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Source. Theorem 5, p. 457, 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. The form $F$ is the one fixed in Section 1, p. 432.

Read depth. Claims checked: the statement was read clause by clause on p. 457, the proof (Sections 16--22, pp. 449--457) for its structure. Nothing here is independently reviewed.

Statement

Let $F(x,y)$ be a cubic binary form with integer coefficients and with only simple linear factors (Section 1, p. 432).

Theorem 5 (p. 457). Let $A$ and $B$ be real numbers with $A<B$, let $G$ be the angle

$$A\le\frac yx\le B\quad\text{or}\quad A\le\Bigl(\frac yx\Bigr)^{-1}\le B$$

about the origin, and let $\gamma>0$. Then there is a positive number $t_0(A,B,\gamma)$ such that for every integer $t\ge t_0(A,B,\gamma)$ there is an integer $k$ with

$$0<|k|\le e^{\gamma t^4}$$

for which the conditions $F(x,y)=k$, $(x,y)\in G$ have at least $t$ different solutions $x=p_i$, $y=q_i$ ($i=1,\ldots,t$) with finite integer coordinates.

Proof pointer

Sections 16--22, pp. 449--457. The construction of Theorem 1 is kept. Sections 17--19 show, treating separately the case where two of the points coincide, that a positive proportion of the points with arguments $(3m+1)u$, $-(3m+2)u$ fall on any given arc of the curve (an equidistribution argument), and the lemma of Section 20 puts at least $t+3$ of them in $G$. The denominator bounds of Section 22 then give the bound on $|k|$ as in Theorem 1.

Dependencies

The construction behind Theorem 1 (Sections 1--14).

Bears on

• Problem 829: only through Theorem 6, which the paper obtains by specializing $F$ and $G$; that page states the relation.