Algebra & Number Theory

Unlikely intersections in semiabelian surfaces

Daniel Bertrand and Harry Schmidt

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider a family, depending on a parameter, of multiplicative extensions of an elliptic curve with complex multiplications. They form a 3-dimensional variety G which admits a dense set of special curves, known as Ribet curves, which strictly contains the torsion curves. We show that an irreducible curve W in G meets this set Zariski-densely only if W lies in a fiber of the family or is a translate of a Ribet curve by a multiplicative section. We further deduce from this result a proof of the Zilber–Pink conjecture (over number fields) for the mixed Shimura variety attached to the threefold G , when the parameter space is the universal one.

Article information

Source
Algebra Number Theory, Volume 13, Number 6 (2019), 1455-1473.

Dates
Received: 6 November 2018
Revised: 8 April 2019
Accepted: 14 May 2019
First available in Project Euclid: 21 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.ant/1566353015

Digital Object Identifier
doi:10.2140/ant.2019.13.1455

Mathematical Reviews number (MathSciNet)
MR3994572

Zentralblatt MATH identifier
07103981

Subjects
Primary: 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]
Secondary: 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 11G50: Heights [See also 14G40, 37P30] 11U09: Model theory [See also 03Cxx]

Keywords
semiabelian varieties complex multiplication Zilber–Pink conjecture mixed Shimura varieties heights $o$-minimality Ribet sections

Citation

Bertrand, Daniel; Schmidt, Harry. Unlikely intersections in semiabelian surfaces. Algebra Number Theory 13 (2019), no. 6, 1455--1473. doi:10.2140/ant.2019.13.1455. https://projecteuclid.org/euclid.ant/1566353015


Export citation

References

  • F. Barroero, “CM relations in fibered powers of elliptic families”, J. Inst. Math. Jussieu (online publication August 2017).
  • F. Barroero and L. Capuano, “Unlikely intersections in families of abelian varieties and the polynomial Pell equation”, preprint, 2018.
  • D. Bertrand, “Minimal heights and polarizations on group varieties”, Duke Math. J. 80:1 (1995), 223–250.
  • D. Bertrand, “Special points and Poincaré bi-extensions”, preprint, 2011.
  • D. Bertrand, “Unlikely intersections in Poincaré biextensions over elliptic schemes”, Notre Dame J. Form. Log. 54:3-4 (2013), 365–375.
  • D. Bertrand and B. Edixhoven, “Pink's conjecture on unlikely intersections and families of semi-abelian varieties”, preprint, 2019.
  • D. Bertrand, D. Masser, A. Pillay, and U. Zannier, “Relative Manin–Mumford for semi-Abelian surfaces”, Proc. Edinb. Math. Soc. $(2)$ 59:4 (2016), 837–875.
  • G. Binyamini, “Bezout-type theorems for differential fields”, Compos. Math. 153:4 (2017), 867–888.
  • A. Chambert-Loir, “Géométrie d'Arakelov et hauteurs canoniques sur des variétés semi-abéliennes”, Math. Ann. 314:2 (1999), 381–401.
  • Z. Gao, “A special point problem of André–Pink–Zannier in the universal family of Abelian varieties”, Ann. Sc. Norm. Super. Pisa Cl. Sci. $(5)$ 17:1 (2017), 231–266.
  • Z. Gao, “Towards the André–Oort conjecture for mixed Shimura varieties: the Ax–Lindemann theorem and lower bounds for Galois orbits of special points”, J. Reine Angew. Math. 732 (2017), 85–146.
  • P. Habegger and J. Pila, “O-minimality and certain atypical intersections”, Ann. Sci. Éc. Norm. Supér. $(4)$ 49:4 (2016), 813–858.
  • M. Hindry, “Autour d'une conjecture de Serge Lang”, Invent. Math. 94:3 (1988), 575–603.
  • E. Hrushovski and A. Pillay, “Effective bounds for the number of transcendental points on subvarieties of semi-abelian varieties”, Amer. J. Math. 122:3 (2000), 439–450.
  • O. Jacquinot and K. A. Ribet, “Deficient points on extensions of abelian varieties by $\mathbb{G}_m$”, J. Number Theory 25:2 (1987), 133–151.
  • G. Jones and H. Schmidt, “Pfaffian definitions of Weierstrass elliptic functions”, preprint, 2017.
  • G. Jones and H. Schmidt, “Effective relative Manin–Mumford for families of $\mathbb{G}_m$-extensions of an elliptic curve”, in preparation.
  • G. O. Jones and M. E. M. Thomas, “Effective Pila–Wilkie bounds for unrestricted Pfaffian surfaces”, preprint, 2018.
  • S. Lang, Fundamentals of Diophantine geometry, Springer, 1983.
  • D. Masser and U. Zannier, “Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings”, J. Eur. Math. Soc. $($JEMS$)$ 17:9 (2015), 2379–2416.
  • M. McQuillan, “Division points on semi-abelian varieties”, Invent. Math. 120:1 (1995), 143–159.
  • R. Pink, “A common generalization of the conjectures of André–Oort, Manin–Mumford, and Mordell–Lang”, preprint, 2005, https://tinyurl.com/rpinkaom.
  • G. Rémond, “Une remarque de dynamique sur les variétés semi-abéliennes”, Pacific J. Math. 254:2 (2011), 397–406.
  • J. H. Silverman, “Heights and the specialization map for families of abelian varieties”, J. Reine Angew. Math. 342 (1983), 197–211.
  • E. Viada, “The intersection of a curve with algebraic subgroups in a product of elliptic curves”, Ann. Sc. Norm. Super. Pisa Cl. Sci. $(5)$ 2:1 (2003), 47–75.