18.787 Lecture Plan

This page will be updated throughout the semester, as I post more notes, fiddle with the topics, and update the calendar. You may also want to download the current compiled form of the notes (or version 1), but beware that these are currently quite unstable. (The TeX source is even more unstable, but let me know if you want it anyway.)

Here are the units covered in the lecture notes. Topics in boldface are certain to be included; either the notes are already posted (click on the topic for a PDF file) or they are already in progress. Topics not in boldface are subject to change, based on audience interest and my own whims.

By popular demand, the TeX sources are also available (except for the last few notes, which are copied straight out of the compiled file), but you will need the preamble file.

• References [tex] (may change often)
• Introduction to the course [tex]: for motivation, sketch of an example of Dwork illustrating how a solution to a p-adic differential equation governs numbers of points on elliptic curves over finite fields.
• Formalism of differential algebra [tex]: mostly definitions and notation, plus the cyclic vector theorem over a field.
• Absolute values [tex]: review of properties of nonarchimedean absolute values on fields and vector spaces.
• Newton polygons [tex]: Newton polygons for ordinary polynomials as well as polynomials twisted by a differential operator.
• Regular and irregular singularities [tex]: application of nonarchimedean techniques to the study of meromorphic singularities of differential equations over the complex numbers.
• Discs and annuli [tex]: the rings of power series convergent on closed discs and annuli, and basic properties.
• Convergence of solutions of p-adic differential equations [tex]: introduction of a key invariant, the generic radius of convergence, for a p-adic differential module on a disc or annulus. We also define the generic radii of subsidiary convergence.
• Master theorems for slope factorization [tex]: a general version of the slope factorization theorem.
• Frobenius antecedents and Frobenius descendants [tex]: two important constructions which let us study the generic and subsidiary radii more effectively.
• A little p-adic numerical analysis [tex]: some facts about the dependence of the eigenvalues of a p-adic matrix on the entries of the matrix.
• Convexity and monotonicity for subsidiary radii [tex]: how the generic and subsidiary radii vary as we consider discs of varying radii around a fixed center.
• Decomposition by subsidiary radii [tex]: how extra information about subsidiary radii can be used to decompose a differential module on a disc or annulus.
• A little difference algebra [tex]: basic formalism of difference algebra, particularly over a complete nonarchimedean field, including a generalization of the Dieudonné-Manin classification theorem.
• Frobenius structures on p-adic differential equations [tex]: application of difference algebra to the structure of p-adic differential equations, especially Picard-Fuchs equations.
• Quasiunipotent differential modules: [tex] some important examples of p-adic differential modules with Frobenius structures, arising from Galois representations in positive characteristic, plus a statement of the p-adic local monodromy theorem.
• The p-adic local monodromy theorem: more on this important theorem, including outlines of the two basic proof techniques (using Frobenius slopes, and using the decomposition theorem).
• Effective convergence bounds: some explicit bounds on the series coefficients in the solution of a differential system, without and with Frobenius structures.
• p-adic exponents: a survey of the theory of p-adic exponents introduced by Christol and Mebkhout.
• p-adic Hodge theory: an illustration of how to use p-adic differential equations to make nontrivial assertions about p-adic Galois representations over local fields in mixed characteristic.
• Rigid cohomology: an illustration of how to use p-adic differential equations to make nontrivial assertions about p-adic cohomology.

In the Sections column of the following table, "Dwork" means Dwork-Gerotto-Sullivan, An Introduction to G-Functions. "Notes" are links to PDF files in the above list.