Period Determinant of an Irregular Connection over an Elliptic Curve

In this article, we calculate the period of an irregular singular connection ∇ = d+dy on the legendre curve U : y = x(x−1)(x−λ). We calculate its de Rham cohomology and the cycles in the homology of the dual connection and describe the period matrix. Terasoma’s work is introduced to approximate the direct image connection π∗(∇) by a sequence of regular connections as an intermediate step where π : U → A = Spec k[y], (x, y) 7→ y. Finally, we will make the comparison of the period obtained by this approximation and that ∇ over U .