Mathematical Institute of the University of Bonn

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 "Rigid analytic geometry II" SS25

I have reserved Gr. Hs. (Wegelerstr. 10!) Thu. 18:00-20:00 for this lecture. This can be changed if this is desired and a consensus for a new date is reached among the participants. My current intention is to continue my lecture from the previous term, although I am to some extent flexible about the prerequisites. A basic knowledge of classical rigid analytic geometry up to and including the proof of Tate acyclicity will however be required. For those who did not attend my lectures in the previous term I scheduled an introduction lecture for Wednesday April 16 14:00 N0.008. Another lecture of this type is scheduled for April 30 14:00 in N0.008. The lectures on April 17 and 24 have also been moved to N0.008 since the blackboards in Gr. Hs. are currently hard to use.

Lecture "Class field theory" SS25

The lecture is scheduled for Tuesdays and Fridays Room 1.008 Endenicher Allee~60. I will mostly follow the articles of Serre and Tate in "Algebraic number theory" edited by Cassels and Fröhlich. This means that a cohomological approach will be used and some previous knowledge of basic homological algebra is absolutely required. The basic theory of global fields (i. e, algebraic number fields and one variable function fields over finite fields), including their rings of adeles, is also absolutely required. The knowledge from my lecture in the previous term will of course be sufficient. Knowledge of the first two exposes (by Cassels and Fröhlich) of the aforementioned book will be sufficient. A knowledge of derived functors is not necessary as the results needed for the lectures can be provided in the exercises. The exercise sheets will be published on this home page:

• Sheet 1 (corrected version), due Tuesday April 15.
• Sheet 2, due Tuesday April 22.
• Sheet 3, due Tuesday April 29.

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:

• Teil I.
• Teil II.
• Teil III.

Selected Publications

• On the spaces F^s_pq of Triebel-Lizorkin type: pointwise multipliers and spaces on domains, Math. Nachr. 125 (1986), 113-149.
• (with Yu. I. Manin and Yu. Tschinkel) Rational points of bounded height on Fano varieties, Invent. Math. 95(1989), 421-435
• (with T. Runst) Regular elliptic boundary value problems in Besov-Triebel-Lizorkin spaces. Math. Nachr. 174 (1995), 113-149.
• Harmonic analysis in weighted L₂-spaces, Ann. Sci. École Norm. Sup. (4), 31(1998), 181-279
• (with J. Schwermer), A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups, Math. Ann. 311(1998), 765-790.
• On the singularities of residual Eisenstein series, Invent. Math. 138(1999), 307-317
• (With T. Kleinjung, F. Morain and T. Wirth) Proving the primality of very large numbers with fastECPP, in Algorithmic number theory, Lecture Notes in Comput. Sci., 3076, 2004, pages 194-207.
• (With T. Kleinjung), Continued fractions and lattice sieving. In: Proceedings SHARCS 2005
• (with K. Aoki, T. Kleinjung, A. Lenstra, D. Osvik) A kilobit special number field sieve factorization, in Advances in cryptology. ASIACRYPT 2007, Lecture Notes in Comput. Sci., 4833, 2007, pages 1-12.
• A topological model for some summand of the Eisenstein cohomology of congruence subgroups, in Eisenstein series and applications, Progr. Math., 258, 2008, pages 27-85.