By Pierre E. Cartier, Bernard Julia, Pierre Moussa, Pierre Vanhove

ISBN-10: 3540303073

ISBN-13: 9783540303077

The relation among arithmetic and physics has a protracted heritage, within which the function of quantity thought and of different extra summary elements of arithmetic has lately develop into extra prominent.More than ten years after a primary assembly in 1989 among quantity theorists and physicists on the Centre de body des Houches, a moment 2-week occasion thinking about the wider interface of quantity thought, geometry, and physics.This ebook is the results of that fascinating assembly, and collects, in 2 volumes, prolonged types of the lecture classes, by way of shorter texts on distinct issues, of eminent mathematicians and physicists.

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**Extra resources for Frontiers in Number Theory, Physics, and Geometry II**

**Sample text**

0 in the upper half-plane. The values in the lower half-plane are obtained from D(¯ √z ) = −D(z). 0149 . . , attained at the point (1 + i 3)/2. The function D(z), which was discovered by D. Wigner and S. Bloch (cf. [1]), has many other beautiful properties. In particular: (II) D(z), which is a real-valued function on C, can be expressed in terms of a function of a single real variable, namely D(z) = z 1 D +D 2 z¯ 1 − 1/z 1 − 1/¯ z +D 1/(1 − z) 1/(1 − z¯) (2) The Dilogarithm Function 11 which expresses D(z) for arbitrary complex z in terms of the function D(eiθ ) = ℑ[Li2 (eiθ )] = ∞ sin nθ .

Equals π mN / |d| times a rational number. √ If r2 > 0, then no such simple result holds. , π2 D e2πi/7 + D e4πi/7 − D e6πi/7 . ζQ(√−7) (2) = √ 3 7 Thus the values of ζF (2) for imaginary quadratic ﬁelds can be expressed in closed form in terms of values of the Bloch-Wigner function D(z) at algebraic arguments z. By using the ideas of the last section we can prove a much stronger statement. Let√O denote the √ring of integers of F (this is the Z-lattice in C spanned by 1 and −a or (1 + −a)/2, depending whether d = −4a or d = −a).

Z3 can be brought into the form {∞, 0, 1, z} by the action of some element of SL2 (C) on P1 (C), and the group SL2 (C) acts on H3 by isometries. The (anti-)symmetry properties of D under permutations of the zi are obvious from the geometric interpretation (7), since renumbering the vertices leaves ∆ unchanged but may reverse its orientation. Formula (6) is also an immediate consequence of (7), since the ﬁve tetrahedra spanned by four at a time of z0 , . . , z4 ∈ P1 (C), counted positively or negatively as in (6), add up algebraically to the zero 3-cycle.

### Frontiers in Number Theory, Physics, and Geometry II by Pierre E. Cartier, Bernard Julia, Pierre Moussa, Pierre Vanhove

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