Thomae type formulas for general cyclic covers of
CP1
Let X be an odd degree general cyclic cover of CP1 ramified
at m points, λ1 ...λm and genus g . We define a class of non
positive divisors of degree g − 1 on X , such that the standard
theta function doesn’t vanish on their image in J(X ), the
Jacobian of X . Using Accola’s and Nakayashiki’s ideas we
show that up to a certain determinant of non standard periods
of X , the value of theta functions at these divisors is an
explicit polynomial in the branch points of X . These formulas
generalize the formulas by Bershadsky and Radul for the non
singular cyclic case using Quantum field theory. For cyclic
covers of degree 2(hyper elliptic curves) Thomae used
Riemann’s ideas to publish these formulas in 1866 and 1870.
All are welcome to attend, though some knowledge of
Riemann surface and theta functions will be assumed.