Other Writings

This page is overflow from my preprints and publications page.

Contents

• Contents
• Dissertation and related papers
• Orphans
• Expository/Other
• Key to format types

Dissertation and related papers

Below you will find my dissertation (which I have not published) and a series of related papers based on the dissertation and some subsequent work (also not published). Most of these results are subsumed by the preprints and papers above, but for the few that are not, here are the relevant links. In particular, the fourth paper culminates in a proof of a weak form of Crew's conjecture; for the strong form (and a unified treatment), see "A p-adic local monodromy theorem" above.

Formats	Title and Citation
dvi, ps.gz	Descent theorems for overconvergent F-crystals, Ph.D. thesis, Massachusetts Institute of Technology, 2000.
arXiv	Unipotency and semistability of overconvergent F-crystals, arXiv:math.AG/0102173.
arXiv	Descent of morphisms of overconvergent F-crystals, arXiv:math.AG/0105244.
arXiv	The Newton polygons of overconvergent F-crystals, arXiv:math.AG/0106192.
arXiv	Quasi-unipotence of overconvergent F-crystals, arXiv:math.AG/0106193.

Orphans

These are bits and pieces which for various reasons did not end up going through the usual publication process.

• A formulation of difference Galois theory, arXiv:math/0610574v1 (2006). This is mostly just a note to myself, clarifying how one can use Yves André's common Tannakian formulation of differential and difference Galois theory to formulate the Galois correspondence for difference fields whose fields of constants need not be algebraically closed.
• Basis discrepancies for extensions of valued fields, arXiv:math/0604212v1 (2006). This is a bit of valuation theory that originally came up in semistable reduction, but did not end up getting used for that purpose.
• Algebraic generalized power series and automata, arXiv:math.RA/0110089 [arXiv]. I rewrote and expanded this paper; see "Finite automata and algebraic extensions of function fields".
• Weil numbers in prime cyclotomic fields [tex, dvi, ps.gz]. Hendrik Lenstra points out that the conjecture in this preprint is originally due to Raphael Robinson, and that John Loxton gave a proof. However, Loxton's bound is far from optimal; someday I might add an improvement of it to the paper and un-downgrade it back to normal paper status.
• Crew's Euler characteristic formula fails for nonzero slopes [tex, dvi, ps.gz, magma], (version of 14 May 2003).
  The magma file is a program that verifies the computations asserted in the paper.
• Number fields with squarefree discriminant and prescribed signature [tex, dvi, ps.gz], version of 15 May 2003. I am rewriting this paper under the new title "A construction of polynomials with squarefree discriminants."
• When is (xy+1)(yz+1)(zx+1) a square? [tex], Mathematics Magazine 71 (1998), 61-63.
• Eichler-Shimura without Wedderburn [tex] (last revised 31 Oct 1997), notes from a contributed talk at the Topics in Number Theory conference at Penn State (August 1997).
• Computing highly composite numbers (numbers with more divisors than any smaller number; see Achim Flammenkamp's HCN page for the current state of the art):
  • A note describing an algorithm [tex].
  • My ANSI C implementation of said algorithm (not completely rigorous, as it does not perform interval arithmetic on the floating point logarithms).
  • The first 1000 HCNs according to said program. (Thanks to Donald Siano for verifying this table by an independent computation.)
  • The first 5000 HCNs. (There is a table of this length in the literature, computed by Guy Robin, but I haven't seen it.)
  • The first 10000 HCNs. This table checks against Flammenkamp's longer tables (see link above).
• A Mathematica code snippet for computing p-orderings of sets of integers. The notion of a p-ordering comes from the paper "P-orderings and polynomial functions on arbitrary subsets of Dedekind rings" by Manjul Bhargava, Journal für die reine und angewandte Mathematik 490 (1997) 101-127.

Expository/Other

• D. Wallner, E. Harder, and R. Agee, Key management for multicast: Issues and architectures, Internet RFC 2627, June 1999 (text). I am not an author on this paper; it is listed here because it is based on work I did with Noam Shazeer and Salil Vadhan at the 1995 NSA Director's Summer Program.
• Comments on: G.S. Bhat and C.D. Savage, Balanced Gray codes, Electronic Journal of Combinatorics 3(1) (1996) R25 (comment posted 9 Apr 1997), MR 1715389 (2000g:05009) [original article, comment, backstory]
• Is this number prime? [tex, dvi, ps.gz, pdf], notes from a talk I gave at the Berkeley Math Circle November 3, 2002.
• Periods for the fundamental group [tex, dvi, ps.gz, pdf], notes from Deligne's talks at the 2002 Arizona Winter School (in progress; last revised 20 Mar 2003).
• Fixed Points and Fair Divisions [tex], notes from a talk in MIT's Pure Math Lunchbox Seminar (a/k/a the graduate student seminar). I gave another version of this talk at the Berkeley Math Circle in November 2001. See Francis Su's home page for more on this topic, including some Java applets.
• The Weil Conjectures I: Elliptic Curves [tex] and The Weil Conjectures II: Curves [tex], notes from talks in MIT's Baby Algebraic Geometry Seminar. I never did get to give the natural third talk in the series...
• The EGA crib sheet has temporarily moved to the web page for my course 18.726 (Algebraic Geometry).
• A sketch of the Eichler-Shimura correspondence [tex] (last revised 15 Feb 1997).
• Quantum mechanics and the fall of public-key cryptography [tex], an overview based on several talks I gave at Harvard. (Last revised 16 Feb 1997.) See also the expanded version of Peter Shor's original paper.
• My senior honors thesis, Complex multiplication and explicit class field theory [tex, ps.gz]. (Revised to correct an error found by Kamal Khuri-Makdisi; further errata to be posted someday, when I find the list.)
• Modular curves, supersingular elliptic curves, and good Goppa codes [tex], written for Math 256 (Theory of Error-Correcting Codes), taught by Noam Elkies (Harvard, fall 1995). (Noam gave me a list of errata, which I'll post if I ever find it again.)

Key to format types

• tex: TeX file (plain text)
• dvi: DVI file (view with xdvi)
• pdf: PDF file (view with Acrobat Reader, ggv, etc.)
• ps.gz: compressed Postscript file (uncompress with gunzip, then view with GhostScript; note that your browser may uncompress automatically)
• mgm.tar: tar file whose entries are Magma source files
• arXiv: link to arXiv e-print server (follow link, then download in your favorite format)
• journal: link to article at journal's web site (follow link if permitted at your institution)
• ppt: PowerPoint presentation (view with OpenOffice Impress!)

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