Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

TWO ZAGIER-LIFTS

  • Kang, Soon-Yi (Department of Mathematics Kangwon National University)
  • Received : 2016.11.28
  • Accepted : 2017.02.28
  • Published : 2017.05.15
Download PDF
⟨ Previous Next ⟩

Abstract

Zagier lift gives a relation between weakly holomorphic modular functions and weakly holomorphic modular forms of weight 3/2. Duke and Jenkins extended Zagier-lifts for weakly holomorphic modular forms of negative-integral weights and recently Bringmann, Guerzhoy and Kane extended them further to certain harmonic weak Maass forms of negative-integral weights. New Zagier-lifts for harmonic weak Maass forms and their relation with Bringmann-Guerzhoy-Kane's lifts were discussed earlier. In this paper, we give explicit relations between the two different lifts via direct computation.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. K. Bringmann, P. Guerzhoy, and B. Kane, Shintani lifts and fractional derivatives for harmonic weak Maass forms, Adv. Math. 255 (2014), 641-671. https://doi.org/10.1016/j.aim.2014.01.015
  2. J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45-90. https://doi.org/10.1215/S0012-7094-04-12513-8
  3. W. Duke and P. Jenkins, Integral traces of singular values of weak Maass form, Algebra and Number Theory 2 (2008), no. 5, 573-593. https://doi.org/10.2140/ant.2008.2.573
  4. W. Duke, O. Imamoglu, and A. Toth, Cycle integrals of the j-function and mock modular forms, Ann. Math. 173 (2011), no. 2, 947-981. https://doi.org/10.4007/annals.2011.173.2.8
  5. B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of L-series, II, Math. Ann. 278 (1987), 497-562. https://doi.org/10.1007/BF01458081
  6. D. Jeon, S.-Y. Kang, and C. H. Kim, Weak Maass-Poincare series and weight 3/2 mock modular forms, J. Number Theory 133 (2013), no. 8, 2567-2587. https://doi.org/10.1016/j.jnt.2013.01.011
  7. D. Jeon, S.-Y. Kang, and C. H. Kim, Zagier-lift type arithmetic in harmonic weak Maass forms, J. Number Theory 169 (2016), 227-249. https://doi.org/10.1016/j.jnt.2016.05.012
  8. W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237-268. https://doi.org/10.1007/BF01455989
  9. D. Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I, Irvine, CA, 1998, International Press Lecture Series 3, Part I (Int. Press, Somerville, MA, 2002) 211-244.