## Papers

### 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.

### Non-Linear Codes

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 *E _{p-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,

*f*. One of the interesting phenomena this fundamental differential modular form posseses is isogeny covariance. The following two papers give explicit formulas for

_{jet}*f*, the first paper modulo

_{jet}*p*and the second paper modulo

^{2}*p*, using different techniques in each case to compute

*f*.

_{jet}
*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,
*E _{p-1}*, is not contained in the zero locus of the reduction modulo

*p*of the differential modular form

*f*.

_{jet}
*Zero Loci of Differential Modular Forms*, J. Number Theory,
** 98**, no 1, 47-54, 2003. file.dvi.

## Notes

### Formal Groups

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]

## Maple Code

The following are web pages showing old Maple files from the various
computations of *f _{jet}*. 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

*f*modulo

_{jet}*p*is not included in any of these and was completed much later. (As of 2000, the formula resulting from computing

^{2}*f*modulo

_{jet}*p*was still 300 pages long. Working to get the formula for

^{2}*f*modulo

_{jet}*p*down to 5 pages took some time.)

^{2}Maple Computations of Kodaira-Spencer Class

## Slides

### ACA 2003, North Carolina, July 2003

This talk which presented the computation of *f _{jet}*
modulo

*p*included many lengthy formulas, all of which are included in the slides (file.pdf).

^{2}### 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.