Prof. Dr. Jens Franke

Material related to Mihailescu's CIDE primality proof

In a first use of these ideas of Mihailescu, certificates for the Leyland numbers 311063+633110 and 86562929+29298656 were calculated in late 2012. The description of the format, together with a (in my opinion) complete mathematical proof that it is indeed a valid primality proof, is here. While the terminology in fmt-0.1.pdf has been chosen to be disjoint from the terminology of the Mihailescu preprints quoted there, all crucial ideas are Mihailescu's.

Lecture Advanced Algebra I (Rigid Analytic Geometry) SS26

An introduction to rigid analytic geometry and adic spaces will be given. Both the formalism of adic spaces and the classical formalism via Grothendieck topologies will be introduced but the bigger emphasis is on the first, more modern machinery, with many proofs for the classical machinery moved to the exercises or omitted altogether. After an initial lecture with some motivation I will give a brief but self-contained introduction to the theory of spectral spaces, to the extent to which this is necessary to understand adic spaces. This treatment will be filter-based and somewhat different from what is found in most textbooks. This is then applied to the valuation spectrum, whose treatment will also be self-contained. After some remarks about the Grothendieck topology based machinery and van der Put's method of introducing additional points to it, I will discuss topological rings, their spaces of contiuous valuations and present some material needed for the proof of Tate acyclicity. After this general treatment of topological rings the special case of affinoid algebras over a p-adic field is dealt with. This then applied to the proof of Tate acyclicity.

The material presented in this term is usually learned by mathematicians having same background in scheme theory as it is typically presented in the lectures "Algebraic Geometry I/II" at this university. This, however, is not an absolute prerequisite for the lectures I offer. The prerequisites are similar to the ones for "Algebraic Geometry I": Commutative and homological algebra, the Zariski spectrum of a ring as well as some knowledge of general topology. A knowledge of p-adic numbers and similar local fields will also be required. Knowledge of the basic theory of holomorphic functions of one complex variable will ne needed to understand some motivational examples and possibly also to solve some (very few) of the exercises. The lectures will likely be more demanding than the typical "Algebraic Geometry I" as the rings under consideration will usually also carry topologies, adding an additional layer of complication. Another complication is the fact that in scheme theory the sheafiness and cohomology of the structrue presheaf are well understood while in the case of adic spaces this is an open problem, although the cases relevant for the classical theory or for perfectoid spaces are settled. Because of the large amount of material covered, it will be necessary to move a considerable number of proofs to the exercise module.

While the filter based approach to general topology will briefly be introduced in the first two weeks of lectures, proofs will be terse and it will be an advantage to already have a knowledge of filters. The introduction I gave in my Algebra II lecture in the previous term is completely sufficient and will essentially be repeated at a higher speed in the first few lectures. The needed material is also mostly covered by sections I.5.[1,2,5,7] and on I.6.[1,2] of the topology textbook by Horst Schubert. This book is said to have an english translation but I cannot verify that the section numbers I gave are correct for this translation. The textbook of Boto Querenburg also has a nice section on filters although it is a bit more terse. To my knowledge no english translation of this book exists.

Knowledge of the machinery of derived functors is also desirable although a brief outline may be given in the lecture.

I plan to follow this by a second lecture in the following term but have not yet made a final decision about whether this will be Advanced Algebra II or a fifth year lecture. In the former case, Algebraic Geometry I/II will not be a strict prerequisite (but is very highly recommended) and exercises may be used to present some proofs omitted in the lectures. In the second case I will add Algebraic Geometry I/II to the list of prerequisites. This lecture series may cover things like more material on Tate acyclicity, possibly including the Buzzard-Verberkmoes theorem needed for the treatment of perfectoid spaces, globalization (the lectures in this term being limited to the affine case), the Krull dimension of the adic space corresponding to a classial rigid analytic space, as well as general remarks about the comparison of machineries (classical, Berkovich, Huber, formal schemes).

Lectures are scheduled for Mondays 14:00ct and Thursdays 8:00ct N0.0008. I am willing to move them to a different date if this is desired by the participants, especially so if this means that we can use Großer Hörsaal with its large blackboard space and if it saves the Thursday lectures which will otherwise be lost because of the feasts of Corpus Christi and of the Ascension.

Lecture Advanced Topics in Number Theory (Weil I) SS26

Based upon my Etale Cohomology lectures of the previous term I plan to give an introduction to the first Deligne paper proving the part of the Weil conjectures remaining after the work of Dwork and Grothendieck: The proof of the Riemann hypothesis for the congruence zeta function.

Very good knowledge of Algebraic Geometry I/II is an absolute prerequisite, as is some familiarity with abelian categories and homological algebra. Knowledge of etale cohomology as introduced in the previous term is also needed: Definition of the etale site, constructible sheaves, cohomology, and proper base change. Knowledge of the proetale site is only needed if it will be the machinery used for introducing l-adic sheaves and their cohomology. This will only be done if all participants which otherwise have the required knowledge can agree to it.

The lectures will start by a fast forward through the material on etale cohomology I had to omit in the previous term: Smooth base change, relation to the classical cohomology of the space of C-valued points, Poincare duality, l-adic cohomology and the Lefschetz trace formula. In this, proofs will mostly be omitted. After this, Lefschetz pencils will be introduced, also omitting most proofs. After this, enough time will be left for a reasonably careful presentation of the Weil I proof. If more time is left I will return to the initial part of the lecture and present some omitted proofs, but this can certainly only be done for a few of them.

The lecture has been scheduled for Mondays 16:00ct and Thursdays 12:00ct in N0.008 although moving it may be an option if the participants can agree on this.

Seminar "Geometrische Konstruktionen und transzendente Zahlen."

Das Seminar fand erstmalig im Sommersemester 2016 für Studenten des zweiten Semesters statt. Um einen guten Anschluß an die Vorlesung "Lineare Algebra I" sicherzustellen, diente ein von mir selbst verfaßter Text als Grundlage des Seminares. Dieser soll hier weiterhin zur Verfüfung gestellt werden.

Sprechstunden

In der vorlesungsfreien Zeit sind die Sprechstunden nach Vereinbarung.

Vorlesungen "Mathematik für Physiker I-III"

Die Javascript-Programme zu den Anwesenheitsübungen dieser Vorlesungen, die ich zwischen 2008 und 2011 gehalten habe, sind weiterhin online:

Selected Publications