Hermite's Constant for Function Fields
In this paper we formulate an analog of Hermite's constant for function fields, state a conjectural value for this analog and prove our conjecture for various cases. In particular this relates to the first minima of twisted heights over function fields.
Hermite's Constant for Function Fields (with J. Thunder), Canad. J. Math., Accepted 2010, in press.
The following paper was inspired by a work of Noam D. Elkies. In particular it generalizes Elkies' construction of error-correcting nonlinear codes found in [Elkies]. The generalization produces a precise average code size over codes in the new construction and the result is a larger family of codes with similar transmission rates and error detection rates to the nonlinear codes found in Elkies' paper. There is also a connection between these nonlinear codes and solutions to simple homogeneous linear equations defined over the function field of a curve.
Non-Linear Codes from Points of Bounded Height (with J. Thunder), Finite Fields Appl., 13, no. 2, 281-292, 2007. [Click to Download]
Arithmetic Difference Operators
A. Buium describes four classes of operators that may be used to ``enlarge usual algebraic geometry". Two of these operators are ideally suited for arithmetic purposes, specifically p-derivations, an arithmetic analog of a derivation, and &pi-difference operators, an arithmetic difference operator that in fact lies morally somewhere between a usual derivation and a p-derivation. This paper details the basic theory of arithmetic difference operators noting the many parallels to and some differences from the theory in the case of p-derivations.
Geometry of Arithmetic Difference Operators, 2005. [Click to Download]
Differential Modular Forms
When algebraic geometry is expanded to include differential operators and arithmetic analogs of differential operators, differential algebraic geometry is the result. In particular this new theory, introduced by Alexandru Buium, includes differential modulular forms. The theory of p-adic modular forms initiated by Serre, Dwork, and Katz "lives" on the complement (in the p-adic completion of the appropriate modular curve) of the zero locus of the Eisentstein form Ep-1. The most interesting phenomena in the theory of differential modular forms takes place on the complement of the zero locus of a fundamental differential modular form, fjet. One of the interesting phenomena this fundamental differential modular form posseses is isogeny covariance. The following two papers give explicit formulas for fjet, the first paper modulo p2 and the second paper modulo p, using different techniques in each case to compute fjet.
Computing Isogeny Covariant Differential Modular Forms, Math. Comp., 74, no. 250, 905-926, 2005. file.pdf.
Isogeny Covariant Differential Modular Forms Modulo p, Compositio Mathematica, 128, no. 1, 17-34, August 2001. file.dvi.
This paper shows that for p not congruent to one modulo 12, the zero locus of the reduction modulo p of the Eisenstein form, Ep-1, is not contained in the zero locus of the reduction modulo p of the differential modular form fjet.
Zero Loci of Differential Modular Forms, J. Number Theory, 98, no 1, 47-54, 2003. file.dvi.
This white paper contains some supplementary material to the theory of formal groups over discrete valuation rings. Namely it discusses formal groups in 2n variables. The material contained is without doubt common knowledge, though not necessarily explicitly written in the texts in the references. The approach is exactly that found in chapter four of The Arithmetic of Elliptic Curves by J. Silverman. [Click to Download]
The following are web pages showing old Maple files from the various computations of fjet. All are written with and by version 7 or version 5 of Maple. They provide many of the intermediate formulas of the computations not included in the papers. Note that most of last part of the computation of fjet modulo p2 is not included in any of these and was completed much later. (As of 2000, the formula resulting from computing fjet modulo p2 was still 300 pages long. Working to get the formula for fjet modulo p2 down to 5 pages took some time.)
ACA 2003, North Carolina, July 2003
This talk which presented the computation of fjet modulo p2 included many lengthy formulas, all of which are included in the slides (file.pdf).
AMS/MAA Special Session at the Joint Mathematics Meeting, New Orleans, January 2001
These slides are here in part as an example of writing slides in latex as of 2001. The latex source files are broken up into an "outside wrapper file" and a "content file". file.tex, file.ltx, file.dvi.