I am an assistant professor at the University of Notre Dame.
University of Notre Dame
Department of Mathematics
275 Hurley Hall
Notre Dame, IN 46556
Email:
ajorza nd edu
CV:
cv.pdf

Research

Research:
  • On symmetric power L-invariants of Iwahori level Hilbert modular forms (pdf, ), with Rob Harron. To appear in the American Journal of Mathematics.
    We compute L-invariants for symmetric powers of non-CM Iwahori level Hilbert modular forms in terms of logarithmic derivatives of Hecke eigenvalues on eigenvarieties.
  • p-adic Families and Galois Representations for GSp(4) and GL(2) (pdf, ), Math. Research Letters 19 (2012), no 05, 1-10.
    We prove local-global compatibility for Iwahori level Siegel modular forms by combining a previous result (up to a quadratic twist) with p-adic families. We deduce information at p and l for two dimensional Galois representations on quadratic imaginary fields.
  • Lagrangian hyperplanes in holomorphic symplectic varieties (arXiv, appendix, code). With Benjamin Bakker. To appear in CEJM.
  • Galois representations for holomorphic Siegel modular forms (pdf, ), DOI 10.1007/s00208-012-0811-3, Mathematische Annalen: Volume 355, Issue 1 (2013), Page 381-400.
    We prove many cases of local-global compatibility (up to a quadratic twist) for holomorphic Siegel modular forms.
  • Crystalline representations for $\operatorname{GL}(2)$ over quadratic imaginary fields (pdf, ), my thesis.
    If $\pi$ is an irreducible admissible regular algebraic cuspidal representation of $\textrm{GL}(2)$ over a quadratic imaginary field and $v$ is an unramified place of $K$ where $\pi_v$ and $\pi_{v^c}$ are unramified principal series with distinct Satake parameters, we show that the Galois representation associated to $\pi$ is crystalline at $v$.
  • Higher rank stable pairs on K3 surfaces(arXiv, ). Communications in Number Theory and Physics, Volume 6, Number 4 (2012). With Benjamin Bakker.
    Virtual curve counts have been defined for threefolds by integration against virtual classes on moduli spaces of stable maps (Gromov-Witten theory), ideal sheaves (Donaldson-Thomas theory), and stable pairs (Pandharipande-Thomas theory). The first two theories are proven to be equivalent for toric threefolds, and all three are conjecturally equivalent for arbitrary threefolds. One may ask whether there is such a correspondence for surfaces. In particular, the Gromov-Witten theory of $K3$ surfaces has recently been computed by Maulik, Pandharipande, and Thomas; it is governed by quasimodular forms and is closely related to invariants obtained from the moduli spaces of rank $r = 0$ stable pairs with $n = 1$ sections. We compute the Hodge polynomials of the moduli spaces of stable pairs for higher rank $r \geq 0$ and level $n \geq 1$, and explore the modularity properties and relationship to Gromov-Witten theory.
  • Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves (pdf), Math. Comp. 78 (2009), no. 268, 2397--2425. With G. Grigorov, S. Patrikis, W. Stein, C. Tarniţǎ.
Expository:
  • The Birch and Swinnerton-Dyer conjecture for abelian varieties over number fields (pdf), my senior thesis at Harvard University, 2005. Errata.

Lecture Notes

Teaching

Current/Future Teaching:
  • Math 20550, Calculus 3, Notre Dame, Spring 2016.
  • Math 40520, Theory of Numbers, Notre Dame, Fall 2015.
  • Math 60220, Basic Algebra 2: Graduate Algebra, Notre Dame, Spring 2015.
  • Math 60210, Basic Algebra 1: Graduate Algebra, Notre Dame, Fall 2014.
  • Math 10560, Calculus 2: Integration, series, differential equations, Notre Dame, Fall 2014.
  • Math 80220, Topics in Algebra 2: Introduction to Algebraic Number Theory, Notre Dame, Spring 2014.
  • Math 20550, Calculus 3, Notre Dame, Spring 2014.
  • Math 10350, Calculus A for Life Sciences, Notre Dame, Fall 2013.
  • Math 5c, Galois Theory and Representations of Finite Groups, Caltech, Spring 2013.
  • Math 160c, Applications of Global Class Field Theory, Caltech, Spring 2013. Lecture notes
  • Math 1a (Section 1), Freshman Mathematics, Caltech, Fall 2011.
  • Math 160b, Local Class Field Theory, Caltech, Winter 2012.
  • Math 162b, p-adic Galois Representations, Caltech, Winter 2012. Lecture notes
  • Math 5c, Galois Theory and Representations of Finite Groups, Caltech, Spring 2011.
  • Math 160b, Local Class Field Theory, Caltech, Winter 2011
  • Math 203, Multivariate Calculus, Princeton, Fall 2009
  • Math 103, Calculus, Princeton, Fall 2007
  • Math 217, Honors Linear Algebra, Princeton, Spring 2010
  • Math 453, Analytic Number Theory, Princeton, Spring 2009
  • Math 322, Galois Theory, Princeton, Fall 2008
  • Math 202, Linear Algebra, Princeton, Spring 2008
  • Math 217, Honors Linear Algebra, Princeton, Spring 2007
  • Math 215, Honors Real Analysis, Princeton, Fall 2006
  • Math 129, Algebraic Number Theory, Harvard, Spring 2005
  • Math 250, Graduate Algebra, Harvard, Fall 2004
  • Math 130, Topology, Harvard, Spring 2004
  • Math 112, Real Analysis, Harvard, Fall 2003
  • Math 55b, Honors Linear Algebra and Analysis, Harvard, Spring 2003
  • Math 55a, Honors Linear Algebra and Analysis, Harvard, Fall 2002