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Denote by C the elliptic curve y = x(x − 1)(x− s) over the function field Q(s). It degenerates only over the points s = 0, 1,∞ of P and so defines a fiber bundle over C \ {0, 1} with fibers being elliptic curves. Let Fs denote the fiber over the point s. The maps ι : (x, y, s) 7→ (1 − x, iy, 1 − s) and τ : (x, y, s) 7→ (x̄, ȳ, s̄) are topological automorphisms of C. Note ι = −1, τ = 1. The fundamental groupoid of the base space C \ {0, 1} acts on the family of homology groups H(Fs,Z), which are all abstractly isomorphic to Z ⊕ Z. Choosing the base point s = 1/2, there are two generators p and q for the topological fundamental group based at s = 1/2 given by loops around 0 and 1 in the s-plane.
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تاریخ انتشار 2004