Like a bird on the wire Like a drunk in a midnight choir I have tried in my way to be free1

Stephen Smale

Description

Niveau: Supérieur, Doctorat, Bac+8
Like a bird on the wire, Like a drunk in a midnight choir I have tried in my way to be free1. To Stephen Smale, at his 80 - th birthday TURNING WASHINGTON'S HEURISTICS IN FAVOR OF VANDIVER'S CONJECTURE PREDA MIHA˘ILESCU Abstract. A famous conjecture bearing the name of Vandiver states that h+p = 1 in the p - cyclotomic extension of Q. Heuris- tics arguments of Washington, which have been briefly exposed in [La], p. 261 and [Wa], p. 158 suggest that the Vandiver conjecture should be false, if certain conditions of statistical independence are fulfilled. In this note we assume that Greenberg's conjecture is true for the p?th cyclotomic extensions and prove an elementary consequence of the assumption that Vandiver's conjecture fails for a certain value of p: the result indicates that there are deep cor- relations between this fact and the defect ??i(p), where i(p) is like usual the irregularity index of p, i.e. the number of Bernoulli numbers B2k ? 0 mod p, 1 < k < (p ? 1)/2. As a consequence, if one combines the various assumptions in Washington's heuristics, these turn, on base of the present result, into an argument in favor of the Vandiver's conjecture. 1. Introduction Let p be an odd prime and K = Q[?] be the p?th cyclotomic field and G = Gal (K/Q).

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Informations

• Publié par: mijec
• Nombre de lectures: 33
• Langue: English

Extrait

Like a bird on the wire, Like a drunk in a midnight choir 1 I have tried in my way to be free .

To Stephen Smale, at his80- th birthday

TURNING WASHINGTON’S HEURISTICS IN FAVOR OF VANDIVER’S CONJECTURE

˘ PREDA MIHAILESCU

Abstract.A famous conjecture bearing the name of Vandiver + states thath= 1 in thep- cyclotomic extension ofQ. Heuris- p tics arguments of Washington, which have been briefly exposed in [La], p. 261 and [Wa], p. 158 suggest that the Vandiver conjecture should be false, if certain conditions of statistical independence are fulfilled. In this note we assume that Greenberg’s conjecture is true for thep−th cyclotomic extensions and prove an elementary consequence of the assumption that Vandiver’s conjecture fails for a certain value ofpresult indicates that there are deep cor-: the − relations between this fact and the defectλ i(p), wherei(p) is like usual the irregularity index ofpthe number of Bernoulli, i.e. numbersB 2k ≡0 modp,1< k <(p−1)/a consequence, if2. As one combines the various assumptions in Washington’s heuristics, these turn, on base of the present result, into an argument in favor of the Vandiver’s conjecture.

1.Introduction Letpbe an odd prime andK=Q[ζ] be thep−th cyclotomic field andG(= Gal K/Q). IfXis a finite abelian group, we denote by X p itsp- Sylow group; letA=C(K) p , thep- Sylow subgroup of the +−+− class groupC(K) andh , hthe sizes ofArespectivelyA. In a + letter to Kronecker from 1857, Kummer refers top-has anoch zu beweisender Satz, a theorem yet to prove (see also [Wa], p. 158). The fact was stated later as a conjecture by Vandiver. In [La], p. 261 Washington gives an heuristic argument which sug- gests that there might be an asymptotic amount ofO(log log(N)) of − primesp≤Nfor whichλ(A) =i(pwhere) + 1, i(p) is the irregularity ∞ index ofpthe number of Bernoulli numbers, i.e. B 2k ,1< k <(p−1)/2 1 Leonard Cohen:Bird on the wire. Date: Version 2.0 September 8, 2010. 1