MAHLERS MEASURE AND SPECIAL VALUES OF L-FUNCTIONS

If P(x(1),...,x(n),) is a polynomial with integer coefficients, the Ma hler measure M(P) of P is defined to be the geometric mean of \P\ Over the n-torus T-n. For n = 1, M(P) is an algebraic integer, but for n > 1, there is reason to believe that M(P) is usually transcendental. Fo r example, Smyth showed that log M(1 + x + y) = L'(chi(-3), -1), where chi(-3) is the odd Dirichlet character of conductor 3. Here we will d escribe some examples for which it appears that log M(P(x, y)) = rL'(E , O), where E is an elliptic curve and r is a rational number, often e ither an integer or the reciprocal of an integer. Most of the formulas we discover have been verified numerically to high accuracy but not r igorously proved.