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The square-free sieve and the rank of elliptic curves. (English) Zbl 0725.11027

There is a commonly held opinion that “almost all” elliptic curves E over \({\mathbb{Q}}\) have rank 0 or 1 (the rank of the group of rational points on E). The authors study the twists \(E_ D:DY^ 2=X^ 3+AX+B\) of a given curve by a squarefree integer D. Assuming a “parity conjecture” (a consequence of the conjectures of Birch-Swinnerton-Dyer and Taniyama- Weil) they show that at least \(x^{1/2-\epsilon}\) of those \(E_ D\) with \(| D| \leq x\) have rank \(\geq 2\) (and even).
The proof requires them to produce points on the curve, which they do by arranging that the binary form \(D=V(U^ 3+AUV^ 2+BV^ 3)\) is squarefree, using a technique of C. Hooley [Applications of Sieve Methods to the Theory of Numbers, Cambridge Tracts 70 (1976; Zbl 0327.10044)], relating to squarefree values of cubic polynomials in one variable. They deduce an unconditional result concerning the analytic rank (order of vanishing of an L-function) when E is a modular curve.
After seeing a preliminary version of this paper, the reviewer [Q. J. Math. (to appear)] developed a technique that the authors could have used which directly tackles the representation of squarefree D’s by binary forms of degree up to six. They announce that this will be applied in forthcoming work of Top and Stewart which will establish an unconditional (but weaker) version of the authors’ result.

MSC:

11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11N35 Sieves

Citations:

Zbl 0327.10044
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