Temporary Assistant Professor
Department of Mathematics
University of Hawaii at Manoa
Honolulu, HI, USA 96822
Office: Physical Sciences Building (PSB) 303
E-mail: (my last name) (at) math (dot) hawai (dot) edu
I am a postdoc at the University of Hawaii. My areas of research are number theory and algebraic geometry. In particular, I am interested in developing geometric tools to study classical and p-adic modular forms, modular forms of half-integral weight, harmonic Maass forms and vector-valued modular forms.
I am currently organizing the UH Number Theory seminar .
|8. Indecomposable 3-dimensional representations of the modular group and their modular forms (with T. Hartland, C. Marks and D. Yepez), submitted, 2017.
We classify the three-dimensional representations of the modular group that are reducible but indecomposable, and their associated spaces of holomorphic vector-valued modular forms. We then demonstrate how such representations may be employed to compute periods of modular curves. This technique obviates the use of Hecke operators, and therefore provides a method for studying noncongruence modular curves as well as congruence.
|7. Vector bundles and modular forms for Fuchsian groups of genus zero (with C. Franc), submitted, 2017.
This article lays the foundations for the study of modular forms transforming with respect to representations of Fuchsian groups of genus zero. More precisely, we define geometrically weighted graded modules of such modular forms, where the graded structure comes from twisting with all isomorphism classes of line bundles on the corresponding compactified modular curve, and we study their structure by relating it to the structure of vector bundles over orbifold curves of genus zero. We prove that these modules are free whenever the Fuchsian group has at most two elliptic points. For three or more elliptic points, we give explicit constructions of indecomposable vector bundles of rank two over modular orbifold curves, which give rise to non-free modules of geometrically weighted modular forms.
|6. The transformation laws of algebraic theta functions, preprint, 2016.
We give a comprehensive treatment of the transformation laws of theta functions from an algebro-geometric perspective, that is, in terms of moduli of abelian schemes. This is accomplished by introducing geometric notions of theta-descent structures, metaplectic stacks, and bundles of half-forms, whose analytic incarnations underlie different aspects of the classical transformation laws. As an application, we lay the foundations for a geometric theory of modular forms of half-integral weight and, more generally, for modular forms taking values in the Weil representation. We discuss further applications to the algebraic theory of Jacobi forms and to the theory of determinant bundles of abelian schemes.
|5. Generating weights for the Weil representation attached
to an even order cyclic quadratic module (with C. Franc and Gene S. Kopp), J. Number Theory 180, 474-497, 2016.
We develop geometric methods to study the generating weights of free modules of vector-valued modular forms of half-integral weight, taking values in a complex representation of the metaplectic group. We then compute the generating weights for modular forms taking values in the Weil representation attached to cyclic quadratic modules of order 2p^r, where p is a prime greater than 3. We also show that the generating weights approach a simple limiting distribution as p grows, or as r grows and p remains fixed.
|4. A geometric perspective on p-adic properties of mock modular forms (with F. Castella), Res. Math. Sci., (2017) 4:5, 2016.
Bringmann, Guerzhoy and Kane have shown how to correct mock modular forms by a certain linear combination of the Eichler integral of their shadows in order to obtain p-adic modular forms in the sense of Serre. In this paper, we give a new proof of their results (for good primes p) by employing the geometric theory of harmonic Maass forms developed by the first author and the theory of overconvergent modular forms due to Katz and Coleman.
|3. The algebraic functional equation of Riemann's theta function , submitted, 2016.
We give an algebraic analog of the functional equation of Riemann's theta function. More precisely, we define a `theta multiplier' line bundle over the moduli stack of principally polarized abelian schemes with theta characteristic and prove that its dual is isomorphic to the determinant bundle over the moduli stack. We do so by explicitly computing with Picard groups over the moduli stack. This is all done over the ring R=Z[1/2,i]: passing to the complex numbers, we recover the classical functional equation.
|2. Vector valued modular forms and the modular orbifold of elliptic curves, (with C. Franc), Int. J. Number Theory, vol. 13(1), 2015.
This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of SL_2(Z) play in the holomorphic theory of vector valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line P(4, 6).
|1. Harmonic weak Maass forms: a geometric approach, Math. Ann., vol. 360(1-2), pp. 489-517, 2014.
The purpose of the present work is to provide a geometric framework for the study of the Fourier coefficients of harmonic weak Maass forms, a space of smooth modular forms first introduced by Bruinier and Funke in the context of singular theta lifts. In this geometric framework harmonic weak Maass forms arise from the construction of differentials whose classes are exact in certain de Rham cohomology groups attached to modular forms. We show how this new interpretation naturally leads to strengthenings of the theorems of Bruinier, Ono and Rhoades, by answering in the affirmative conjectures about the field of definitions of Fourier coefficients of harmonic weak Maass forms. Moreover, as part of our geometric framework, we describe a geometric interpretation for the Shimura-Maass lowering operator analogous to the description of the Shimura-Maass raising operator given by Katz. We also produce Eichler-Shimura-style isomorphisms for the de Rham cohomology attached to modular forms, generalizing results of Bringmann, Guerzhoy, Kent and Ono to any level and field of definition.