Kummer–Vandiver conjecture

In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime p does not divide the class number h_K of the maximal real subfield $K=\mathbb{Q}(\zeta_p)^+$ of the p-th cyclotomic field. The conjecture was first made by Ernst Kummer in 1849 December 28 and 1853 April 24 in letters to Leopold Kronecker, reprinted in (Kummer 1975, pages 84, 93, 123–124), and independently rediscovered around 1920 by Philipp Furtwängler and Harry Vandiver (1946, p. 576),

As of 2011, there is no particularly strong evidence either for or against the conjecture and it is unclear whether it is true or false, though it is likely that counterexamples are very rare.

Background

The class number h of the cyclotomic field $\mathbb{Q}(\zeta_p)$ is a product of two integers h₁ and h₂, called the first and second factors of the class number, where h₂ is the class number of the maximal real subfield $K=\mathbb{Q}(\zeta_p)^+$ of the p-th cyclotomic field. The first factor h₁ is well understood and can be written explicitly in terms of Bernoulli numbers, and is usually rather large. The second factor h₂ is not well understood and seems hard to compute explicitly.

Kummer showed that if a prime p does not divide the class number h, then Fermat's last theorem holds for exponent p.

Kummer also showed that if p divides the second factor, then it also divides the first factor. In particular the Kummer–Vandiver conjecture holds for regular primes.

Evidence for and against Vandiver's conjecture

Kummer verified Vandiver's conjecture for p less than 200, and Vandiver extended this to p less than 600. Joe Buhler, Richard Crandall, and Reijo Ernvall et al. (2001) verified it for p < 12 million. Harvey (2008) extended this to primes less than 2³⁷, about 10¹¹.

Washington (1996, p. 158) describes an informal probability argument, based on rather dubious assumptions about the equidistribution of class numbers mod p, suggesting that the number of primes less than x that are exceptions to Vandiver's conjecture might grow like (1/2)log log x. This grows extremely slowly, and suggests that the computer calculations do not provide much evidence for Vandiver's conjecture: for example, the probability argument (combined with the calculations for small primes) suggests that one should only expect about 1 counterexample in the first 10¹⁰⁰ primes, suggesting that it is unlikely any counterexample will be found by further brute force searches even if there are an infinite number of exceptions.

Mihăilescu (2010) gave a refined version of Washington's heuristic argument, suggesting that Vandiver's conjecture is probably true.

Schoof (2003) gave conjectural calculations of the class numbers of real cyclotomic fields for primes up to 10000, which strongly suggest that the class numbers are not randomly distributed mod p. They tend to be quite small and are often just 1. For example, assuming the generalized Riemann hypothesis, the class number of the real cyclotomic field for the prime p is 1 for p<163, and divisible by 4 for p=163.

Consequences of Vandiver's conjecture

Kurihara (1992) showed that the conjecture is equivalent to a statement in the algebraic K-theory of the integers, namely that K_n(Z) = 0 whenever n is a multiple of 4. In fact from Vandiver's conjecture and the norm residue isomorphism theorem follows a full conjectural calculation of the K-groups for all values of n; see Quillen–Lichtenbaum conjecture for details.

See also

• regular and irregular primes

References

• Buhler, Joe; Crandall, Richard; Ernvall, Reijo; Metsänkylä, Tauno; Shokrollahi, M. Amin (2001), Irregular primes and cyclotomic invariants to 12 million, in Bosma, Wieb, "Computational algebra and number theory. Proceedings of the 2nd International Magma Conference held at Marquette University, Milwaukee, WI, May 12–16, 1996.", Journal of Symbolic Computation 31 (1): 89–96, doi:10.1006/jsco.1999.1011, ISSN 0747-7171, MR1806208 
• Ghate, Eknath (2000), "Vandiver's conjecture via K-theory", in Adhikari, S. D.; Katre, S. A.; Thakur, Dinesh, Cyclotomic fields and related topics, Proceedings of the Summer School on Cyclotomic Fields held in Pune, June 7–30, 1999, Bhaskaracharya Pratishthana, Pune, pp. 285–298, MR1802389, http://www.math.tifr.res.in/~eghate/vandiver.pdf 
• Kummer, Ernst Eduard (1975), Weil, André, ed., Collected papers. Volume 1: Contributions to Number Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-06835-0, MR0465760 
• Kurihara, Masato (1992), "Some remarks on conjectures about cyclotomic fields and K-groups of Z", Compositio Mathematica 81 (2): 223–236, ISSN 0010-437X, MR1145807, http://www.numdam.org/item?id=CM_1992__81_2_223_0 
• Mihăilescu, Preda (2010), Turning Washington's heuristics in favor of Vandiver's conjecture, arXiv:1011.6283 
• Schoof, René (2003), "Class numbers of real cyclotomic fields of prime conductor", Mathematics of Computation 72 (242): 913–937, doi:10.1090/S0025-5718-02-01432-1, ISSN 0025-5718, MR1954975 
• Vandiver, H. S. (1946), "Fermat's last theorem. Its history and the nature of the known results concerning it", The American Mathematical Monthly 53: 555–578, ISSN 0002-9890, JSTOR 2305236, MR0018660 
• Washington, Lawrence C. (1996). Introduction to Cyclotomic Fields. Springer. ISBN 0-387-94762-0. 

External links

• Harvey, David (2008), Large-scale verification of Vandiver's conjecture, Slides for a talk, http://www.cims.nyu.edu/~harvey/talks/vandiver-talk.pdf