Cornelius Greither

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Cornelius Greither (born 1956) [1] is a German mathematician specialising in Iwasawa theory and the structure of Galois modules. [2]

Education and career

Greither completed his PhD in 1983 at the Ludwig-Maximilians-Universität München under the supervision of Bodo Pareigis: [3] his thesis bears the title Zum Kürzungsproblem kommutativer Algebren. [4] He habilitated in 1988 at same university, with thesis title Cyclic Galois extensions and normal bases. [4]

In 1992, Greither proved the Iwasawa main conjecture for abelian number fields in the case. [5] [6] In 1999, together with D. R. Rapogle, K. Rubin, and A. Srivastav, he proved a converse to the Hilbert–Speiser theorem. [7]

Greither was a full professor at the Universität der Bundeswehr München. He retired in 2022, [8] and he is now an emeritus. [9] [10]

Greither is on the editorial boards of the journals Archivum mathematicum Brno, [2] [11] New York Journal of Mathematics, [2] [12] as well as the Journal de Théorie des Nombres Bordeaux. [2] [13] Until 2014, he was an associate editor of Annales mathématiques du Québec. [2] [14]

Related Research Articles

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In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently, Ralph Greenberg has proposed an Iwasawa theory for motives.

In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.

In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbersQp (where p is any prime number), or the field of formal Laurent series Fq((T)) over a finite field Fq.

In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any finite abelian extension of Q, which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields.

In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form . The Kronecker–Weber theorem provides a partial converse: every finite abelian extension of Q is contained within some cyclotomic field. In other words, every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients. For example,

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.

In algebraic number theory, the Hilbert class fieldE of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K.

In number theory, the Stark conjectures, introduced by Stark and later expanded by Tate (1984), give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin L-function associated with a Galois extension K/k of algebraic number fields. The conjectures generalize the analytic class number formula expressing the leading coefficient of the Taylor series for the Dedekind zeta function of a number field as the product of a regulator related to S-units of the field and a rational number.

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In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles. The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.

Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group G of a certain arithmetic variety X, or some related geometric object, can help to restore X. The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck. As introduced in Esquisse d'un Programme the latter were about how topological homomorphisms between two arithmetic fundamental groups of two hyperbolic curves over number fields correspond to maps between the curves. These Grothendieck conjectures were partially solved by Hiroaki Nakamura and Akio Tamagawa, while complete proofs were given by Shinichi Mochizuki.

In algebraic number theory, the Gras conjecture relates the p-parts of the Galois eigenspaces of an ideal class group to the group of global units modulo cyclotomic units. It was proved by Mazur & Wiles (1984) as a corollary of their work on the main conjecture of Iwasawa theory. Kolyvagin (1990) later gave a simpler proof using Euler systems.

In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by Emil Artin as an expression appearing in the functional equation of an Artin L-function.

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<span class="mw-page-title-main">Moshe Jarden</span> Israeli mathematician

Moshe Jarden is an Israeli mathematician, specialist in field arithmetic.

Greenberg's conjecture is either of two conjectures in algebraic number theory proposed by Ralph Greenberg. Both are still unsolved as of 2021.

References

  1. ↑ "981058766320560". Biblioteka Narodowa. Retrieved 3 October 2021.
  2. 1 2 3 4 5 "Univ.-Prof. Dr. Cornelius Greither" . Retrieved 3 October 2021.
  3. ↑ Cornelius Greither at the Mathematics Genealogy Project
  4. 1 2 "Nichtreferierte Publikationen" . Retrieved 3 October 2021.
  5. ↑ Greither, Cornelius (1992). "Class groups of abelian fields, and the main conjecture". Annales de l'Institut Fourier. 42 (3): 449–499. doi: 10.5802/aif.1299 .
  6. ↑ Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields (2 ed.). Springer. p. 372.
  7. ↑ Greither, Cornelius; Replogle, Daniel R.; Rubin, Karl; Srivastav, Anupam (1999), "Swan modules and Hilbert–Speiser number fields", Journal of Number Theory, 79: 164–173, doi: 10.1006/jnth.1999.2425
  8. ↑ "Universitätsprofessur (W3) für Mathematik - Universität der Bundeswehr München - academics". 2021-10-03. Archived from the original on 2021-10-03. Retrieved 2023-12-28.
  9. ↑ "Ehemalige Mitarbeiter". Universität der Bundeswehr München (in German). Retrieved 2023-12-28.
  10. ↑ "Univ.-Prof. Dr. Andreas Nickel". Universität der Bundeswehr München (in German). Retrieved 2023-12-28.
  11. ↑ "Masaryk University, Archivum Mathematicum". emis.impa.br. Archived from the original on 2021-10-03.
  12. ↑ "NYJM Editorial Board".
  13. ↑ "Journal de Théorie des Nombres de Bordeaux".
  14. ↑ "ASMQ - Editorial Board".
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