We prove that, under certain additional assumptions, the endomorphism ring of the Jacobian of a curve yl = f(x) contains a maximal commutative subring isomorphic to the ring of algebraic integers of the lth cyclotomic field. Here l is an odd prime dividing the degree n of the polynomial f and different from the characteristic of the algebraically closed ground field; moreover, n ≥ 9. The additional assumptions stipulate that all coefficients of f lie in some subfield K over which its (the polynomial's) Galois group coincides with either the full symmetric group Sn or with the alternating group An.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory