1999
DOI: 10.1090/s0025-5718-99-01058-3
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Zeta functions of a class of elliptic curves over a rational function field of characteristic two

Abstract: Abstract. We show how to calculate the zeta functions and the orders |X| of Tate-Shafarevich groups of the elliptic curves with equation Y 2 + XY = X 3 + αX 2 + const · T −k over the rational function field Fq(T ), where q is a power of 2. In the range q = 2, k ≤ 37, α ∈ F 2 [T −1 ] odd of degree ≤ 19, the largest values obtained for |X| are 47 2 (one case), 39 2 (one case) and 27 2 (three cases).We observe and discuss a remarkable pattern for the distributions of signs in the functional equation and of fudge … Show more

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