189-726B: Modular Forms II Assignment 12

Due: Wednesday, April 16

1. Let E be an elliptic curve over the ring R = k[λ][1/P (λ)], where P ∈ k[λ] and k is a field. Prove that the Gauss-Manin connection

∇ : H_{dR}^{1}(E/R)⟶H_{dR}^{1}(E/R) ⊗ Ω^{1}(R/k)

satisfies the Leibniz rule for differentiation:

∇(fω) = ω ⊗ (df) + f∇(ω),

for all f ∈ R and ω ∈ H_{dR}^{1}(E/R).

2. Let E be the Legendre family

y^{2} = x(x − 1)(x − λ)

of elliptic curves defined over the ring R = k[λ][\frac{1}{λ(λ−1)}], and let ω = \frac{dx}{y} be the generator of the module of regular differentials on E. Compute the class

∇_{λ}ω = ⟨∇(ω), \frac{d}{dλ}⟩,

(where the pairing on the right is the natural duality between differential forms and derivations). Express this class as a linear combination, with coefficients in R, of the generators ω and η = xω for H_{dR}^{1}(E/R).