## Abstract

We investigate so-called “higher” Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier–Ehlen–Yang and Bruinier–Schwagenscheidt. We give a series representation of the lift in terms of Gauss hypergeometric functions, and evaluate the lift as the constant term of a Fourier series involving the Rankin–Cohen bracket of harmonic Maass forms and theta functions. Using the higher Siegel lifts, we obtain a vector-valued analogue of Mertens’ result stating that the Rankin–Cohen bracket of the holomorphic part of a harmonic Maass form of weight \(\frac{3}{2}\) and a unary theta function, plus a certain form, is a holomorphic modular form. As an application of these results, we offer a novel proof of a conjecture of Cohen which was originally proved by Mertens, as well as a novel proof of a theorem of Ahlgren and Kim, each in the scalar-valued case.

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## Notes

Note that the authors there use a different signature convention than the current paper.

The paper [3] also used the higher Millson theta lift.

For the \(j=0\) case Ahlgren and Kim explicitly showed equality with the quasi-modular Eisenstein series \(\frac{1}{12}E_2\).

## References

Abramowitz, M., Stegun, I.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. In: National Bureau of Standards Applied Mathematics Series, vol. 55 (1964)

Ahlgren, S., Kim, B.: Congruences for a mock modular form on \(\text{ SL}_{2}({\mathbb{Z}})\) and the smallest parts function. J. Number Theory

**189**, 81–89 (2018)Alfes-Neumann, C., Bringmann, K., Males, J., Schwagenscheidt, M.: Cycle integrals of meromorphic modular forms and coefficients of harmonic Maass forms. J. Math. Anal. Appl. (

**to appear**)Alfes-Neumann, C., Bringmann, K., Schwagenscheidt, M.: On the rationality of cycle integrals of meromorphic modular forms. Math. Ann.

**376**(1–2), 243–266 (2020)Andrews, G.: The number of smallest parts in the partitions of n. J. Reine Angew. Math.

**624**, 133–142 (2008)Bringmann, K.: On the explicit construction of higher deformations of partition statistics. Duke Math. J.

**144**(2), 195–233 (2008)Bruinier, J.H.: Borcherds products on O(2, l) and Chern classes of Heegner divisors. In: Lecture Notes in Mathematics, vol. 1780. Springer, Berlin (2002)

Bruinier, J.H., Ehlen, S., Yang, T.: CM values of higher automorphic Green functions on orthogonal groups. Invent. Math.

**225**(3), 693–785 (2021)Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J.

**125**, 45–90 (2004)Bruinier, J.H., Funke, J., Imamoglu, Ö.: Regularized theta liftings and periods of modular functions. J. Reine Angew. Math.

**703**, 43–93 (2015)Bruinier, J., Schwagenscheidt, M.: Theta lifts for Lorentzian lattices and coefficients of mock theta functions. Math. Z.

**297**(3-4), 1633–1657 (2020)Bruinier, J., Yang, T.: Faltings heights of CM cycles and derivatives of L-functions. Invent. Math.

**177**(3), 631–681 (2009)Cohen, H.: Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann.

**217**(3), 271–285 (1975)Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York–Toronto-London (1954). Based, in part, on notes left by Harry Bateman

Li, Y., Schwagenscheidt, M.: Mock modular forms with integral fourier coefficients. Adv. Math. (

**to appear**)Mertens, M.: Eichler–Selberg type identities for mixed mock modular forms. Adv. Math.

**301**, 359–382 (2016)Mertens, M.: Mock modular forms and class number relations. Res. Math. Sci.

**1**, 16 (2014)NIST digital library of mathematical functions. http://dlmf.nist.gov/, Release 1.1.5 of 2022-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain (eds.)

Zagier, D.: Modular forms and differential operators. Proc. Indian Acad. Sci. Math. Sci.

**104**(1), 57–75 (1994)Zagier, D.: Nombres de classes et formes modulaires de poids 3/2. C. R. Acad. Sci. Paris Sér. A-B

**281**(21), A883–A886 (1975). (**French, with English summary**)

## Acknowledgements

The author thanks Markus Schwagenscheidt for many insightful conversations on the contents of the paper, in particular suggesting the connection to Theorems 1.2 and 1.3, as well as useful comments on previous versions of the paper. The author would also like to thank Jan Bruinier and Andreas Mono for helpful comments on an earlier draft of the paper. The research conducted for this paper is supported by the Pacific Institute for the Mathematical Sciences (PIMS). The research and findings may not reflect those of the Institute.

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Males, J. Higher Siegel theta lifts on Lorentzian lattices, harmonic Maass forms, and Eichler–Selberg type relations.
*Math. Z.* **301**, 3555–3569 (2022). https://doi.org/10.1007/s00209-022-03023-6

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DOI: https://doi.org/10.1007/s00209-022-03023-6