---

Source. Theorem 1, p. 447, 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 and the hypotheses on $F$ were read clause by clause on pp. 432 and 447, the proof for its structure. Nothing here is independently reviewed.

Statement

Let

$$F(x,y)=a_0x^3+a_1x^2y+a_2xy^2+a_3y^3$$

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

Theorem 1 (p. 447). For every $\gamma>0$ there is a positive number $t_0(\gamma)$ such that for every integer $t\ge t_0(\gamma)$ there is an integer $k$ with

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

which $F$ represents in at least $t$ different ways, $k=F(p_h,q_h)$ ($h=1,\ldots,t$), with integers $p_h,q_h$.

So the number $A(k)$ of integer solutions of $F(x,y)=k$, finite for each $k\ne0$ by Thue's theorem when $F$ is irreducible (p. 431), is not bounded. The introduction (p. 431) states that there are infinitely many $k$ with $A(k)\ge\sqrt[4]{\log k}$. For the forms $x^3+y^3$ and $xy(x+y)$ with positive variables, the paper proves bounds of this shape as Theorem 6 and Theorem 7, through the angle version Theorem 5. The introduction also says that the paper cannot prove similar results for primitive solutions ($x,y$ coprime), and that whether their number is bounded is still open (pp. 431--432).

Proof pointer

Sections 1--14, pp. 432--447. The curve $C:F(x,y)=1$ has genus 1 and is uniformized by elliptic functions; from a point with elliptic argument $u_1$ the chord-and-tangent construction gives the points with arguments $(3m+1)u_1$ and $-(3m+2)u_1$, $m=0,\ldots,n-1$. Sections 4--7 choose a lattice point $(x_1,y_1)$ with $\max(|x_1|,|y_1|)\le65n^3$ off finitely many lines through the origin, so that on the similar curve $C(k_1)$, $k_1=F(x_1,y_1)\ne0$, the $2n$ points built from it are distinct and finite; then $0<|k_1|\le c_5(65n^3)^3$ (Section 14). Sections 8--13 bound the denominators of their coordinates by a recursion; with $Z$ the least common multiple of the denominators, the $2n$ points scaled by $Z$ are lattice points on $C(Z^3k_1)$, and the bounds give $0<|k|\le e^{16\gamma n^4}$ for $n\ge n_0(\gamma)$. Put $t=2n$.

Dependencies

None outside the paper's own Sections 1--14.

Bears on

• Problem 829: only through Theorem 6, on sums of two cubes of positive integers, which the paper derives from the angle version Theorem 5; that page states the relation. Theorem 1 itself counts solutions in integers of either sign and proves no upper bound.