papers and scraps

Alternate formats are available upon request.

preprints
(pdf) Analytic families of finite-slope Selmer groups. Submitted.

(pdf) Cyclotomic Iwasawa theory of motives. Submitted.

(pdf) Cohomology of arithmetic families of (φ,Γ)-modules. With K.S. Kedlaya and L. Xiao. Submitted.

publications
(pdf) Parity-induced Selmer Growth For Symplectic, Ordinary Families. Journal of Number Theory 130 (2010), pp.1024–1047.

expository material
(pdf) Harder–Narasimhan theory. Works out in detail an exercise given by L. Fargues.

(pdf) Basic Dieudonné Theory. Applications are given to classification, and to abelian varieties. (My minor thesis, one of Harvard's Ph.D. requirements.)

(pdf) Some Selmer Groups. Basics of Selmer groups of finite Galois modules over Q, a couple of specific cases, and the weak Mordell–Weil theorem. (My "Phase 2" paper for MIT's undergraduate writing requirement.)

lecture notes
(pdf) Degeneration of the Hodge–de-Rham spectral sequence. (In the MIT STAGE.)

(pdf) Potentially semistable deformation rings (following Kisin). (Joint with David Geraghty, in a seminar at Harvard on the work of Colmez and Kisin on the Fontaine–Mazur conjecture.)

(pdf) Logarithmic structures on schemes. (In the MIT BAGS.)

(pdf) A "cheat sheet" on what a cohomological duality theorem (according to Grothendieck) looks like. Later lectures, describing Nekovar's Selmer complexes, are described in an appendix to my thesis. (Lecture series to grad students at Harvard.)

(pdf) "What is Iwasawa theory about (in my opinion)?" (To grad students at Harvard.)

(pdf) Deligne's method of attaching l-adic representations to Hecke eigenforms: a very quick look. (To grad students at Harvard.)

(pdf) Dwork's proof of the rationality of the zeta function of a variety over a finite field, with lip service to Katz's thesis. (In the MIT STAGE.)

(pdf) Grothendieck's algebraic de Rham theorem for schemes of finite type over the complex numbers. (In the MIT STAGE.)

(pdf) The easy case of Iwasawa's main conjecture, via Stieckelberger's theorem. (For a reading group at M.I.T. on Iwasawa theory of cyclotomic fields.)

(pdf) "Many Twisted Interpolations, Part I". Describes the form of the (Dirichlet) p-adic L-functions, and drawing several scenarios in which one can construct them. (In the Harvard trivial notions seminar.)

(pdf) Notes from the first half of a seminar offered to counselors at PROMYS 2001, on modular forms. Covers congruence subgroups, group actions, elliptic-curves-with-structure, modular forms for congruence subgroups, modular forms with character.

(pdf) Formal power series, the functions they define, and the multiplicative structure of the p-adic field. (In the PROMYS 2002 p-adics counselor seminar.)

(pdf) Hendrik Lenstra's elliptic curve algorithm for factoring integers. Beware of some vagueness in the language, and of some modifications needed in characteristics 2 and 3 which I did not mention. (At XMASCON 2001, intended for computer h4x0rz, NOT mathematicians.)

translations (my apologies to the authors for what errors I have introduced)
(pdf) J.-P. Serre, Congruences et formes modulaires (d'après H. P. F. Swinnerton-Dyer), Séminaire Bourbaki 1971/72, no. 416. Serre's study of modular forms mod p.

(pdf) J.-P. Serre, Classees des corps cyclotomiques (d'après K. Iwasawa), Séminaire Bourbaki (Décembre 1958). Serre's (at that time original) approach to Iwasawa's early theory.

(pdf) J.-P. Serre, Endomorphismes complètement continus des espaces de Banach p-adiques from Publ. Math. I.H.E.S., no. 12. Serre's simplification of Dwork's theory of p-adic analysis.

(pdf) P. Deligne, Formes Modulaires et représentations l-adiques, Séminaire Bourbaki 1968/1969, no. 355. Deligne's famous Bourbaki talk where he (1) associates Galois representations to Hecke eigenforms of arbitrary weight, (2) as a consequence reducing the Ramanujan–Petersson conjectures to the Weil conjectures.

leftovers...
(pdf) Analytic families of finite-slope Selmer groups. Version 8-Mar-2010. This document was canibalized by the above papers "Analytic..." and "Cyclotomic...". Some arguments in this version do not appear in the above final versions for space reasons.

(pdf) Triangulordinary Selmer Groups. Version 16-May-2008. This paper was part of my Ph.D. dissertation and will not be submitted; its material will be incorporated into other papers.

(pdf) Here are some notes I wrote as an undergraduate, in an attempt to understand the basics of generalized Selmer groups. These notes are wrong in two ways. First, Tate–Shafarevich group is not a Selmer group. Second, I only prove that an ideal class group injects into a Selmer group, so the open question should be, what is the quotient? My complete lack of knowing these things at the time of writing is obvious if you read the notes. Heh.