Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 057, 13 pages      arXiv:1802.04976      https://doi.org/10.3842/SIGMA.2018.057
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4

Ian Kiming a and Nadim Rustom b
a) Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
b) Department of Mathematics, Koç University, Rumelifeneri Yolu, 34450, Sariyer, Istanbul, Turkey

Received February 28, 2018, in final form June 04, 2018; Published online June 13, 2018

Abstract
We work out a non-trivial example of lifting a so-called weak eigenform to a true, characteristic $0$ eigenform. The weak eigenform is closely related to Ramanujan's tau function whereas the characteristic $0$ eigenform is attached to an elliptic curve defined over ${\mathbb Q}$. We produce the lift by showing that the coefficients of the initial, weak eigenform (almost all) occur as traces of Frobenii in the Galois representation on the $4$-torsion of the elliptic curve. The example is remarkable as the initial form is known not to be liftable to any characteristic $0$ eigenform of level $1$. We use this example as illustrating certain questions that have arisen lately in the theory of modular forms modulo prime powers. We give a brief survey of those questions.

Key words: congruences between modular forms; Galois representations.

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References

  1. Bellaïche J., Chenevier G., Families of Galois representations and Selmer groups, Astérisque (2009), xii+314 pages.
  2. Bellaïche J., Khare C., Level 1 Hecke algebras of modular forms modulo $p$, Compos. Math. 151 (2015), 397-415.
  3. Calegari F., Emerton M., The Hecke algebra $T_k$ has large index, Math. Res. Lett. 11 (2004), 125-137, math.NT/0311367.
  4. Camporino M., Pacetti A., Congruences between modular forms modulo prime powers, Rev. Mat. Iberoamericana, to appear, arXiv:1312.4925.
  5. Carayol H., Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, in $p$-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, MA, 1991), Contemp. Math., Vol. 165, Amer. Math. Soc., Providence, RI, 1994, 213-237.
  6. Chen I., Kiming I., On the theta operator for modular forms modulo prime powers, Mathematika 62 (2016), 321-336, arXiv:1301.3087.
  7. Chen I., Kiming I., Rasmussen J.B., On congruences mod ${\mathfrak p}^m$ between eigenforms and their attached Galois representations, J. Number Theory 130 (2010), 608-619, arXiv:0809.3622.
  8. Chen I., Kiming I., Wiese G., On modular Galois representations modulo prime powers, Int. J. Number Theory 9 (2013), 91-113, arXiv:1105.1918.
  9. Grosswald E., Representations of integers as sums of squares, Springer-Verlag, New York, 1985.
  10. Hatada K., Congruences of the eigenvalues of Hecke operators, Proc. Japan Acad. Ser. A Math. Sci. 53 (1977), 125-128.
  11. Jensen C.U., Ledet A., Yui N., Generic polynomials. Constructive aspects of the inverse Galois problem, Mathematical Sciences Research Institute Publications, Vol. 45, Cambridge University Press, Cambridge, 2002.
  12. Jochnowitz N., Congruences between systems of eigenvalues of modular forms, Trans. Amer. Math. Soc. 270 (1982), 269-285.
  13. Katz N.M., Higher congruences between modular forms, Ann. of Math. 101 (1975), 332-367.
  14. Kiming I., The classification of certain low-order $2$-extensions of a field of characteristic different from $2$, C. R. Math. Rep. Acad. Sci. Canada 11 (1989), 57-61.
  15. Kiming I., On the asymptotics of the number of $\overline p$-core partitions of integers, Acta Arith. 80 (1997), 127-139.
  16. Kiming I., On the existence of $\overline p$-core partitions of natural numbers, Quart. J. Math. Oxford Ser. (2) 48 (1997), 59-65.
  17. Kiming I., Olsson J.B., Congruences like Ramanujan's for powers of the partition function, Arch. Math. (Basel) 59 (1992), 348-360.
  18. Kiming I., Rustom N., Wiese G., On certain finiteness questions in the arithmetic of modular forms, J. Lond. Math. Soc. 94 (2016), 479-502, arXiv:1408.3249.
  19. Kolberg O., Congruences for Ramanujan's function $\tau (n)$, Arbok Univ. Bergen Mat.-Natur. Ser. 1962 (1962), 8 pages.
  20. The LMFDB Collaboration: The $L$-functions and Modular Forms Database, http://www.lmfdb.org.
  21. Ono K., Robins S., Wahl P.T., On the representation of integers as sums of triangular numbers, Aequationes Math. 50 (1995), 73-94.
  22. Petersson H., Modulfunktionen und quadratische Formen, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 100, Springer-Verlag, Berlin, 1982.
  23. Rustom N., Filtrations of dc-weak eigenforms, Acta Arith. 180 (2017), 297-318, arXiv:1603.02884.
  24. Rustom N., Congruences modulo prime powers of Hecke eigenvalues in level $1$, arXiv:1710.00928.
  25. Swinnerton-Dyer H.P.F., On $l$-adic representations and congruences for coefficients of modular forms, in Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Lecture Notes in Math, Vol. 1350, Springer, Berlin, 1973, 1-55.
  26. Taixés i Ventosa X., Wiese G., Computing congruences of modular forms and Galois representations modulo prime powers, in Arithmetic, Geometry, Cryptography and Coding Theory 2009, Contemp. Math., Vol. 521, Amer. Math. Soc., Providence, RI, 2010, 145-166, arXiv:0909.2724.
  27. Tsaknias P., On higher congruences of modular Galois representations, Ph.D. Thesis, The University of Sheffield, 2009.
  28. Tsaknias P., Wiese G., Topics on modular Galois representations modulo prime powers, in Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, Editors G. Böckle, W. Decker, G. Malle, Springer, Cham, 2017, 741-763, arXiv:1612.05017.

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