fbpx
Wikipedia

Brumer–Stark conjecture

November 13, 2023

The Brumer–Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger's theorem about the factorization of Gauss sums. It is named after Armand Brumer and Harold Stark.

It arises as a special case (abelian and first-order) of Stark's conjecture, when the place that splits completely in the extension is finite. There are very few cases where the conjecture is known to be valid. Its importance arises, for instance, from its connection with Hilbert's twelfth problem.

Statement of the conjecture edit

Let K/k be an abelian extension of global fields, and let S be a set of places of k containing the Archimedean places and the prime ideals that ramify in K/k. The S-imprimitive equivariant Artin L-function θ(s) is obtained from the usual equivariant Artin L-function by removing the Euler factors corresponding to the primes in S from the Artin L-functions from which the equivariant function is built. It is a function on the complex numbers taking values in the complex group ring C[G] where G is the Galois group of K/k. It is analytic on the entire plane, excepting a lone simple pole at s = 1.

Let μK be the group of roots of unity in K. The group G acts on μK; let A be the annihilator of μK as a Z[G]-module. An important theorem, first proved by C. L. Siegel and later independently by Takuro Shintani, states that θ(0) is actually in Q[G]. A deeper theorem, proved independently by Pierre Deligne and Ken Ribet, Daniel Barsky, and Pierrette Cassou-Noguès, states that Aθ(0) is in Z[G]. In particular, Wθ(0) is in Z[G], where W is the cardinality of μK.

The ideal class group of K is a G-module. From the above discussion, we can let Wθ(0) act on it. The Brumer–Stark conjecture says the following:[1]

Brumer–Stark Conjecture. For each nonzero fractional ideal   of K, there is an "anti-unit" ε such that

  1.  
  2. The extension   is abelian.

The first part of this conjecture is due to Armand Brumer, and Harold Stark originally suggested that the second condition might hold. The conjecture was first stated in published form by John Tate.[2]

The term "anti-unit" refers to the condition that |ε|ν is required to be 1 for each Archimedean place ν.[1]

Progress edit

The Brumer Stark conjecture is known to be true for extensions K/k where

In 2020,[5] Dasgupta and Kakde proved the Brumer–Stark conjecture away from the prime 2.[6] In 2023, a full proof of the conjecture has been announced.[7]

Function field analogue edit

The analogous statement in the function field case is known to be true, having been proved by John Tate and Pierre Deligne in 1984,[8] with a different proof by David Hayes in 1985.[9][10]

References edit

  1. ^ a b c Lemmermeyer, Franz (2000). Reciprocity laws. From Euler to Eisenstein. Springer Monographs in Mathematics. Berlin: Springer-Verlag. p. 384. ISBN 3-540-66957-4. MR 1761696. Zbl 0949.11002.
  2. ^ a b Tate, John, Brumer–Stark–Stickelberger, Séminaire de Théorie des Nombres, Univ. Bordeaux I Talence, (1980-81), exposé no. 24.
  3. ^ Tate, John, "Les Conjectures de Stark sur les Fonctions L d'Artin en s=0", Progress in Mathematics, Birkhauser, 47, MR 0782485
  4. ^ Sands, J. W. (1984), "Galois groups of exponent 2 and the Brumer–Stark conjecture", J. Reine Angew. Math., 349 (1): 129–135, doi:10.1515/crll.1984.349.129, S2CID 116102479
  5. ^ arXiv:2010.00657
  6. ^ Dasgupta, Samit; Kakde, Mahesh (2023). "On the Brumer-Stark Conjecture and Refinements". Annals of Mathematics. 197 (1): 289–388.
  7. ^ arXiv:2310.16399
  8. ^ Tate, John (1984). Les conjectures de Stark sur les fonctions L d'Artin en s = 0. Progress in Mathematics. Vol. 47. Boston, MA: Birkhäuser.
  9. ^ Hayes, David R. (1985). "Stickelberger elements in function fields". Compositio Mathematica. 55 (2): 209–239.
  10. ^ Rosen, Michael (2002), "15. The Brumer-Stark conjecture", Number theory in function fields, Graduate Texts in Mathematics, vol. 210, New York, NY: Springer-Verlag, ISBN 0-387-95335-3, Zbl 1043.11079

November 13, 2023
brumer, stark, conjecture, conjecture, algebraic, number, theory, giving, rough, generalization, both, analytic, class, number, formula, dedekind, zeta, functions, also, stickelberger, theorem, about, factorization, gauss, sums, named, after, armand, brumer, h. The Brumer Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions and also of Stickelberger s theorem about the factorization of Gauss sums It is named after Armand Brumer and Harold Stark It arises as a special case abelian and first order of Stark s conjecture when the place that splits completely in the extension is finite There are very few cases where the conjecture is known to be valid Its importance arises for instance from its connection with Hilbert s twelfth problem Contents 1 Statement of the conjecture 2 Progress 3 Function field analogue 4 ReferencesStatement of the conjecture editLet K k be an abelian extension of global fields and let S be a set of places of k containing the Archimedean places and the prime ideals that ramify in K k The S imprimitive equivariant Artin L function 8 s is obtained from the usual equivariant Artin L function by removing the Euler factors corresponding to the primes in S from the Artin L functions from which the equivariant function is built It is a function on the complex numbers taking values in the complex group ring C G where G is the Galois group of K k It is analytic on the entire plane excepting a lone simple pole at s 1 Let mK be the group of roots of unity in K The group G acts on mK let A be the annihilator of mK as a Z G module An important theorem first proved by C L Siegel and later independently by Takuro Shintani states that 8 0 is actually in Q G A deeper theorem proved independently by Pierre Deligne and Ken Ribet Daniel Barsky and Pierrette Cassou Nogues states that A8 0 is in Z G In particular W8 0 is in Z G where W is the cardinality of mK The ideal class group of K is a G module From the above discussion we can let W8 0 act on it The Brumer Stark conjecture says the following 1 Brumer Stark Conjecture For each nonzero fractional ideal a displaystyle mathfrak a nbsp of K there is an anti unit e such that a W 8 0 e displaystyle mathfrak a W theta 0 varepsilon nbsp The extension K e 1 W k displaystyle K left varepsilon frac 1 W right k nbsp is abelian The first part of this conjecture is due to Armand Brumer and Harold Stark originally suggested that the second condition might hold The conjecture was first stated in published form by John Tate 2 The term anti unit refers to the condition that e n is required to be 1 for each Archimedean place n 1 Progress editThe Brumer Stark conjecture is known to be true for extensions K k where K Q is cyclotomic this follows from Stickelberger s theorem 1 K is abelian over Q 3 K k is a quadratic extension 2 K k is a biquadratic extension 4 In 2020 5 Dasgupta and Kakde proved the Brumer Stark conjecture away from the prime 2 6 In 2023 a full proof of the conjecture has been announced 7 Function field analogue editThe analogous statement in the function field case is known to be true having been proved by John Tate and Pierre Deligne in 1984 8 with a different proof by David Hayes in 1985 9 10 References edit a b c Lemmermeyer Franz 2000 Reciprocity laws From Euler to Eisenstein Springer Monographs in Mathematics Berlin Springer Verlag p 384 ISBN 3 540 66957 4 MR 1761696 Zbl 0949 11002 a b Tate John Brumer Stark Stickelberger Seminaire de Theorie des Nombres Univ Bordeaux I Talence 1980 81 expose no 24 Tate John Les Conjectures de Stark sur les Fonctions L d Artin en s 0 Progress in Mathematics Birkhauser 47 MR 0782485 Sands J W 1984 Galois groups of exponent 2 and the Brumer Stark conjecture J Reine Angew Math 349 1 129 135 doi 10 1515 crll 1984 349 129 S2CID 116102479 arXiv 2010 00657 Dasgupta Samit Kakde Mahesh 2023 On the Brumer Stark Conjecture and Refinements Annals of Mathematics 197 1 289 388 arXiv 2310 16399 Tate John 1984 Les conjectures de Stark sur les fonctionsLd Artin ens 0 Progress in Mathematics Vol 47 Boston MA Birkhauser Hayes David R 1985 Stickelberger elements in function fields Compositio Mathematica 55 2 209 239 Rosen Michael 2002 15 The Brumer Stark conjecture Number theory in function fields Graduate Texts in Mathematics vol 210 New York NY Springer Verlag ISBN 0 387 95335 3 Zbl 1043 11079 Retrieved from https en wikipedia org w index php title Brumer Stark conjecture amp oldid 1182434477, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.