Index theory on curves

Galois Theory and Torsion Points on Curves

We begin with a brief history of the problem of determining the set of points of a curve that map to torsion points of the curve’s Jacobian. Let K be a number field, and suppose that X/K is an algebraic curve of genus g ≥ 2. Assume, furthermore, that X is embedded in its Jacobian variety J via a K-rational Albanese map i; thus there is a K-rational divisor D of degree one on X such that i = iD ...

On the K-theory of Elliptic Curves

Let A be the coordinate ring of an affine elliptic curve (over an infinite field k) of the form X − {p}, where X is projective and p is a closed point on X. Denote by F the function field of X. We show that the image of H•(GL2(A), Z) in H•(GL2(F ), Z) coincides with the image of H•(GL2(k), Z). As a consequence, we obtain numerous results about the K-theory of A and X. For example, if k is a num...

Introduction of Singularity Theory on Curves

On Hodge Theory of Singular Plane Curves

The dimensions of the graded quotients of the cohomology of a plane curve complement U = P \ C with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on H(U,C).

On the Index of the Heegner Subgroup of Elliptic Curves

Let E be an elliptic curve of conductor N and rank one over Q. So there is a non-constant morphism X 0 (N) −→ E defined over Q, where X 0 (N) = X0(N)/wN and wN is the Fricke involution. Under this morphism the traces of the Heegner points of X 0 (N) map to rational points on E. In this paper we study the index I of the subgroup generated by all these traces on E(Q). We propose and also discuss ...