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CMS Jeffery-Williams Lecture / Conférence Jeffery-Williams de la
SMC
(David Boyd, Organizer)
- DAVID BOYD, University of British Columbia, Vancouver, British Columbia
Mahler's measure, hyperbolic geometry and the dilogarithm
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If P(x,y) is a polynomial with integer coefficients, then the Mahler
measure m(P) is defined to be the average of log|P| over the
torus. That is
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m(P) = |
ó
õ
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1
0
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ó
õ
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1
0
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log|P(e2pi s,e2pi t)| ds dt. |
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The starting point of our lecture will be the remarkable formula
of Smyth, proved in the late 1970's:
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m(1 + x + y) = |
3 Ö3
4p
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L(c-3,2), |
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where L(c-3,s) is the Dirichlet L-series for the odd
character of conductor 3.
Chinburg asked the natural question whether there are other similar
formulas for other Dirichlet characters. We will show how this
question is connected with hyperbolic geometry and to K-theory via
the theory of the Bloch group, and use this to produce many similar
formulas (some conjectural and some proved) involving Dirichlet
L-functions and L-functions of elliptic curves.
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