$\def\Z{\mathbb Z} \def\R{\mathbb R} \def\C{\mathbb C} \def\Q{\mathbb Q} \def\bs{\backslash} \def\p{\mathfrak p} \def\OF{\mathfrak o} \def\GL{\rm GL} \def\PGL{\rm PGL} \def\SL{\rm SL} \def\SO{\rm SO} \def\Symp{\rm Sp} \def\Spec{\rm{Spec}} \def\GSp{\rm GSp} \def\PGSp{\rm PGSp} \def\meta{\widetilde{\rm SL}} \def\sgn{{\rm sgn}} \def\St{{\rm St}} \def\triv{1} \def\Mp{\rm Mp} \def\val{\rm val} \def\Irr{\rm Irr} \def\Norm{\rm N} \def\Mat{\rm M} \def\Gal{\rm Gal} \def\GSO{\rm GSO} \def\GO{\rm GO} \def\OO{\rm O} \def\Ind{\rm Ind} \def\ind{\rm ind} \def\cInd{\mathrm{c}\text{-}\mathrm{Ind}} \def\Trace{\rm T} \def\trace{\rm tr} \def\Real{\rm Real} \def\Hom{\rm Hom} \def\SSp{\rm Sp} \def\EO{\mathfrak{o}_E} \def\new{\rm new} \def\vl{\rm vol} \def\disc{\rm disc} \def\pisw{\pi_{\scriptscriptstyle SW}} \def\pis{\pi_{\scriptscriptstyle S}} \def\piw{\pi_{\scriptscriptstyle W}} \def\trip{\rm trip} \def\Left{\rm Left} \def\Right{\rm Right} \def\sroot{\Delta} \def\im{\rm im} \def\A{\mathbb A}$
Jennifer Johnson-Leung

Assistant Professor
University of Idaho

## Contact

jenfns@uidaho.edu
Jennifer Johnson-Leung
Department of Mathematics
PO Box 441103
University of Idaho
Moscow ID 83844-1103
USA

### Office Hours:

303 Brink Hall
Thursday 12-2pm and by appointment.

## Research

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 Swinnerton-Dyer conjecture.
In addition, I spend some time thinking about Math Education(Show/Hide MathEd Summary)
As a Co-PI 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.
If you are interested in doing research with me, please read my advice for students if you are interested in doing research with me or if you need a letter of recommendation.

### Mathematics Publications

• Johnson-Leung, J., & Roberts, B. (2014). Fourier Coefficients for Twists of Siegel Paramodular Forms. preprint. (Show/Hide Abstract)
• Johnson-Leung, 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 Atkin-Lehner involution for primes $\ell\neq p$.
• Johnson-Leung, 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.
• Johnson-Leung, 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 non-archimedean 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 well-known $\GL(2)$ twisting operator.
• Johnson-Leung, 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 $s<0$.
• Johnson-Leung, 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 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 $\mathbb{Q}$.
• H. Grundman, J. Johnson-Leung, K. Lauter, A. Salerno, B. Viray, and E. Wittenborn. (2011). Embeddings of Quartic CM fields and Intersection theory on the Hilbert Modular Surface, in WIN--Women 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 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.
• Johnson-Leung, 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.
• Johnson-Leung, 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 Galois-equivariant $L$-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 Johnson-Leung, J. (2015) Generic Example Proving Criteria for All submitted.
• Adams, A. and Johnson-Leung, J. (2014) Justification as Exploration in Understanding Divisibility Rules. preprint.
• Johnson-Leung, J. and Ely, R. (2015) Learning Trajectories in Linear Algebra. in preparation.

## Teaching–Spring 2015

• Algebraic Geometry (558) syllabus
• Linear Algebra (330) syllabus
• Course in Arithmetic Reading Group (504)
I use bblearn for my course web pages.

## Bio

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.