A construction of local points on elliptic curves over modular curves

Let K be the function field of a curve over a finite field of characteristic p, v a place of K, and Kv the completion of K at v. If E is an elliptic curve over K then by the Mordell-Weil theorem E(K) is a finitely generated abelian group. On the other hand E(Kv) is quite large: for example, if E has split multiplicative reduction at v, then E(Kv) has a subgroup of finite index isomorphic as topological group to the direct product of countably many copies of Zp, the p-adic integers. In this paper we study these groups when K is the function field of a modular curve, v is a certain (cuspidal) place, and E is the “universal” elliptic curve over K. We will give an explicit construction using modular forms of a finite rank Zp-module S and an injection φ : S → E(Kv); the rank of S is (at most) the order of vanishing of the Hasse-Weil L-function of E over K. The main result (Theorem 5) says that φ(S) contains a finite index subgroup of the image of the natural inclusion ι : E(K) → E(Kv). (The injection φ is constructed using a particularly nice logarithm from a finite index subgroup of E(Kv) to a certain group of formal power series with coefficients in a ring of Witt vectors. That φ(S) contains a finite index subgroup of ι(E(K)) is proven by exploiting the action of Hecke operators on E(K), on a group closely related to E(Kv), and on a certain cohomology group.) The very explicit nature of the map φ suggests the possibility constructing points in E(K) itself. What is needed is to specify the Z-module φ(ι(E(K))) inside the Zp-module S. (This construction would be in the same spirit as a result of Rubin [4].) In the last section of the paper we propose a candidate for this module when the L-function of E vanishes to order precisely 1 at the center of its critical strip. We remark that our conjectural construction of global points uses neither Heegner points nor Drinfeld modules. The main theorem also has an interesting consequence for automorphic forms over Q which we can state without any further preliminaries.