On 𝑝-adic harmonic Maass functions

Griffin, Michael

doi:10.1090/tran/8105

On $p$-adic harmonic Maass functions
HTML articles powered by AMS MathViewer

by Michael J. Griffin PDF

Trans. Amer. Math. Soc. 373 (2020), 7019-7066 Request permission

Abstract:

Modular and mock modular forms possess many striking $p$-adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology of modular curves. In the setting of over-convergent $p$-adic modular forms, Candelori and Castella showed this leads to $p$-adic analogs of harmonic Maass forms.

In this paper we take an analytic approach to construct $p$-adic analogs of harmonic Maass forms of weight $0$ with square free level. Although our approaches differ, where the two theories intersect the forms constructed are the same. However our analytic construction defines these functions on the full super-singular locus as well as on the ordinary locus.

As with classical harmonic Maass forms, these $p$-adic analogs are connected to weight $2$ cusp forms and their modular derivatives are weight $2$ weakly holomorphic modular forms. Traces of their CM values also interpolate the coefficients of half-integer weight modular and mock modular forms. We demonstrate this through the construction of $p$-adic analogs of two families of theta lifts for these forms.

References

Claudia Alfes, Formulas for the coefficients of half-integral weight harmonic Maaßforms, Math. Z. 277 (2014), no. 3-4, 769–795. MR 3229965, DOI 10.1007/s00209-014-1278-6
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
Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. MR 1625724, DOI 10.1007/s002220050232
Kathrin Bringmann, Pavel Guerzhoy, and Ben Kane, Mock modular forms as $p$-adic modular forms, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2393–2410. MR 2888211, DOI 10.1090/S0002-9947-2012-05525-5
Kathrin Bringmann, Pavel Guerzhoy, and Ben Kane, Half-integral weight $p$-adic coupling of weakly holomorphic and holomorphic modular forms, Res. Number Theory 1 (2015), Paper No. 26, 13. MR 3501010, DOI 10.1007/s40993-015-0027-1
Kathrin Bringmann and Ken Ono, Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series, Math. Ann. 337 (2007), no. 3, 591–612. MR 2274544, DOI 10.1007/s00208-006-0048-0
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
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 Hendrik Bruinier and Ken Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms, Adv. Math. 246 (2013), 198–219. MR 3091805, DOI 10.1016/j.aim.2013.05.028
Jan H. Bruinier, Ken Ono, and Robert C. Rhoades, Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues, Math. Ann. 342 (2008), no. 3, 673–693. MR 2430995, DOI 10.1007/s00208-008-0252-1
Jan Hendrik Bruinier and Oliver Stein, The Weil representation and Hecke operators for vector valued modular forms, Math. Z. 264 (2010), no. 2, 249–270. MR 2574974, DOI 10.1007/s00209-008-0460-0
Luca Candelori, Towards a p-adic theory of harmonic weak Maass forms, M.Sc. Thesis, McGill University, 2010.
Luca Candelori, Harmonic weak Maass forms of integral weight: a geometric approach, Math. Ann. 360 (2014), no. 1-2, 489–517. MR 3263171, DOI 10.1007/s00208-014-1043-5
Luca Candelori and Francesc Castella, A geometric perspective on $p$-adic properties of mock modular forms, Res. Math. Sci. 4 (2017), Paper No. 5, 15. MR 3619605, DOI 10.1186/s40687-017-0095-z
W. Duke, Ö. Imamoḡlu, and Á. Tóth, Cycle integrals of the $j$-function and mock modular forms, Ann. of Math. (2) 173 (2011), no. 2, 947–981. MR 2776366, DOI 10.4007/annals.2011.173.2.8
W. Duke, Ö. Imamoḡlu, and Á. Tóth, Real quadratic analogs of traces of singular moduli, Int. Math. Res. Not. IMRN 13 (2011), 3082–3094. MR 2817687, DOI 10.1093/imrn/rnq159
William Duke and Paul Jenkins, Integral traces of singular values of weak Maass forms, Algebra Number Theory 2 (2008), no. 5, 573–593. MR 2429454, DOI 10.2140/ant.2008.2.573
B. Dwork, The $U_{p}$ operator of Atkin on modular functions of level $2$ with growth conditions, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 57–67. MR 0332659
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
Pavel Guerzhoy, On Zagier’s adele, Res. Math. Sci. 1 (2014), Art. 7, 19. MR 3344174, DOI 10.1186/2197-9847-1-7
Pavel Guerzhoy, Zachary A. Kent, and Ken Ono, $p$-adic coupling of mock modular forms and shadows, Proc. Natl. Acad. Sci. USA 107 (2010), no. 14, 6169–6174. MR 2630103, DOI 10.1073/pnas.1001355107
Haruzo Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 231–273. MR 868300
M. Hövel, Automorphe formen mit Singularit’́aten auf dem hyperbolischen Raum, 2012.
Ben Kane and Matthias Waldherr, Explicit congruences for mock modular forms, J. Number Theory 166 (2016), 1–18. MR 3486258, DOI 10.1016/j.jnt.2016.01.029
Nicholas M. Katz, $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 69–190. MR 0447119
Alison Miller and Aaron Pixton, Arithmetic traces of non-holomorphic modular invariants, Int. J. Number Theory 6 (2010), no. 1, 69–87. MR 2641715, DOI 10.1142/S1793042110002818
Douglas Niebur, A class of nonanalytic automorphic functions, Nagoya Math. J. 52 (1973), 133–145. MR 337788
Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191–268 (French). MR 0404145
Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
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
J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (French). MR 646366
D. Zagier, Eisenstein series and the Riemann zeta function, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Institute of Fundamental Research, Bombay, 1981, pp. 275–301. MR 633666
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