Skip to main content

Rational points on algebraic curves in infinite towers of number fields

The Ramanujan Journal volume 60, pages 809–824 (2023)Cite this article

Abstract

We study a natural question in the Iwasawa theory of algebraic curves of genus \(>1\). Fix a prime number p. Let X be a smooth, projective, geometrically irreducible curve defined over a number field K of genus \(g>1\), such that the Jacobian of X has good ordinary reduction at the primes above p. Fix an odd prime p and for any integer \(n>1\), let \(K_n^{(p)}\) denote the degree-\(p^n\) extension of K contained in \(K(\mu _{p^{\infty }})\). We prove explicit results for the growth of \(\#X(K_n^{(p)})\) as \(n\rightarrow \infty \). When the Jacobian of X has rank zero and the associated adelic Galois representation has big image, we prove an explicit condition under which \(X(K_{n}^{(p)})=X(K)\) for all n. This condition is illustrated through examples. We also prove a generalization of Imai’s theorem that applies to abelian varieties over arbitrary pro-p extensions.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Bloom, S.: The square sieve and a Lang–Trotter question for generic abelian varieties. J. Number Theory 191, 119–157 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Coates, J.: Galois Cohomology of Elliptic Curves. Lecture Notes at the Tata Institute of Fundamental Research No. 88. Tata Institute of Fundamental Research (2000)

  3. Coates, J., Greenberg, R.: Kummer theory for abelian varieties over local fields. Invent. math. 124(1), 129–174 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cojocaru, A.C., Davis, R., Silverberg, A., Stange, K.E.: Arithmetic properties of the Frobenius traces defined by a rational abelian variety (with two appendices by J. P. Serre). Int. Math. Res. Not. 2017(12), 3557–3602 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Delbourgo, D., Lei, A.: Estimating the growth in Mordell–Weil ranks and Shafarevich–Tate groups over Lie extensions. Ramanujan J. 43(1), 29–68 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Faltings, G.: Finiteness theorems for abelian varieties over number fields. Invent. math. 73(3), 349–366 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  7. Harris, M.: Systematic growth of Mordell–Weil groups of abelian varieties in towers of number fields. Invent. math. 51(2), 123–141 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hung, P.-C., Lim, M.F.: On the growth of Mordell–Weil ranks in \( p \)-adic Lie extensions. Asian J. Math. 24(4), 549–570 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Imai, H.: A remark on the rational points of abelian varieties with values in cyclotomic \(\mathbb{Z}_p\)-extensions. Proc. Jpn Acad. 51(1), 12–16 (1975)

    MATH  Google Scholar 

  10. Kato, K., et al.: p-adic Hodge theory and values of zeta functions of modular forms. Astérisque 295, 117–290 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Lei, A., Ponsinet, G.: On the Mordell–Weil ranks of supersingular abelian varieties in cyclotomic extensions. Proc. Am. Math. Soc. B 7(1), 1–16 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lei, A., Sprung, F.: Ranks of elliptic curves over \({\mathbb{Z}_p^2}\)-extensions. Isr. J. Math. 236, 1–24 (2020)

    Article  MATH  Google Scholar 

  13. Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Invent. math. 18(3–4), 183–266 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  14. Murty, V.K.: Modular Forms and the Chebotarev Density Theorem II. London Mathematical Society Lecture Note Series, pp. 287–308. London Mathematical Society (1997)

  15. Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields, vol. 323. Springer, Berlin (2013)

    MATH  Google Scholar 

  16. O’Meara, O.T.: Symplectic Groups, vol. 16. American Mathematical Society, Providence (1978)

    MATH  Google Scholar 

  17. Pink, R.: \(\ell \)-Adic algebraic monodromy groups, cocharacters, and the Mumford Tate conjecture. J. reine angew. Math. 495, 187–237 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ray, A.: Asymptotic growth of Mordell–Weil ranks of elliptic curves in noncommutative towers. arXiv preprint (2021). arXiv:2109.07457

  19. Raynaud, M.: Courbes sur une variété abélienne et points de torsion. Invent. math. 71(1), 207–233 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ribet, K.: Torsion points of abelian varieties in cyclotomic extensions. Enseign. Math. 27, 315–319 (1981)

    Google Scholar 

  21. Rohrlich, D.E.: On L-functions of elliptic curves and cyclotomic towers. Invent. math. 75(3), 409–423 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schneider, P.: p-Adic height pairings. II. Invent. math. 79(2), 329–374 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  23. Serre, J.-P.: Letter to M. F. Vigneras, January 1st, 1983. In: Œuvres, Collected Papers. IV. Springer, Berlin (2000)

  24. Serre, J.-P.: Resume des cours de 1985–1986. Annuaire du College de france. In: Oeuvres. Collected Papers. IV (2000)

  25. Washington, L.C.: Introduction to Cyclotomic Fields, vol. 83. Springer, New York (1997)

    MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank Jeffrey Hatley, Antonio Lei, Larry Washington, and Tom Weston for helpful comments. The author is especially grateful to Ananth N. Shankar for insightful suggestions including the idea used in the proof of Theorem 3.4.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Anwesh Ray

Authors
  1. Anwesh Ray
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anwesh Ray.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ray, A. Rational points on algebraic curves in infinite towers of number fields. Ramanujan J 60, 809–824 (2023). https://doi.org/10.1007/s11139-022-00583-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-022-00583-3

Keywords

  • Iwasawa theory
  • Rational points of algebraic curves

Mathematics Subject Classification

  • Primary
  • 11G10
  • 11G30
  • 11R23
  • Secondary
  • 14H40