Dynamics of class groups

Host Institution:

La Trobe University

Title of Seminar:

Dynamics of class groups

Speaker's Name:

Prof Alexander Stolin

Speaker's Institution:

University of Gothenburg, Sweden

Time and Date:

Friday 14 September, 2012, 2:30pm (AEST)

Seminar Abstract:

In his attempt to prove that the equation xp+yp=zp (where p is a prime number) has no integer solutions, Kummer introduced the class groups of the finite extensions of the field ℚ of rational numbers. It turned out that the p-Sylow components Sn of the class class groups Cl Qζn, where ζn is a primitite pn+1-th root of unity, play an important role in Kummer's theory.

Later, Iwasawa studied the groups Sn and proved the number of elements of Sn for large n equals pIn, where In=μ pn+λ n+ ν. Here μ, λ and ν are the so-called Iwasawa numbers. It was proved in the 80’s that μ=0.

The aim of my talk is a more detailed description of the function In. It turns out that in some sense, μ does not vanish completely.

More precisely, under a certain condition related to Bernoulli numbers, in the interval [0, logpλ] we have In=μ pn. After the critical value N=logpλ, In=n-Nλ+IN.

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