Advertisement

The Ramanujan Journal

, Volume 46, Issue 3, pp 835–862 | Cite as

Newforms with rational coefficients

  • David P. Roberts
Article

Abstract

We consider the set of classical newforms with rational coefficients and no complex multiplication. We study the distribution of quadratic twist-classes of these forms with respect to weight k and minimal level N. We conjecture that for each weight \(k \ge 6\), there are only finitely many classes. In large weights, we make this conjecture effective: in weights \(18 \le k \le 24\), all classes have \(N \le 30\); in weights \(26 \le k \le 50\), all classes have \(N \in \{2,6\}\); and in weights \(k \ge 52\), there are no classes at all. We study some of the newforms appearing on our conjecturally complete list in more detail, especially in the cases \(N=2\), 3, 4, 6, and 8, where formulas can be kept nearly as simple as those for the classical case \(N=1\).

Keywords

Modular form Newform Weight Level Maeda conjecture 

Mathematics Subject Classification

11F30 (Primary) 11F80 (Secondary) 

Notes

Acknowledgements

The author thanks the conference organizers for the opportunity to speak at Automorphic forms: theory and computation at King’s College London, in September 2016. This paper grew out of the first half of the author’s talk. The list of newforms drawn up here is applied in [19], which is an expanded version of the second half.

References

  1. 1.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3-4), 235–265 (1997). Computational Algebra and Number Theory (London, 1993)Google Scholar
  2. 2.
    Bosman, J.: Polynomials for projective representations of level one forms. In: Computational Aspects of Modular Forms and Galois Representations. Annals of Mathematics Studies, vol. 176, pp. 159–172. Princeton University Press, Princeton, NJ (2011)Google Scholar
  3. 3.
    Breuil, C., Conrad, B., Diamond, F., Taylor, R.: On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Am. Math. Soc. 14(4), 843–939 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, I., Kiming, I., Rasmussen, J.B.: On congruences mod \({\mathfrak{p}}^m\) between eigenforms and their attached Galois representations. J. Number Theory 130(3), 608–619 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dieulefait, L., Tsaknias, P.: Possible connection between a generalized Maeda’s conjecture and local types. arXiv:1608.05285 (2016)
  6. 6.
    DiPippo, S.A., Howe, E.W.: Real polynomials with all roots on the unit circle and abelian varieties over finite fields. J. Number Theory 73(2), 426–450 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ghitza, A., McAndrew, A.: Experimental evidence for Maeda’s conjecture on modular forms. Tbil. Math. J. 5(2), 55–69 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gouvêa, F.Q., Yui, N.: Rigid Calabi-Yau threefolds over \(\mathbb{Q}\) are modular. Expos. Math. 29(1), 142–149 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hida, H., Maeda, Y.: Non-abelian base change for totally real fields. Pac. J. Math. (Special Issue), 181, pp 189–217 (1997). Olga Taussky-Todd: in memoriamGoogle Scholar
  10. 10.
    Jones, J.W., Roberts, D.P.: A database of local fields. J. Symb. Comput. 41(1), pp. 80–97 (2006). http://math.asu.edu/~jj/localfields
  11. 11.
    Jones, J.W., Roberts, D.P.: A database of number fields. LMS J. Comput. Math. 17(1), pp. 595–618 (2014). http://hobbes.la.asu.edu/NFDB
  12. 12.
    Kedlaya, K., Medvedovsky, A.: Mod \(2\) linear algebra and tabulation of rational eigenforms (in preparation)Google Scholar
  13. 13.
    Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture. I. Invent. Math. 178(3), 485–504 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture. II. Invent. Math. 178(3), 505–586 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Koblitz, N.: Introduction to Elliptic Curves and Modular Forms, 2nd edn. Graduate Texts in Mathematics, vol. 97. Springer, New York (1993)Google Scholar
  16. 16.
    Meyer, C.: Modular Calabi-Yau threefolds. Fields Institute Monographs, vol. 22. American Mathematical Society, Providence, RI (2005)zbMATHGoogle Scholar
  17. 17.
    Paranjape, K., Ramakrishnan, D.: Modular forms and Calabi-Yau varieties. In: Arithmetic and Geometry. London Mathematical Society Lecture Note Series, vol. 420, pp. 351–372. Cambridge University Press, Cambridge (2015)Google Scholar
  18. 18.
    Ramanujan, S.: On certain arithmetical functions [Trans. Cambridge Philos. Soc. 22 (1916), no. 9, 159–184]. In: Collected Papers of Srinivasa Ramanujan, pp. 136–162. AMS Chelsea Publ., Providence, RI (2000)Google Scholar
  19. 19.
    Roberts, D.P.: \({P}{G}{L}_2({F}_\ell )\) number fields with rational companion forms. Arxiv, November (2016)Google Scholar
  20. 20.
    Schütt, M.: CM newforms with rational coefficients. Ramanujan J. 19(2), 187–205 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Serre, J.-P.: Sur les représentations modulaires de degré 2 de Gal\((\overline{\rm Q}/{\rm Q})\). Duke Math. J. 54(1), 179–230 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Stein, W.: Modular Forms, A Computational Approach. Graduate Studies in Mathematics, vol. 79. American Mathematical Society, Providence, RI (2007). With an appendix by Paul E. GunnellsGoogle Scholar
  23. 23.
    Swinnerton-Dyer, H.P.F.: On \(l\)-adic representations and congruences for coefficients of modular forms. In: Modular Functions of One Variable, III (Proceedings of International Summer School, University of Antwerp, 1972), pp. 1–55. Lecture Notes in Mathematics, vol. 350. Springer, Berlin (1973)Google Scholar
  24. 24.
    Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebras. Ann. Math. (2) 141(3), 553–572 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    The LMFDB Collaboration: The L-functions and modular forms database (2016). http://www.lmfdb.org
  26. 26.
    Tsaknias, P.: A possible generalization of Maeda’s conjecture. In: Computations with Modular Forms. Contributions in Mathematical and Computational Sciences, vol. 6, pp. 317–329. Springer, Cham (2014)Google Scholar
  27. 27.
    Watkins, M.: Some heuristics about elliptic curves. Exp. Math. 17(1), 105–125 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. (2) 141(3), 443–551 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yui, N.: Modularity of Calabi-Yau varieties: 2011 and beyond. In: Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds. Fields Institute Communications, vol. 67, pp. 101–139. Springer, New York (2013)Google Scholar

Copyright information

© Springer Science+Business Media New York 2018

Authors and Affiliations

  1. 1.Division of Science and MathematicsUniversity of Minnesota-MorrisMorrisUSA

Personalised recommendations