Szpiro’s conjecture when the denominator of the \(j\)-invariant is small

Hector Pasten

doi:10.56754/0719-0646.2802.383

DOI:

https://doi.org/10.56754/0719-0646.2802.383

Abstract

We prove Szpiro's conjecture for elliptic curves over the rationals having \(j\)-invariant with denominator of logarithmic size with respect to its numerator.

Keywords

Szpiro’s conjecture , elliptic curves , discriminant , conductor

Mathematics Subject Classification:

11G05 , 11G50

Hector Pasten Departamento de Matemáticas, Pontificia Universidad Católica de Chile. Facultad de Matemáticas, 4860 Av. Vicuña Mackenna, Macul, RM, Chile. https://orcid.org/0000-0002-7789-8459

Pages: 383-389
Date Published: 2026-05-31
Vol. 28 No. 2 (2026)

S. Chan, “Almost all elliptic curves with prescribed torsion have Szpiro ratio close to the expected value,” 2024, arXiv: 2407.13850.

M. Hindry and J. H. Silverman, “The canonical height and integral points on elliptic curves,” Invent. Math., vol. 93, no. 2, pp. 419–450, 1988, doi: 10.1007/BF01394340.

D. Lorenzini, “Models of curves and wild ramification,” Pure Appl. Math. Q., vol. 6, no. 1, pp. 41–82, 2010, doi: 10.4310/PAMQ.2010.v6.n1.a3.

D. W. Masser, “Abcological anecdotes,” Mathematika, vol. 63, no. 3, pp. 713–714, 2017, doi: 10.1112/S0025579317000146.

J.-F. Mestre and J. Oesterlé, “Courbes de Weil semi-stables de discriminant une puissance m-ième.” J. Reine Angew. Math., vol. 400, pp. 173–184, 1989, doi: 10.1515/crll.1989.400.173.

M. R. Murty and H. Pasten, “Modular forms and effective Diophantine approximation,” J. Number Theory, vol. 133, no. 11, pp. 3739–3754, 2013, doi: 10.1016/j.jnt.2013.05.006.

H. Pasten, “Shimura curves and the abc conjecture,” J. Number Theory, vol. 254, pp. 214–335, 2024, doi: 10.1016/j.jnt.2023.07.002.

F. Pazuki, “Modular invariants and isogenies,” Int. J. Number Theory, vol. 15, no. 3, pp. 569–584, 2019, doi: 10.1142/S1793042119500295.

F. Pellarin, “On an explicit upper bound for the degree of an isogeny between two elliptic curves,” Acta Arith., vol. 100, no. 3, pp. 203–243, 2001, doi: 10.4064/aa100-3-1.

J. Pesenti and L. Szpiro, “Inequality for the discriminant of elliptic pencils with arbitrary reductions,” Compos. Math., vol. 120, no. 1, pp. 83–117, 2000, doi: 10.1023/A:1001736823128.

J.-P. Serre, Local fields, ser. Grad. Texts Math. Springer, Cham, 1979, vol. 67.

J. H. Silverman, “Heights and elliptic curves,” in Arithmetic Geometry. Springer New York, 1986, pp. 253–265, doi: 10.1007/978-1-4613-8655-1_10.

J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, ser. Grad. Texts Math. New York, NY: Springer-Verlag, 1994, vol. 151.

L. Szpiro, “Présentation de la théorie d’Arakélov,” in Current Trends in Arithmetical Algebraic Geometry, ser. Contemporary Mathematics. Providence, RI:American Mathematical Society, 1987, vol. 67, pp. 279–293.

J. Tate, “Algorithm for determining the type of a singular fiber in an elliptic pencil.” in Modular functions of one variable. IV. Proceedings of the international summer school, University of Antwerp, RUCA, July 17 – August 3, 1972. Berlin: Springer, 1975, pp. 33–52.