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Licensed Unlicensed Requires Authentication Published by De Gruyter October 14, 2015

Vector valued theta functions associated with binary quadratic forms

Stephan Ehlen ORCID logo
From the journal Forum Mathematicum


We study the space of vector valued theta functions for the Weil representation of a positive definite even lattice of rank two with fundamental discriminant. We work out the relation of this space to the corresponding scalar valued theta functions of weight one and determine an orthogonal basis with respect to the Petersson inner product. Moreover, we give an explicit formula for the Petersson norms of the elements of this basis.

MSC 2010: 11F11; 11F27; 11E16

Communicated by Jan Bruinier

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: BR-2163/2-1

Funding statement: This work was partly supported by DFG grant BR-2163/2-1.

The results of this article are also contained in the author’s thesis [8]. I would like to thank my advisor Jan Bruinier for his constant support and helpful comments on an earlier version of this paper. Moreover, I thank the anonymous referee for carefully reading the manuscript and providing helpful comments.


[1] Borcherds R. E., Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. 10.1007/s002220050232Search in Google Scholar

[2] Bruinier J. H., Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors, Lecture Notes in Math. 1780, Springer, Berlin, 2002. 10.1007/b83278Search in Google Scholar

[3] Bruinier J. H., Hilbert modular forms and their applications, The 1-2-3 of Modular Forms, Universitext, Springer, Berlin (2008), 105–179. 10.1007/978-3-540-74119-0_2Search in Google Scholar

[4] Bruinier J. H. and Bundschuh M., On Borcherds products associated with lattices of prime discriminant, Ramanujan J. 7 (2003), no. 1–3, 49–61. 10.1007/978-1-4757-6044-6_5Search in Google Scholar

[5] Bruinier J. H. and Yang T., Faltings heights of CM cycles and derivatives of L-functions, Invent. Math. 177 (2009), no. 3, 631–681. 10.1007/s00222-009-0192-8Search in Google Scholar

[6] Bundschuh M., Ü,ber die Endlichkeit der Klassenzahl gerader Gitter der Signatur (2,n) mit einfachem Kontrollraum, Ph.D. thesis, Universität Heidelberg, 2001. Search in Google Scholar

[7] Duke W. and Li Y., Harmonic maass forms of weight 1, Duke Math. J. 164 (2015), no. 1, 39–113. 10.1215/00127094-2838436Search in Google Scholar

[8] Ehlen S., CM, values of regularized theta lifts, Ph.D. thesis, TU Darmstadt, 2013. Search in Google Scholar

[9] Hofmann E. F. W., Automorphic products on unitary groups, Ph.D. thesis, TU Darmstadt, 2011. Search in Google Scholar

[10] Kani E., The space of binary theta series, Ann. Sci. Math. Québec 36 (2012), no. 2, 501–534. Search in Google Scholar

[11] Kitaoka Y., Arithmetic of Quadratic Forms, Cambridge Tracts in Math. 106, Cambridge University Press, Cambridge, 1993. 10.1017/CBO9780511666155Search in Google Scholar

[12] Kneser M., Quadratische Formen, Springer, Berlin, 2002. 10.1007/978-3-642-56380-5Search in Google Scholar

[13] Kudla S. S., Integrals of Borcherds forms, Compos. Math. 137 (2003), no. 3, 293–349. 10.1023/A:1024127100993Search in Google Scholar

[14] Neukirch J., Algebraische Zahlentheorie, Springer, Berlin, 2007. Search in Google Scholar

[15] Scheithauer N., The Weil representation of SL2() and some applications, Int. Math. Res. Not. IMRN 8 (2009), 1488–1545. 10.1093/imrn/rnn166Search in Google Scholar

[16] Scheithauer N., Some constructions of modular forms for the Weil representation of SL2(), preprint 2011, Search in Google Scholar

[17] Schwagenscheidt M. and Völz F., Lifting newforms to vector valued modular forms for the Weil representation, Int. J. Number Theory 11 (2015), no. 7, 2199–2219. 10.1142/S1793042115500980Search in Google Scholar

[18] Shimura G., Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, 1994. Search in Google Scholar

[19] Strömberg F., Weil representations associated with finite quadratic modules, Math. Z. 275 (2013), no. 1–2, 509–527. 10.1007/s00209-013-1145-xSearch in Google Scholar

[20] Zagier D. B., Zetafunktionen und Quadratische Körper, Hochschultext, Springer, Berlin, 1981. 10.1007/978-3-642-61829-1Search in Google Scholar

Received: 2015-2-17
Revised: 2015-7-25
Published Online: 2015-10-14
Published in Print: 2016-9-1

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