By D. Mumford
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10] D. ZAGIER: Green’s functions of quotients of the upper half-plane (in preparation). The Dilogarithm Function 21 Notes on Chapter I. A The comment about “too little-known” is now no longer applicable, since the dilogarithm has become very popular in both mathematics and mathematical physics, due to its appearance in algebraic K-theory on the one hand and in conformal ﬁeld theory on the other. Today one needs no apology for devoting a paper to this function. B From the point of view of the modern theory, the arguments of the dilogarithm occurring in these eight formulas are easy to recognize: they are the totally real algebraic numbers x (oﬀ the cut) for which x and 1 − x, × if non-zero, belong to the same rank 1 subgroup of Q , or equivalently, for which [x] is a torsion element of the Bloch group.
What is relevant is that this group has been studied by Borel , who showed that it is isomorphic (modulo torsion) to Zr2 and that there is a canonical homomorphism, the “regulator mapping,” from it into Rr2 such that the co-volume of the image is a non-zero rational multiple of |d|1/2 ζF (2)/π 2r1 +2r2 ; moreover, it is known that under the identiﬁcation of K3ind (F ) with BF this mapping D corresponds to the composition BF → (BC )r2 → Rr2 , where the ﬁrst arrow comes from using the r2 embeddings F ⊂ C (α → α(i) ).
16 Don Zagier We give an example element of the Bloch group. For con√ of a non-trivial √ −1 − −7 1 − −7 , β= . Then venience, set α = 2 2 √ √ √ √ 1 + −7 1 − −7 −1 + −7 5 − −7 2 · ∧ + ∧ 2 2 4 4 α2 1 ∧ = β 2 ∧ α − β ∧ α2 = 2 · β ∧ α − 2 · β ∧ α = 0 , = 2 · (−β) ∧ α + β β so 2 1+ √ 2 −7 + √ −1 + −7 4 ∈ BC . (13) This example should make it clear why non-trivial elements of BC can only arise from algebraic numbers: the key relations 1 + β = α and 1 − β −1 = α2 /β in the calculation above forced α and β to be algebraic.
Abelian Varieties by D. Mumford