolm Rupert
 Daniel Reiss