Tran, Manh Hung Uniform bounds for rational points on complete intersections of two quadric surfaces. (English) Zbl 1434.11081 Acta Arith. 186, No. 4, 301-318 (2018). From the text: “We give uniform upper bounds for the number of rational points of height at most \(B\) on non-singular complete intersections of two quadrics in \(\mathbb{P}^3\) defined over \(\mathbb{Q}\). To do this, we combine determinant methods with descent arguments.Let \(C\) be a non-singular complete intersection of two quadrics in \(\mathbb P^3\) defined by\[ q(x_0, x_1, x_2, x_3) = r(x_0, x_1, x_2, x_3) = 0, \] where \(q\) and \(r\) are quadratic forms in \(\mathbb Z[x_0, x_1, x_2, x_3]\). Thus \(C\) is of genus 1 and related to elliptic curves. We want to find uniform upper bounds for the counting function\[ N(B) := \# \{P\in C(Q): H(P)\le B\}, \] with the naive height function \(H(P):= \max \{\vert x_0\vert; \vert x_1\vert, \vert x_2\vert, \vert x_3\vert\}\) for \(P = [x_0, x_1, x_2, x_3]\) with coprime integer values of \(x_0, x_1, x_2, x_3\): The first result is the following.Theorem 1.1. Let \(C\) be a non-singular complete intersection of two quadrics in \(\mathbb P^3\) and \(r\) be the rank of the Jacobian \(\mathrm{Jac}(C)\). Then for any \(B\ge 3\) and any positive integer \(m\) we have\[ N(B) \ll m^r(B^{1/(2m^2)} + m^2) \log B\] uniformly in \(C\), with an implied constant independent of \(m\). The proof follows the same strategy as in [D. R. Heath-Brown and D. Testa, Sci. China, Math. 53, No. 9, 2259–2268 (2010; Zbl 1229.11052)] on non-singular cubic curves where the authors combine D. R. Heath-Brown’s \(p\)-adic determinant method in [Ann. Math. (2) 155, No. 2, 553–598 (2002; Zbl 1039.11044)] with descent theory. But we follow the author’s approach in [J. Number Theory 189, 138–146 (2018; Zbl 1401.11104)] and replace the \(p\)-adic determinant method by P. Salberger’s global determinant method [Ann. Sci. Éc. Norm. Supér. 38, No. 1, 93–115 (2005; Zbl 1110.14020)]. Taking \(m = 1 + [\sqrt{\log B}] \) we obtain the following result:Corollary 1.2. Under the condition above we have\[ N(B) \sim d_C(\log B)^{2+r/2}, \] uniformly in \(C\).This estimate should be compared with the classical non-uniform bound of Néron \[ N(B) \sim d_C(\log B)^{r/2}, \] where \(d_C\) is a constant depending on \(C\). The upper bounds in Theorem 1.1 are uniform in the sense that the implicit constants only depend on the rank of the Jacobian. We also use another approach to improve the uniformity and establish upper bounds which do not depend on the rank of \(\mathrm{Jac}(C)\). In this direction, Heath-Brown [loc. cit.] obtained the bound \(N(B) \ll_{\varepsilon} B^{1/2+\varepsilon}\) by using his \(p\)-adic determinant method. Salberger [loc. cit.] proved a slightly better estimate \(N(B)\ll B^{1/2} \log B\). The aim of this paper is to improve these bounds for a class of such curves \(C\) in \(\mathbb P^3\) by using Theorem 1.1 and a refinement of the \(p\)-adic determinant method. We prove the following theorem.Theorem 1.3. Let \(\delta < 3/392\) and \(C\) be a non-singular complete intersection in \(\mathbb P^3\) defined by two simultaneously diagonal quadratic forms \(q\) and \(r\), where \[ q(x_0, x_1, x_2, x_3) = a_0x_0^2+ a_1x_1^2+ a_2x_2^2+ a_3x_3^2, \] \[ r(x_0, x_1, x_2, x_3) = b_0x_0^2+ b_1x_1^2+ b_2x_2^2+ b_3x_3^2 \] with integral coefficients \(a_i\), \(b_i\). Then \[ N(B) \ll B^{1/2-\delta}, \] where the implicit constant depends solely on \(\delta\) and not on the coefficients of \(q\) or \(r\). This class contains examples of elliptic curves with arbitrary \(j\)-invariants.” Reviewer: Olaf Ninnemann (Uffing am Staffelsee) MSC: 11D25 Cubic and quartic Diophantine equations 11D45 Counting solutions of Diophantine equations 11G05 Elliptic curves over global fields 14G05 Rational points Keywords:elliptic curves; Diophantine equations; descent; determinant method; rational points Citations:Zbl 1229.11052; Zbl 1039.11044; Zbl 1401.11104; Zbl 1110.14020 PDFBibTeX XMLCite \textit{M. H. Tran}, Acta Arith. 186, No. 4, 301--318 (2018; Zbl 1434.11081) Full Text: DOI arXiv References: [1] S. Y. An, S. Y. Kim, D. C. Marshall, S. H. Marshall, W. G. McCallum and A. R. Perlis, Jacobians of genus one curves, J. Number Theory 90 (2001), 304-315. · Zbl 1066.14035 [2] N. Broberg, A note on a paper by R. Heath-Brown: “The density of rational points on curves and surfaces”, J. Reine Angew. Math. 571 (2004), 159-178. · Zbl 1053.11027 [3] A. Brumer and K. 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