Geodesic
Reprint: On the lattice points on curves of genus $1$ (1935)
Documenta mathematica, Mahler Selecta (2019), pp. 399-436.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $F(x,y)$ be a cubic binary form with integer coefficients that is irreducible over the field of rational numbers, and let $k\neq 0$ be an integer. Further, let $A(k)$ be the number of pairs of integers $(x,y)$ satisfying $F(x,y)=k$. Here, Mahler proves that $A(k)$ is unbounded, and that there are infinitely many integers $k$ such that
$A(k)\geqslant \sqrt[4]{\log k}.$
\par Reprint of the author's paper [Proc. Lond. Math. Soc. (2) 39, 431--466 (1935; Zbl 0012.15006; JFM 61.0146.02); corrigendum ibid. 40, 558 (1936; JFM 61.1055.02)].
Classification : 11-03, 11D41, 11G99, 14H25 
@article{DOCMA_2019__S1__a22,
     author = {Mahler, Kurt},
     title = {Reprint: {On} the lattice points on curves of genus \(1\) (1935)},
     journal = {Documenta mathematica},
     pages = {399--436},
     publisher = {mathdoc},
     volume = {Mahler Selecta},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2019__S1__a22/}
}
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Mahler, Kurt. Reprint: On the lattice points on curves of genus \(1\) (1935). Documenta mathematica, Mahler Selecta (2019), pp. 399-436. http://geodesic.mathdoc.fr/item/DOCMA_2019__S1__a22/