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Source. Theorem 11, pp. 461--462, of Kurt Mahler, On the lattice points on curves of genus 1, Proc. London Math. Soc. (2) 39 (1935), 431--466, the edition named on the source card.

Read depth. Claims checked: the statement was read clause by clause on pp. 461--462, its derivation from Theorem 10 (Section 25, p. 461) for its structure. Nothing here is independently reviewed.

Statement

Theorem 11 (pp. 461--462). For every integer $t\ge1$ and every rational number $J$ there is a cubic curve with absolute invariant $J$ on which lie at least $t$ different points with integer coordinates, and which is defined by an equation of the form

$$Ay^2+Bx^3+Cx+D=0$$

with integer coefficients.

Here the absolute invariant of $y^2=4x^3-g_2x-g_3$ is $J=g_2^3/(g_2^3-27g_3^2)$ (p. 461). The introduction (p. 432) sets the theorem against Siegel's theorem that each cubic curve of genus 1 with rational coefficients has only finitely many lattice points: curves of one invariant $J$ are birationally equivalent, but their numbers of lattice points are not bounded.

Proof pointer

Section 25, p. 461. Choose rationals $g_2,g_3$ with $g_2^3-27g_3^2\ne0$ and $J=g_2^3/(g_2^3-27g_3^2)$. Theorem 10 (p. 461), a case of Theorem 8, gives a rational $\lambda\ne0$ such that $y^2-(1+\lambda)(4x^3-g_2x-g_3)=0$ has at least $t$ rational points; this curve has invariant $J$, and the substitution $x\mapsto x/Z$, $y\mapsto y/Z$ with $Z$ a common denominator of the $t$ points keeps the invariant and turns them into lattice points.

Dependencies

Theorems 8 and 10 of the paper (pp. 459, 461), which have no pages here.

Bears on

No Erdős problem in the corpus.