Equidistribution Questions for Universal Extensions

Let k be a field, C/k a proper, smooth, geometrically connected curve of genus g ≥ 1 given with a marked rational point 0 ∈ C(k), JC/k := Pic 0 C/k its Jacobian. Concretely, the group JC(k) is the group (under tensor product) of isomorphism classes of invertible sheaves L on C of degree zero. Given a point P ∈ C(k), we denote by I(P ) ⊂ OC the ideal sheaf of functions vanishing at P . Given P1, ..., Pr a finite, possibley empty, list of distinct points in C(k), and D := ∑ i ni[Pi] a divisor of degree zero (i.e., ∑ i ni = 0) supported at these points, we have the invertible sheaf LD := ⊗iI(Pi)i . [This LD is denoted L(−D) in Riemann-Roch notation, and called OC(−D) classically.] If the list is empty, i.e. if D = 0 is the zero divisor, we take L0 := OC . Although not every point in JC(k) need be the isomorphism class of such an LD built of rational points (unless either g = 1 or k is algebraically closed), those that are form a subgroup of JC(k), namely the subgroup generated by all elements of the form I(P )⊗ I(0)−1 with P ∈ C(k). For g = 1, i.e. when C/k is an elliptic curve E/k with origin 0, every element of JE(k) is uniquely of this form (and this bijection of JE(k) with E(k) is what gives E(k) its group structure). Given an invertible sheaf L on C which has degree zero, one has the notion of a connection ∇ on L, namely a k-linear map