Startseite A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
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A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight

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Veröffentlicht/Copyright: 28. Februar 2023
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 35 Heft 2

Abstract

This paper takes the first steps towards a systematic study of additive rigid meromorphic cocycles of higher weight. These were introduced by Darmon and Vonk, who focused on additive cocycles of weight two and their multiplicative lifts. After classifying certain rigid meromorphic cocycles of weight 2 ⁢ k , we construct an explicit holomorphic kernel function realizing a Shimura–Shintani style correspondence from modular forms of weight k + 1 / 2 and level 4 ⁢ p 2 to rigid analytic cocycles of weight 2 ⁢ k on SL 2 ⁡ ( ℤ ⁢ [ 1 / p ] ) .

Keywords: Algebraic number theory; Shimura–Shintani correspondence, theta correspondence; rigid meromorphic cocycles
MSC 2010: 11S99

Communicated by Jan Bruinier

References

[1] R. E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. 10.1007/s002220050232Suche in Google Scholar

[2] J. Bruinier and K. Ono, Heegner divisors, L-functions and harmonic weak Maass forms, Ann. of Math. (2) 172 (2010), no. 3, 2135–2181. 10.4007/annals.2010.172.2135Suche in Google Scholar

[3] Y. Choie and D. B. Zagier, Rational period functions for psl ( 2 , Z ) , A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemp. Math. 143, American Mathematical Society, Providence (1993), 89–108. 10.1090/conm/143/00992Suche in Google Scholar

[4] H. Darmon, Rational Points on Modular Elliptic Curves, CBMS Reg. Conf. Ser. Math. 101, American Mathematical Society, Providence, 2004. 10.1090/cbms/101Suche 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/s00208-020-02086-2Suche in Google Scholar

[6] H. Darmon, A. Pozzi and J. Vonk, The values of the Dedekind–Rademacher cocycle at real multiplication points, preprint (2021), https://arxiv.org/abs/2103.02490. Suche in Google Scholar

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

[8] H. Darmon and J. Vonk, Real quadratic Borcherds products, preprint. 10.4310/PAMQ.2022.v18.n5.a1Suche in Google Scholar

[9] S. Dasgupta and J. Teitelbaum, The p-adic upper half plane, p-adic Geometry, Univ. Lecture Ser. 45, American Mathematical Society, Providence (2008), 65–121. 10.1090/ulect/045/03Suche in Google Scholar

[10] J. Fresnel and M. van der Put, Rigid Analytic Geometry and its Applications, Progr. Math. 218, Birkhäuser, Boston, 2004. 10.1007/978-1-4612-0041-3Suche in Google Scholar

[11] L. Gerritzen and M. van der Put, Schottky Groups and Mumford Curves, Lecture Notes in Math. 817, Springer, Berlin, 1980. 10.1007/BFb0089957Suche in Google Scholar

[12] R. Greenberg and G. Stevens, p-adic L-functions and p-adic periods of modular forms, Invent. Math. 111 (1993), no. 2, 407–447. 10.1007/BF01231294Suche in Google Scholar

[13] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175–198. 10.1007/BF01389166Suche in Google Scholar

[14] W. Kohnen and D. Zagier, Modular forms with rational periods, Modular Forms, Ellis Horwood Ser. Math. Appl. Statist. Oper. Res., Horwood, Chichester (1984), 197–249. Suche in Google Scholar

[15] T. Oda, On modular forms associated with indefinite quadratic forms of signature ( 2 , n - 2 ) , Math. Ann. 231 (1977/78), no. 2, 97–144. 10.1007/BF01361138Suche in Google Scholar

[16] L. Orton, An elementary proof of a weak exceptional zero conjecture, Canad. J. Math. 56 (2004), no. 2, 373–405. 10.4153/CJM-2004-018-4Suche in Google Scholar

[17] P. Schneider, Rigid-analytic L-transforms, Number Theory, Noordwijkerhout 1983, Lecture Notes in Math. 1068, Springer, Berlin (1984), 216–230. 10.1007/BFb0099455Suche in Google Scholar

[18] M. Schwagenscheidt, Borcherds lifts of harmonic Maass forms and modular integrals, Math. Ann. 375 (2019), no. 3–4, 1615–1647. 10.1007/s00208-019-01903-7Suche in Google Scholar

[19] J. Stopple, Theta series for indefinite quadratic forms over real number fields, Acta Arith. 72 (1995), no. 4, 299–309. 10.4064/aa-72-4-299-309Suche in Google Scholar

[20] J. T. Teitelbaum, Values of p-adic L-functions and a p-adic Poisson kernel, Invent. Math. 101 (1990), no. 2, 395–410. 10.1007/BF01231508Suche in Google Scholar

[21] M.-F. Vignéras, Séries thêta des formes quadratiques indéfinies, Séminaire Delange–Pisot–Poitou, 17e année: 1975/76, Théorie des nombres: Fasc. 1, Secrétariat Math., Paris (1977), Exp. No. 20. 10.1007/BFb0065303Suche in Google Scholar

[22] D. Zagier, Modular forms associated to real quadratic fields, Invent. Math. 30 (1975), no. 1, 1–46. 10.1007/BF01389846Suche in Google Scholar
Received: 2022-08-11
Revised: 2022-12-11
Published Online: 2023-02-28
Published in Print: 2023-03-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. On the asymptotics of the shifted sums of Hecke eigenvalue squares
  3. On the splitting conjecture in the hybrid model for the Riemann zeta function
  4. Almost Dedekind domains without radical factorization
  5. Groups in which the centralizer of any non-central primary element is maximal
  6. GKM actions on cohomogeneity one manifolds
  7. Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
  8. Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
  9. Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
  10. Measures of noncompactness of interpolated polynomials
  11. The homotopy category of acyclic complexes of pure-projective modules
  12. The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
  13. Sharp Li–Yau inequalities for Dunkl harmonic oscillators
  14. A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
  15. Free polynilpotent groups and the Magnus property
https://doi.org/10.1515/forum-2022-0235
Schlagwörter für diesen Artikel
Algebraic number theory; Shimura–Shintani correspondence, theta correspondence; rigid meromorphic cocycles

Artikel in diesem Heft

  1. Frontmatter
  2. On the asymptotics of the shifted sums of Hecke eigenvalue squares
  3. On the splitting conjecture in the hybrid model for the Riemann zeta function
  4. Almost Dedekind domains without radical factorization
  5. Groups in which the centralizer of any non-central primary element is maximal
  6. GKM actions on cohomogeneity one manifolds
  7. Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
  8. Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
  9. Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
  10. Measures of noncompactness of interpolated polynomials
  11. The homotopy category of acyclic complexes of pure-projective modules
  12. The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
  13. Sharp Li–Yau inequalities for Dunkl harmonic oscillators
  14. A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
  15. Free polynilpotent groups and the Magnus property
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