Shintani theta lifts of harmonic Maass forms

Alfes-Neumann, Claudia; Schwagenscheidt, Markus

doi:10.1090/tran/8265

Shintani theta lifts of harmonic Maass forms
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Trans. Amer. Math. Soc. 374 (2021), 2297-2339 Request permission

Abstract:

We define a regularized Shintani theta lift which maps weight $2k+2$ ($k \in \mathbb {Z}, k \geq 0$) harmonic Maass forms for congruence subgroups to (sesqui-)harmonic Maass forms of weight $3/2+k$ for the Weil representation of an even lattice of signature $(1,2)$. We show that its Fourier coefficients are given by traces of CM values and regularized cycle integrals of the input harmonic Maass form. Further, the Shintani theta lift is related via the $\xi$-operator to the Millson theta lift studied in our earlier work. We use this connection to construct $\xi$-preimages of Zagier’s weight $1/2$ generating series of singular moduli and of some of Ramanujan’s mock theta functions.

References

Claudia Alfes and Stephan Ehlen, Twisted traces of CM values of weak Maass forms, J. Number Theory 133 (2013), no. 6, 1827–1845. MR 3027941, DOI 10.1016/j.jnt.2012.10.008
Claudia Alfes, Michael Griffin, Ken Ono, and Larry Rolen, Weierstrass mock modular forms and elliptic curves, Res. Number Theory 1 (2015), Paper No. 24, 31. MR 3501008, DOI 10.1007/s40993-015-0026-2
Claudia Alfes-Neumann and Markus Schwagenscheidt, On a theta lift related to the Shintani lift, Adv. Math. 328 (2018), 858–889. MR 3771144, DOI 10.1016/j.aim.2018.02.015
Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
Kathrin Bringmann, Nikolaos Diamantis, and Stephan Ehlen, Regularized inner products and errors of modularity, Int. Math. Res. Not. IMRN 24 (2017), 7420–7458. MR 3801423, DOI 10.1093/imrn/rnw225
Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90. MR 2097357, DOI 10.1215/S0012-7094-04-12513-8
Jan Hendrik Bruinier and Jens Funke, Traces of CM values of modular functions, J. Reine Angew. Math. 594 (2006), 1–33. MR 2248151, DOI 10.1515/CRELLE.2006.034
Jan H. Bruinier, Jens Funke, and Özlem Imamoḡlu, Regularized theta liftings and periods of modular functions, J. Reine Angew. Math. 703 (2015), 43–93. MR 3353542, DOI 10.1515/crelle-2013-0035
Jan H. Bruinier, Jens Funke, Özlem Imamoḡlu, and Yingkun Li, Modularity of generating series of winding numbers, Res. Math. Sci. 5 (2018), no. 2, Paper No. 23, 23. MR 3796415, DOI 10.1007/s40687-018-0140-6
Kathrin Bringmann, Karl-Heinz Fricke, and Zachary A. Kent, Special $L$-values and periods of weakly holomorphic modular forms, Proc. Amer. Math. Soc. 142 (2014), no. 10, 3425–3439. MR 3238419, DOI 10.1090/S0002-9939-2014-12092-2
Kathrin Bringmann, Pavel Guerzhoy, and Ben Kane, Shintani lifts and fractional derivatives for harmonic weak Maass forms, Adv. Math. 255 (2014), 641–671. MR 3167495, DOI 10.1016/j.aim.2014.01.015
Kathrin Bringmann, Pavel Guerzhoy, and Ben Kane, On cycle integrals of weakly holomorphic modular forms, Math. Proc. Cambridge Philos. Soc. 158 (2015), no. 3, 439–449. MR 3335420, DOI 10.1017/S0305004115000055
Jan Bruinier and Ken Ono, Heegner divisors, $L$-functions and harmonic weak Maass forms, Ann. of Math. (2) 172 (2010), no. 3, 2135–2181. MR 2726107, DOI 10.4007/annals.2010.172.2135
Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. MR 1625724, DOI 10.1007/s002220050232
Jan H. Bruinier, Borcherds products on O(2, $l$) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, vol. 1780, Springer-Verlag, Berlin, 2002. MR 1903920, DOI 10.1007/b83278
Jan Hendrik Bruinier and Markus Schwagenscheidt, Algebraic formulas for the coefficients of mock theta functions and Weyl vectors of Borcherds products, J. Algebra 478 (2017), 38–57. MR 3621662, DOI 10.1016/j.jalgebra.2016.12.034
Barry A. Cipra, On the Niwa-Shintani theta-kernel lifting of modular forms, Nagoya Math. J. 91 (1983), 49–117. MR 716787, DOI 10.1017/S0027763000020468
Henri Cohen, Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. MR 382192, DOI 10.1007/BF01436180
Jonathan Crawford, A Singular Theta Lift and the Shimura Correspondence, Ph.D. thesis, Durham University, 2015.
W. Duke, Ö. Imamoḡlu, and Á. Tóth, Kronecker’s first limit formula, revisited, Res. Math. Sci. 5 (2018), no. 2, Paper No. 20, 21. MR 3782448, DOI 10.1007/s40687-018-0138-0
Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
Jens Funke and John Millson, Spectacle cycles with coefficients and modular forms of half-integral weight, Arithmetic geometry and automorphic forms, Adv. Lect. Math. (ALM), vol. 19, Int. Press, Somerville, MA, 2011, pp. 91–154. MR 2906907
B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of $L$-series. II, Math. Ann. 278 (1987), no. 1-4, 497–562. MR 909238, DOI 10.1007/BF01458081
Jeffrey A. Harvey and Gregory Moore, Algebras, BPS states, and strings, Nuclear Phys. B 463 (1996), no. 2-3, 315–368. MR 1393643, DOI 10.1016/0550-3213(95)00605-2
Martin Hövel, Automorphe Formen mit Singularitäten auf dem hyperbolischen Raum, TU Darmstadt Diss., 2012.
Winfried Kohnen, Modular forms of half-integral weight on $\Gamma _{0}(4)$, Math. Ann. 248 (1980), no. 3, 249–266. MR 575942, DOI 10.1007/BF01420529
Winfried Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32–72. MR 660784, DOI 10.1515/crll.1982.333.32
Winfried Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237–268. MR 783554, DOI 10.1007/BF01455989
Svetlana Katok and Peter Sarnak, Heegner points, cycles and Maass forms, Israel J. Math. 84 (1993), no. 1-2, 193–227. MR 1244668, DOI 10.1007/BF02761700
Jennifer Kupka, Mock modular morms and traces of singular moduli, TU Darmstadt Master’s thesis, 2017.
W. Kohnen and D. Zagier, Modular forms with rational periods, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 197–249. MR 803368
Shinji Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Math. J. 56 (1975), 147–161. MR 364106
Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
Takuro Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126. MR 389772
Nils-Peter Skoruppa and Don Zagier, Jacobi forms and a certain space of modular forms, Invent. Math. 94 (1988), no. 1, 113–146. MR 958592, DOI 10.1007/BF01394347
Don Zagier, Nombres de classes et formes modulaires de poids $3/2$, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 21, Ai, A883–A886 (French, with English summary). MR 429750
Don Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998) Int. Press Lect. Ser., vol. 3, Int. Press, Somerville, MA, 2002, pp. 211–244. MR 1977587
S. P. Zwegers, Mock $\theta$-functions and real analytic modular forms, $q$-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 269–277. MR 1874536, DOI 10.1090/conm/291/04907