Vectorial extensions of Jacobians (Q1175676)

A generalized jacobian of a curve represents isomorphism classes of invertible sheaves algebraically equivalent to zero (AEZ) with a trivialization along a closed subscheme. The universal vector extension of its jacobian represents isomorphism classes of AEZ invertible sheaves with a connection. The bridge between these two schemes is a scheme which represents isomorphism classes of AEZ invertible sheaves with a connection and a `` horizontal trivialization'' along a closed subscheme. The author makes explicit the connection between the theory of the universal vectorial extension and the theory of generalized jacobians, for curves over an arbitrary base. In particular, he proves those results needed in his paper ``The universal vectorial bi-extension and \(p\)-adic heights'' [Invent. Math. 103, No. 3, 631-650 (1991)] to verify the equality of Mazur-Tate height and the height defined in his joint paper with \textit{B. H. Gross} ``\(p\)-adic heights on curves'' in Adv. Stud. Pure Math. 17, 73-81 (1989).