Skip to main content
SpringerLink
Log in

Shifted convolution L-series values for elliptic curves

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

Using explicit constructions of the Weierstrass mock modular form and Eisenstein series coefficients, we obtain closed formulas for the generating functions of values of shifted convolution L-functions associated to certain elliptic curves. These identities provide a surprising relation between weight 2 newforms and shifted convolution L-values when the underlying elliptic curve has modular degree 1 with conductor N such that \(\text {genus}(X_0(N)) = 1\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Alfes, M. Griffin, K. Ono, and L. Rolen, Weierstrass mock modular forms and elliptic curves, Res. Number Theory 1 (2015), Art. 24, 31 pp.

  2. O. Beckwith, Asymptotic bounds for special values of shifted convolution dirichlet series, Proc. Amer. Math. Soc. 145 (2017), 2373–2381.

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over \(\mathbb{Q} \): wild \(3\)-adic exercises, J. Amer. Math. Soc. 14 (2001), 843–939.

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Bringmann, M. H. Mertens, and K. Ono, \(p\)-adic properties of modular shifted convolution Dirichlet series, Proc. Amer. Math. Soc. 144 (2016), 1439–1451.

    Article  MathSciNet  MATH  Google Scholar 

  5. K. Bringmann and K. Ono, Coefficients of harmonic Maass forms, In: Partitions, q-Series, and Modular Forms, 23–38, Dev. Math., 23, Springer, New York, 2012.

  6. J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), 45–90.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. H. Bruinier, K. Ono, and R. C. Rhoades, Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues, Math. Ann. 342 (2008), 673–693.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Choi, Poincaré series and the divisors of modular forms, Proc. Amer. Math. Soc. 138 (2010), 3393–3403.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Clemm, Modular forms and Weierstrass mock modular forms, Mathematics 4 (2016), 8 p.

  10. F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics, 228, Springer-Verlag, New York, 2005.

  11. M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957), 267–298.

    Article  MathSciNet  MATH  Google Scholar 

  12. B. H. Gross and D. B. Zagier, Heegner points and derivatives of \(L\)-series, Invent. Math. 84 (1986), 225–320.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Hoffstein, T. A. Hulse, and A. Reznikov, Multiple Dirichlet series and shifted convolutions, J. Number Theory 161 (2016), 457–533.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Guerzhoy, A mixed mock modular solution of the Kaneko-Zagier equation, Ramanujan J. 36 (2015), 149–164.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. H. Mertens and K. Ono, Special values of shifted convolution Dirichlet series, Mathematika 62 (2016), 47–66.

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, In: Current Developments in Mathematics, 2008, 347–454, Int. Press, Somerville, MA, 2009.

  17. R.A. Rankin, Contributions to the theory of Ramanujan’s function \(\tau (n)\) and similar arithmetical functions, Proc. Cambridge Philos. Soc. 35 (1939), 351–372.

    Article  MathSciNet  MATH  Google Scholar 

  18. R. C. Rhoades, Linear relations among Poincaré series via harmonic weak Maaßforms, Ramanujan J. 29 (2012), 311–320.

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Schoeneberg, Elliptic Modular Functions: An Introduction, Die Grundlehren der mathematischen Wissenschaften, Band 203, Springer-Verlag, New York-Heidelberg, 1974.

  20. A. Selberg, V. Bjerknes, and J. Molland, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47–50.

  21. P. G. Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11 (1959), 291–311.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Sturm, Projections of \({\cal{C}} ^\infty \) automorphic forms, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 435–439.

  23. E. T. Whittaker, An expression of certain known functions as generalized hypergeometric functions, Bull. Amer. Math. Soc. 10 (1903), 125–134.

  24. D. Zagier, Modular parametrizations of elliptic curves, Canad. Math. Bull. 28 (1985), 372–384.

  25. S. Zwegers, Mock theta functions, PhD Thesis, Universiteit Utrecht, 2002.

Download references

Acknowledgements

The authors would like to thank the NSF (Grant DMS-1250467) and the Emory REU (especially Dr. Mertens and Professor Ono) for their support. We would also like to thank the anonymous referee for his/her careful review and helpful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Asra Ali

  2. Department of Mathematics, Stanford University, Stanford, CA , 94305, USA

    Nitya Mani

Authors
  1. Asra Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nitya Mani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nitya Mani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ali, A., Mani, N. Shifted convolution L-series values for elliptic curves. Arch. Math. 110, 225–244 (2018). https://doi.org/10.1007/s00013-017-1112-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-017-1112-6

Keywords

Search

Navigation