We calculate the first homology group of the mapping class group with
coefficients in the first rational homology group of the universal abelian Z/LZ-cover of the surface. If the surface has one marked point, then the
answer is \Q^{\tau(L)}, where Ï„(L) is the number of positive divisors of
L. If the surface instead has one boundary component, then the answer is
\Q. We also perform the same calculation for the level L subgroup of the
mapping class group. Set HL​=H1​(Σg​;Z/LZ). If the surface has one
marked point, then the answer is \Q[H_L], the rational group ring of HL​.
If the surface instead has one boundary component, then the answer is \Q.Comment: 32 pages, 10 figures; numerous corrections and simplifications; to
appear in J. Topol. Ana