MATH 894, p-Adic analysis (Reading Course)

Instructor: Nils Bruin
SC K 10507
(778) 782 3794
Textbook: Neal Koblitz p-adic Number, p-adic Analysis, and Zeta-Functions.
Graduate Texts in Mathematics, 58. Springer-Verlag, New York, 1977.
x+122 pp. ISBN: 0-387-90274-0
Meetings: Thursday, 12:30 - 14:20, P 9318 (part of P 9309, next to AQ level stairs entrance)
First meeting January 10
Goal and format: The goal is that the participants learn about p-adic analysis. As a specific goal, we will work towards Dwork's proof of the rationality of zeta functions of hypersurfaces over finite fields.

Each meeting, one of the participants will lecture on a part of the textbook. We will fix the assignment of topics and the lecture schedule on the first meeting, January 10.

Lecturers, as a guide-line, should prepare a lecture of about an hour. Our time slot allows for overrun and discussion.

Grading: Based on lectures, participation, and assignments

Lecture schedule:
Date Lecturer Title Sections
Jan 10 Nils Planning and Introduction I will sketch an outline of the lecture schedule and we will fill it in.
Jan 17 Yue Ostrowski's Theorem I. 1,2
Jan 24 Brett Hensel's Lemma (aka Newton's Method) I. 3,4,5
Jan 31 Ryan Finite field extensions and unramified extensions of Qp III. 1.
Feb 7 Navid Extension of absolute values to finite algebraic extensions III. 2.
Feb 14
reading break
Feb 21 Navid Algebraic closure of Qp III. 3.
Feb 28 Avi Omega (aka Cp) III. 4.
Mar 7 Ryan The Artin-Hasse exponential IV. 1,2.
Mar 14 Brett Newtop polygons, Weierstrass preparation theorem IV. 3,4.
Mar 21 Avi Hypersurfaces and their zeta functions (Statement of Dwork's Theorem) V. 1.
Mar 28 Yue Characters and their liftings; Power series multiplication as linear transformations. V. 2,3.
Apr 4 Avi, Brett, Navid p-adic analytic expression for zeta functions V. 4.
Apr 11 The proof of Dwork's Theorem. V. 5.