Sturm bounds for Siegel modular forms

Abstract

We establish Sturm bounds for degree g Siegel modular forms modulo a prime p, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properties of specializations of Jacobi forms to torsion points. In particular, our approach is completely different from the proofs of the previously known cases g=1,2, which do not extend to the case of general g.

MSC 2010: Primary 11F46; Secondary 11F33

References

  1. 1

    Bruinier, J, Westerholt-Raum, M. Kudla’s modularity conjecture and formal Fourier-Jacobi series. http://arxiv.org/abs/1409.4996.

  2. 2

    Choi, D, Choie, Y, Kikuta, T: Sturm type theorem for Siegel modular forms of genus 2 modulo p. Acta Arith. 158(2), 129–139 (2013).

    MathSciNet  Article  Google Scholar 

  3. 3

    Choi, D, Choie, Y, Richter, O: Congruences for Siegel modular forms. Ann. Inst. Fourier (Grenoble). 61(4), 1455–1466 (2011).

    MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Dewar, M, Richter, O: Ramanujan congruences for Siegel modular forms. Int. J. Number Theory. 6(7), 1677–1687 (2010).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5

    Eichler, M, Zagier, D: The theory of Jacobi forms, Birkhäuser, Boston (1985).

  6. 6

    Faltings, G, Chai, CL: Degeneration of abelian varieties. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 22. Springer, Heidelberg (1990).

    Google Scholar 

  7. 7

    Ono, K: The web of modularity: Arithmetic of the coefficients of modular forms and q-series. In: CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC (2004).

    Google Scholar 

  8. 8

    Poor, C, Ryan, N, Yuen, D: Lifting puzzles in degree four. Bull. Aust. Math. Soc. 80(1), 65–82 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    Poor, C, Yuen, D: Paramodular cusp forms. Math. Comp. 84(293), 1401–1438 (2015).

  10. 10

    Raum, M, Richter, O: The structure of Siegel modular forms mod p and U(p) congruences. Math. Res. Lett. 22(3), 899–928 (2015).

    Article  MathSciNet  Google Scholar 

  11. 11

    Shimura, G: On certain reciprocity-laws for theta functions and modular forms. Acta. Math. 141(1–2), 35–71 (1978).

    MathSciNet  Article  MATH  Google Scholar 

  12. 12

    Stein, WA: Modular forms: a computational approach. In: Graduate Studies in Mathematics. American Mathematical Society, Providence (2007). With an appendix by P. Gunnells.

  13. 13

    Sturm, J: On the congruence of modular forms. In: Number Theory (New York, 1984–1985). Lecture Notes in Math, pp. 275–280. Springer, Springer, Berlin (1987).

    Google Scholar 

  14. 14

    Ziegler, C: Jacobi forms of higher degree. Abh. Math. Sem. Univ. Hamburg. 59, 191–224 (1989).

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

The first author was partially supported by Simons Foundation Grant #200765. The second author thanks the Max Planck Institute for Mathematics for their hospitality. The paper was partially written, while the second author was supported by the ETH Zurich Postdoctoral Fellowship Program and by the Marie Curie Actions for People COFUND Program.

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Correspondence to Olav K Richter.

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The authors declare that they have no competing interests.

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OKR and MW-R performed research and wrote the paper. Both authors read and approved the final manuscript.

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Richter, O.K., Westerholt-Raum, M. Sturm bounds for Siegel modular forms. Res. number theory 1, 5 (2015). https://doi.org/10.1007/s40993-015-0008-4

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Keywords

  • Modular Form
  • Jacobi Form
  • Fourier Series Expansion
  • Torsion Point
  • Siegel Modular Form