Riemann surfaces and abelian varieties with an automorphism of prime order (Q1210442)

The authors compute the number of isomorphism classes of principally polarized abelian varieties of dimension \(g=(p-1)/2\) admitting a complex multiplication of prime order \(p\). The number is equal to \(h(p-1)^{- 1}\sum_{d\in S(p)}\varphi(d)2^{(p-1)/2d}\), where \(S(p)\) is the set of odd positive integers \(d\) which divide \(p-1\), \(\varphi\) denotes the Euler function and \(h\) denotes the ideal class number. Some of these varieties (but not all) can be realized as the Jacobian varieties of compact Riemann surfaces admitting a cyclic involution of order \(p\) with the quotient of genus zero. Each such a Riemann surface can be defined by the equation \(y^ p=x^ a(x-1)\) for some \(1\leq a<p\). The number of their isomorphism classes was computed by S. Lefschetz. It is equal to \((p+1)/6\) if \(p\equiv 2\pmod 3\) and to \((p+5)/6\) otherwise. The authors present another proof of this classical result.