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 I" WS24/25

The lecture is scheduled for Tuesdays 8-10ct and Fridays 10-12ct in room 0.007 Endenicher Allee 60. Rigid analytic geometry will be introduced in the classical framework of Tate and the Grauert school but additional points will be introduced early, using the definition of van der Put. Knowledge of topology, homological algebra and basic commutative algebra at a level similar to the prerequisites for a basic algebraic geometry course is required.

Lecture "Algebra II" WS24/25

The lecture is scheduled for Mondays 16:00-18:00ct and Thursdays 14:00-16:00ct in Kleiner Hörsaal Wegelerstraße 10. The theory of Dedekind rings will be developed and applied to the theory of algebraic number fields and their local fields. In particular, the ideal group and the ideal class group will be introduced and the finiteness of the ideal class group and Dirichlet's theorem about the structure of the group of units will be shown. A good knowledge of Galois theory and the other Introduction to Algebra topics, as well as some familiarity with Noetherian rings and knowledge of the analysis lectures from the first year will be required.

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.