My thesis at U.C. Berkeley dealt with Galois representations associated to modular forms.
Since then, however, my work has gravitated more toward the p-adic geometry of modular curves.
In particular, my primary focus for the first several years out was on computing the stable reduction of the modular curve X0(pn) at the prime p.
Due to the work of various other people, this had been known for n≤2 since about 1990.
Robert Coleman and I computed the stable reduction of X0(p3) in a paper for ANT (see below).
Here are some reasonable pictures of the associated
Stable Reduction Graphs.
As is often the case in math, our abstract work was motivated by many explicit examples, and a few of these have been published as well (see below).
Recently, others have made significant progress on the stable reduction problem.
Most notably, Jared Weinstein has in some sense solved the problem for all n, but from the "local Langlands" perspective.
Takahiro Tsushima has also made significant advances using the methods of Coleman-McMurdy.
This has prompted me to shift gears somewhat into the area of overconvergent modular forms.
My background in rigid-analysis and stable reduction of modular curves gives me a slightly different perspective on this field, and I'm hopeful that as a result I will be able to bring some new insight.
A first paper on the U7 operator (joint with Lloyd Kilford) has now appeared in the LMS JCM (see below).
Other interests include the arithmetic of supersingular elliptic curves (see recent paper and earlier joint work with K. Lauter below), and special properties of Galois representations associated to rational points on certain modular curves.
I also have an enduring interest in explicit equations for modular curves, which has been the basis for the computational aspects of much of my work.
At some point I hope to publish a semi-expository paper on the various methods for obtaining explicit equations which I have come across and/or developed over the years.
My lecture notes on eta product models from the Heilbronn Workshop (see below) have some very useful formulas which were actually coded up by David Loeffler for a SAGE module which comes with the standard download.
Here is a table with some Equations for X0(N)
for a few small values of N (along with useful formulas for moduli-theoretic maps).
- "Explicit Representation of the Endomorphism Rings of Supersingular Elliptic Curves" (preprint, Sage Code)
- "Applying a Galois Transformation to the Roots of a Polynomial" (preprint)
- "Slopes of the U7 Operator Acting on a Space of Overconvergent Modular Forms" (with L. J. P. Kilford), LMS J. Comput. Math. 15 (2012), 113-139 (download, Sage Code)
- "Stable Reduction of X0(p3)" (with R. Coleman), Algebra and Number Theory 4 (4) (2010), 357-431
- "Stable Reduction of X0(81)", Contemp. Math. 463 (2008), 91-109
- "Do the Coefficients of a Modular Form Really 'Encode Arithmetic Data?'" (with Hari Ravindran), Rose-Hulman MSTR 08-01 (download)
- "Fake CM and the Stable Model of X0(Np3)" (with R. Coleman),
Documenta Math. Extra Volume: John H. Coates' Sixtieth Birthday (2006), 261-300
- "Explicit Generators for Endomorphism Rings of Supersingular Elliptic Curves", joint
with Kristin Lauter, preprint posted to online
proceedings for Conference in
Honor of Harold Stark
- "Stable Model of X0(125)," LMS J. Comput. Math. 7 (2004), 21-36
- "Explicit Parameterizations of Ordinary and Supersingular Regions of
X0(pn)," Modular Curves and Abelian Varieties, Progress in
Mathematics 224, 165-179 (Birkhauser, 2004)
- "A Splitting Criterion for Galois Representations Associated to Exceptional Modular
Forms," Ph.D. thesis, U.C. Berkeley (2001) (download)
Notes and Slides from Recent Talks:
- "Elliptic Curves with Non-abelian Entanglements", Maine-Quebec Number Theory Conference (F '15) (slides)
- "How to Transform the Roots of a Polynomial (without finding them first)", Ramapo Math Club, 2014 (slides)
- "A New Algorithm for Computing Endomorphism Rings of Supersingular Elliptic Curves", AMS-MAA Joint Meetings (San Diego, 2013) (slides)
- "Stable Models and Up Slope Calculations," Maine-Quebec Number Theory Conference, U.C. Berkeley Number Theory Seminar (F '11) (slides)
- "Slopes of the U7 Operator Acting on a Space of 7-adic Overconvergent Modular Forms," 24th AFW (Hawaii, 2010) (slides)
- "Stable Reduction of X0(625), with Implications," U. C. Berkeley Number Theory Seminar (S '09) (slides)
- "Eta Products and Models for Modular Curves," Heilbronn Workshop on Computations with Modular Forms (Bristol, UK, F '08) (slides)
- "Explicit Verification of a Theorem of Shimura," 22nd Annual Workshop on Automorphic Forms and Related Topics (Texas A&M, S '08) (slides)
- "Heegner Points on X0(pn)," AMS (sectional) Special Session on
Computational Arithmetic Geometry (San Francisco, 2006)
- "Stable Reduction of X0(p3)," AMS Special Session on
Arithmetic Algebraic Geometry (Atlanta, 2005)
- "Why Number Theorists Care About Elliptic Curves," RHIT Math Colloquium
(notes: part 1, part 2)
- "Explicit Generators for Endomorphism Rings of Supersingular Elliptic Curves,"
Conference in Honor of Harold Stark (Minneapolis, 2004)
- "p-adic Properties of CM j-invariants." University of Rochester Number Theory
Seminar (notes: part 1,