Helfgott, H. A.; Venkatesh, Akshay Integral points on elliptic curves and \(3\)-torsion in class groups. (English) Zbl 1127.14029 J. Am. Math. Soc. 19, No. 3, 527-550 (2006). Let \(E\) be an elliptic curve over a number field \(K\) defined by a Weierstrass equation with coefficient in the integer ring \(\mathcal{O}_K\) of \(K\). Let \(S\) be a finite set of places of \(K\), including all infinite places and all primes dividing the discriminant of \(E\). The key result of the paper asserts that for every \(h_0 \geq 1\) and every choice \(t \in [0,1]\), the number of \(S\)-integer points of \(E(K)\) of canonical height up to \(h_0\) is at most \[ O_{\varepsilon, [K:\mathbb{Q}]} \left(C^s \varepsilon^{-2 (s +[K:\mathbb{Q}])}s^{[K:\mathbb{Q}]}(1+ \log h_0)^2 e^{t[K:\mathbb{Q}]h_0+(\beta(t)+\varepsilon)r} \right), \]for every sufficiently small \(\varepsilon\), where \(r\) is the rank of \(E(K)\) as a \(\mathbb{Z}\)-lattice, \(s = \# S\), \(C\) is an absolute constant, and \[ \beta(t) = \frac{1+f(t)}{2f(t)} \log \frac{1+f(t)}{2f(t)} - \frac{1-f(t)}{2f(t)} \log \frac{1-f(t)}{2f(t)}, f(t) = \frac{\sqrt{(1+t)(3-t)}}{2}, \]for \(t \in [0,1), \beta(1) =0\). In order to prove it the authors subdivide the set of \(S\)-integral points on \(E\) into points modulo suitable ideal in \(\mathcal{O}_K\) with norm about \(e^{t[K:\mathbb{Q}]h_0}\). After some refinement of this partition the points from the same part turn out to be well separated in the Mordell-Weil lattice. The term \(e^{t[K:\mathbb{Q}]h_0}\) arises from the number of parts, and the term \(e^{\beta(t)r}\) is obtained by applying the sphere-packing to each part.Choosing \(t\) to be optimal, the authors derive that for a positive integer \(N\) there are at most \(O(N^{0.22377\ldots})\) elliptic curves over \(\mathbb{Q}\) of conductor \(N\). Furthermore, for every nonzero integer \(D\), at most \(O([D]^{0.44178\ldots})\) elements of the class group of \(\mathbb{Q}[\sqrt{D}]\) are 3-torsion. These bounds improve the known bounds and are obtained by new methods. The paper is concluded by discussion how on can use the developed techniques to bound the number of points on a curve of higher genus without knowing the rank of its Jacobian. Reviewer: Vasyl I. Andriychuk (Lviv) Cited in 3 ReviewsCited in 32 Documents MSC: 11G05 Elliptic curves over global fields 11R29 Class numbers, class groups, discriminants 14G05 Rational points 11R11 Quadratic extensions PDFBibTeX XMLCite \textit{H. A. Helfgott} and \textit{A. Venkatesh}, J. Am. Math. Soc. 19, No. 3, 527--550 (2006; Zbl 1127.14029) Full Text: DOI arXiv References: [1] A. Baker, The Diophantine equation \?²=\?\?³+\?\?²+\?\?+\?, J. London Math. Soc. 43 (1968), 1 – 9. · Zbl 0155.08701 [2] B. Brindza, J.-H. Evertse, and K. Győry, Bounds for the solutions of some Diophantine equations in terms of discriminants, J. Austral. Math. Soc. Ser. A 51 (1991), no. 1, 8 – 26. · Zbl 0746.11018 [3] Armand Brumer and Kenneth Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), no. 4, 715 – 743. · Zbl 0376.14011 [4] E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), no. 2, 337 – 357. · Zbl 0718.11048 [5] Armand Brumer and Joseph H. 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