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Diagonal restriction of Eisenstein series and Kudla–Millson theta lift

  • Romain Branchereau ORCID logo EMAIL logo
Published/Copyright: July 26, 2023

Abstract

We consider the Kudla–Millson theta series associated to a quadratic space of signature ( N , N ) . By combining a “see-saw” argument with the Siegel–Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree N is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon, Pozzi and Vonk to totally real fields.


Communicated by Jan Bruinier


Award Identifier / Grant number: 754362

Funding statement: The author was funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754362.

Acknowledgements

This project was done during my thesis. I thank my advisors Nicolas Bergeron and Luis Garcia for suggesting to me this topic and for their support. I also thank Henri Darmon and Jan Vonk for helpful discussions about this paper.

References

[1] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Grad. Texts in Math. 82, Springer, New York, 1982. 10.1007/978-1-4757-3951-0Search in Google Scholar

[2] R. Branchereau, Diagonal restriction and denominators of some Eisenstein cohomology classes, PhD thesis, PSL Université, Paris, 2022. Search in Google Scholar

[3] R. Branchereau, The Kudla–Millson form via the Mathai–Quillen formalism, submitted. Search in Google Scholar

[4] D. Bump, Automorphic Forms and Representations, Cambridge Stud. Adv. Math. 55, Cambridge University, Cambridge, 1997. 10.1017/CBO9780511609572Search in Google Scholar

[5] H. Darmon, A. Pozzi and J. Vonk, Diagonal restrictions of p-adic Eisenstein families, Math. Ann. 379 (2021), no. 1–2, 503–548. 10.1007/s00208-020-02086-2Search in Google Scholar

[6] H. Darmon and J. Vonk, Singular moduli for real quadratic fields: A rigid analytic approach, Duke Math. J. 170 (2021), no. 1, 23–93. 10.1215/00127094-2020-0035Search in Google Scholar

[7] J. Funke, Heegner divisors and nonholomorphic modular forms, Compos. Math. 133 (2002), no. 3, 289–321. 10.1023/A:1020002121978Search in Google Scholar

[8] J. Funke and J. Millson, The geometric theta correspondence for Hilbert modular surfaces, Duke Math. J. 163 (2014), no. 1, 65–116. 10.1215/00127094-2405279Search in Google Scholar

[9] K. Iwasawa, Hecke’s L-Functions, Springer Briefs Math., Springer, Singapore, 2019. Search in Google Scholar

[10] S. S. Kudla, On the integrals of certain singular theta-functions, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 439–463. Search in Google Scholar

[11] S. S. Kudla, Seesaw dual reductive pairs, Automorphic Forms of Several Variables (Katata 1983), Progr. Math. 46, Birkhäuser, Boston (1984), 244–268. Search in Google Scholar

[12] S. S. Kudla and J. J. Millson, The theta correspondence and harmonic forms. I, Math. Ann. 274 (1986), no. 3, 353–378. 10.1007/BF01457221Search in Google Scholar

[13] S. S. Kudla and J. J. Millson, The theta correspondence and harmonic forms. II, Math. Ann. 277 (1987), no. 2, 267–314. 10.1007/BF01457364Search in Google Scholar

[14] S. S. Kudla and J. J. Millson, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Publ. Math. Inst. Hautes Études Sci. 71 (1990), 121–172. 10.1007/BF02699880Search in Google Scholar

[15] G. Lion and M. Vergne, The Weil Representation, Maslov Index and Theta Series, Progr. Math. 6, Birkhäuser, Boston, 1980. 10.1007/978-1-4684-9154-8Search in Google Scholar

[16] V. Mathai and D. Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986), no. 1, 85–110. 10.1016/0040-9383(86)90007-8Search in Google Scholar

[17] C. Mœ glin, M.-F. Vignéras and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adique, Lecture Notes in Math. 1291, Springer, Berlin, 1987. 10.1007/BFb0082712Search in Google Scholar

[18] J. Neukirch, Algebraic Number Theory, Grundlehren Math. Wiss. 322, Springer, Berlin, 1999. 10.1007/978-3-662-03983-0Search in Google Scholar

[19] C. O’Sullivan, Formulas for non-holomorphic Eisenstein series and for the Riemann zeta function at odd integers, Res. Number Theory 4 (2018), no. 3, Paper No. 36. 10.1007/s40993-018-0129-7Search in Google Scholar

[20] R. Ranga Rao, On some explicit formulas in the theory of Weil representation, Pacific J. Math. 157 (1993), no. 2, 335–371. 10.2140/pjm.1993.157.335Search in Google Scholar

[21] A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211. 10.1007/BF02391012Search in Google Scholar

[22] F. Wielonsky, Séries d’Eisenstein, intégrales toroïdales et une formule de Hecke, Enseign. Math. (2) 31 (1985), no. 1–2, 93–135. Search in Google Scholar

Received: 2022-11-18
Revised: 2023-06-09
Published Online: 2023-07-26
Published in Print: 2023-09-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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