DEGREES OF CURVES IN ABELIAN-VARIETIES

The degree of a curve C in a polarized abelian variety (X, lambda) is the integer d = C.lambda. When C is irreducible and generates X, we find a lower bound on d which depends on n and the degree of the polarization lambda. The smallest possible degree is d = n and is obtained only for a smooth curve in its Jacobian with its principal polarization (Ran, Collino). The cases d = n+1 and d = n+2 are studied. Moreover, when X is simple, it is shown, using results of Smyth on the trace of totally positive algebraic integers, that if d less than or equal to 1.7719 n, then C is smooth and X is isomorphic to its Jacobian. We also get an upper bound on the geometric genus of C iri terms of its degree.