Topology and its Applications 124 (2002) 347–353 Counting curves on elliptic ruled surface Tian-Jun Li a,∗ , Ai-Ko Liu b,1 a Department of Mathematics, Yale University, New Haven, CT 06520, USA b Department of Mathematics, MIT, Cambridge, MA 02139, USA Received 9 August 1999 1. Introduction In this paper, we present some calculation of the Gromov–Witten invariants of S 2 × T 2 . Since the symplectic Gromov–Witten invariants in fact only depend on the deformation class of symplectic forms and we have shown in [10] that there is a unique deformation class on S 2 × T 2 , we merely need to compute the Gromov–Witten invariants for some spe- cific symplectic structure. We will actually pick some Kahler structures in the computation. Let (M,ω) be a symplectic S 2 × T 2 and [S 2 ] and [T 2 ] be the homology classes represented by S 2 × pt and pt × T 2 , respectively, and pair positively with the symplectic form ω. Denote the homology class l [S 2 ]+ d [T 2 ] by A l,d and we simply write A 1,d as A d . Our first result is about the embedded genus one curves of the sequence of classes A l,1 . More precisely, let us define N 1 (A l,1 ) as the number of embedded genus 1 curves in the class A 1,d and passing through 2l points. Theorem 1. The Gromov–Witten invariants N 1 (A l,1 ) = 2. The second result is about the general nodal curves of the sequence of classes A d . Denote N g (A d ) (or n g (A d )) the number of genus g curves in class A d , passing through g + 1 points (or passing through g points and intersecting two circles which generate the first integral homology). It is more illuminating to assemble them into generating functions. To that end, we recall that the quasimodular form G 2 is defined by G 2 =- 1 24 + ∞  k=1 σ k q k where σ k = ∑ d |k d is the partition function. * Corresponding author. Department of Mathematics, Princeton University, Princeton, NJ 08544, USA. E-mail addresses: tli@math.yale.edu (T.-J. Li), akliu@math.mit.edu (A.-K. Liu). 1 Present address: Department of Mathematics, UC Berkeley, Berkeley, CA 94729, USA. 0166-8641/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0166-8641(01)00235-8