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Cyclotomy and an extension of the Taniyama group. (English) Zbl 0591.14001

The “Taniyama group” \(T\) classifies those “motives” that arise from the spectra of number fields or abelian varieties over \(\mathbb Q\) with potential complex multiplication. In this elegant paper, the author constructs a certain extension of \(T\) by a profinite completion of \(2\pi i\mathbb Z\). This extension classifies what the author calls “ulterior motives”. Motives factorize into ulterior motives like (in an analogy given elsewhere by the author) hadrons, e.g., protons, neutrons, factor into quarks. Through the use of ulterior motives, and results of D. Blasius and C. Siegel, the author is able to prove the \(\Gamma\)-hypothesis of Lichtenbaum which gives critical values of certain Hecke \(L\)-series in terms of special values of the classical \(\Gamma\)-function.

MSC:

11G15 Complex multiplication and moduli of abelian varieties
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14L40 Other algebraic groups (geometric aspects)
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References:

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