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LOW RANK SPECIALISATIONS OF ELLIPTIC SURFACES

Part of: Arithmetic algebraic geometry Surfaces and higher-dimensional varieties Curves

Published online by Cambridge University Press:  13 January 2025

MENTZELOS MELISTAS Open the ORCID record for MENTZELOS MELISTAS [Opens in a new window]
MENTZELOS MELISTAS*
Affiliation:
Department of Applied Mathematics, University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands
*
e-mail: mentzmel@gmail.com
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Abstract

Let $E/\mathbb {Q}(T)$ be a nonisotrivial elliptic curve of rank r. A theorem due to Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math. 342 (1983), 197–211] implies that the rank $r_t$ of the specialisation $E_t/\mathbb {Q}$ is at least r for all but finitely many $t \in \mathbb {Q}$. Moreover, it is conjectured that $r_t \leq r+2$, except for a set of density $0$. When $E/\mathbb {Q}(T)$ has a torsion point of order $2$, under an assumption on the discriminant of a Weierstrass equation for $E/\mathbb {Q}(T)$, we produce an upper bound for $r_t$ that is valid for infinitely many t. We also present two examples of nonisotrivial elliptic curves $E/\mathbb {Q}(T)$ such that $r_t \leq r+1$ for infinitely many t.

Keywords

elliptic surfaceelliptic curverankspecialisation

MSC classification

Primary: 14J27: Elliptic surfaces
Secondary: 11G05: Elliptic curves over global fields 14H52: Elliptic curves
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 10
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Aguirre, J., Lozano-Robledo, Á. and Peral, J. C., ‘Elliptic curves of maximal rank’, in: Proceedings of the Segundas Jornadas de Teoría de Números (eds. J. Cilleruelo, E. Gonz\'alez Jim\'enez, A. Quir\'os and X. Xarles) (Revista Matemática Iberoamericana, Madrid, 2008), 1–28.Google Scholar
Bouniakowsky, V., ‘Sur les diviseurs numériques invariables des fonctions rationnelles entières’, Mém. Acad. Sc. St. Pétersbourg 6 (1857), 305–329.Google Scholar
Caro, J. and Pasten, H., ‘On the fibres of an elliptic surface where the rank does not jump’, Bull. Aust. Math. Soc. 108(2) (2023), 276–282.CrossRefGoogle Scholar
Conrad, B., Conrad, K. and Helfgott, H., ‘Root numbers and ranks in positive characteristic’, Adv. Math. 198(2) (2005), 684–731.CrossRefGoogle Scholar
Friedlander, J. and Iwaniec, H., Opera de Cribro, Colloquium Publications, 57 (American Mathematical Society, Providence, RI, 2010).CrossRefGoogle Scholar
Halberstam, H. and Richert, H.-E., Sieve Methods, London Mathematical Society Monographs, 4 (Academic Press, London, 1974).Google Scholar
Husemöller, D., Elliptic Curves, 2nd edn, Graduate Texts in Mathematics, 111 (Springer-Verlag, New York, 2004).Google Scholar
Iwaniec, H., ‘Almost-primes represented by quadratic polynomials’, Invent. Math. 47 (1978), 171–188.CrossRefGoogle Scholar
Kapoor, V., ‘Almost-primes represented by quadratic polynomials’, Master of Science thesis, Simon Fraser University, 2006; arXiv:1910.2019.02885 (2019).Google Scholar
Lemke Oliver, R. J., ‘Almost-primes represented by quadratic polynomials’, Acta Arith. 151(3) (2012), 241–261.CrossRefGoogle Scholar
Neumann, O.. ‘Elliptic curves with prescribed reduction. I’, Math. Nachr. 49 (1971), 106–123.Google Scholar
Richert, H.-E., ‘Selberg’s sieve with weights’, Mathematika 16 (1969), 1–22.CrossRefGoogle Scholar
Setzer, B., ‘Elliptic curves of prime conductor’, J. Lond. Math. Soc. (2) 10 (1975), 367–378.CrossRefGoogle Scholar
Silverman, J. H., ‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math. 342 (1983), 197–211.Google Scholar
Silverman, J. H., ‘Divisibility of the specialization map for families of elliptic curves’, Amer. J. Math. 107 (1985), 555–565.CrossRefGoogle Scholar
Silverman, J. H., Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 151 (Springer-Verlag, New York, 1994).CrossRefGoogle Scholar