p -adic measures attached to Siegel modular forms

Siegfried Böcherer; Claus-Günther Schmidt

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 5, page 1375-1443
  • ISSN: 0373-0956

Abstract

top
We study the critical values of the complex standard- L -function attached to a holomorphic Siegel modular form and of the twists of the L -function by Dirichlet characters. Our main object is for a fixed rational prime number p to interpolate p -adically the essentially algebraic critical L -values as the Dirichlet character varies thus providing a systematic control of denominators of critical values by generalized Kummer congruences. In order to organize this information we prove the existence of p -adic measures such that integration of any Dirichlet character of p -power conductor over the measure yields the suitably normalized critical value of the complex L -function twisted by the Dirichlet character. In a standard manner the p -adic measures naturally define p -adic L -functions which hence p -adically interpolate the normalized critical values.

How to cite

top

Böcherer, Siegfried, and Schmidt, Claus-Günther. "$p$-adic measures attached to Siegel modular forms." Annales de l'institut Fourier 50.5 (2000): 1375-1443. <http://eudml.org/doc/75460>.

@article{Böcherer2000,
abstract = {We study the critical values of the complex standard-$L$-function attached to a holomorphic Siegel modular form and of the twists of the $L$-function by Dirichlet characters. Our main object is for a fixed rational prime number $p$ to interpolate $p$-adically the essentially algebraic critical $L$-values as the Dirichlet character varies thus providing a systematic control of denominators of critical values by generalized Kummer congruences. In order to organize this information we prove the existence of $p$-adic measures such that integration of any Dirichlet character of $p$-power conductor over the measure yields the suitably normalized critical value of the complex $L$-function twisted by the Dirichlet character. In a standard manner the $p$-adic measures naturally define $p$-adic $L$-functions which hence $p$-adically interpolate the normalized critical values.},
author = {Böcherer, Siegfried, Schmidt, Claus-Günther},
journal = {Annales de l'institut Fourier},
keywords = {-adic interpolation; standard -functions; Siegel modular forms; algebraic special -values; -adic measures},
language = {eng},
number = {5},
pages = {1375-1443},
publisher = {Association des Annales de l'Institut Fourier},
title = {$p$-adic measures attached to Siegel modular forms},
url = {http://eudml.org/doc/75460},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Böcherer, Siegfried
AU - Schmidt, Claus-Günther
TI - $p$-adic measures attached to Siegel modular forms
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 5
SP - 1375
EP - 1443
AB - We study the critical values of the complex standard-$L$-function attached to a holomorphic Siegel modular form and of the twists of the $L$-function by Dirichlet characters. Our main object is for a fixed rational prime number $p$ to interpolate $p$-adically the essentially algebraic critical $L$-values as the Dirichlet character varies thus providing a systematic control of denominators of critical values by generalized Kummer congruences. In order to organize this information we prove the existence of $p$-adic measures such that integration of any Dirichlet character of $p$-power conductor over the measure yields the suitably normalized critical value of the complex $L$-function twisted by the Dirichlet character. In a standard manner the $p$-adic measures naturally define $p$-adic $L$-functions which hence $p$-adically interpolate the normalized critical values.
LA - eng
KW - -adic interpolation; standard -functions; Siegel modular forms; algebraic special -values; -adic measures
UR - http://eudml.org/doc/75460
ER -

References

top
  1. [1] A.N. ANDRIANOV, V.L. KALININ, On the analytic properties of standard zeta functions of Siegel modular forms, Math. USSR Sbornik, 35 (1979), 1-17. Zbl0417.10024
  2. [2] A.N. ANDRIANOV, Quadratic Forms and Hecke Operators. Grundlehren der mathematischen Wissenschaften 286, Berlin-Heidelberg-New York, Springer, 1987. Zbl0613.10023
  3. [3] S. BÖCHERER, Über die Fourier-Jacobi-Entwicklung der Siegelschen Eisensteinreihen, Math.Z., 183 (1983), 21-43. Zbl0497.10020
  4. [4] S. BÖCHERER, Über die Fourier-Jacobi-Entwicklung der Siegelschen Eisensteinreihen II, Math.Z., 189 (1985), 81-100. Zbl0558.10022
  5. [5] S. BÖCHERER, Über die Fourierkoeffizienten Siegelscher Eisensteinreihen, Manuscripta Math., 45 (1984), 273-288. Zbl0533.10023
  6. [6] S. BÖCHERER, Über die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modulgruppe, J. reine angew. Math., 362 (1985), 146-168. Zbl0565.10025
  7. [7] S. BÖCHERER, Ein Rationalitätssatz für formale Heckereihen zur Siegelschen Modulgruppe, Abh. Math. Sem. Univ. Hamburg, 56 (1986), 35-47. Zbl0613.10026
  8. [8] U. CHRISTIAN, Selberg's Zeta-, L-, and Eisenstein Series. Lecture Notes in Math. 1030. Berlin-Heidelberg-New York, Springer, 1983. Zbl0519.10018MR85k:11024
  9. [9] U. CHRISTIAN, Maaßsche L-Reihen und eine Identität für Gaußsche Summen, Abh. Math. Sem. Univ. Hamburg, 54 (1984), 163-175. Zbl0556.10021MR86h:11041
  10. [10] P. DELIGNE, Valeurs de fonctions L et périodes d'intégrales, Proc. Symp. Pure Math., 33, Part 2 (1979), 313-346. Zbl0449.10022MR81d:12009
  11. [11] P. FEIT, Poles and residues of Eisenstein series for symplectic and unitary groups, Memoirs AMS, 61 (1986), no 346. Zbl0591.10017MR88a:11049
  12. [12] E. FREITAG, Siegelsche Modulfunktionen. Grundlehren der mathematischen Wissenschaften 254, Berlin-Heidelberg-New York, 1983. Zbl0498.10016MR88b:11027
  13. [13] P.B. GARRETT, M. HARRIS, Special values of triple product L-functions, Amer. J. Math., 115 (1993), 159-238. Zbl0776.11027MR94e:11058
  14. [14] S. GELBART, I. PIATETSKI-SHAPIRO, S. RALLIS, Explicit Constructions of Automorphic L-Functions. Springer Lecture Notes in Math. 1254, Berlin-Heidelberg-New York, Springer, 1987. Zbl0612.10022MR89k:11038
  15. [15] M. HARRIS, Special values of zeta functions attached to Siegel modular forms, Ann Sci. Ecole Norm. Sup., 14 (1981), 77-120. Zbl0465.10022MR82m:10046
  16. [16] L.K. HUA, Harmonic analysis of functions of several complex variables in the classical domains, Transl. Math. Monographs 6, AMS 1963. Zbl0112.07402
  17. [17] Y. KITAOKA, Dirichlet series in the theory of Siegel modular forms, Nagoya Math. J., 95 (1984), 73-84. Zbl0551.10025MR86b:11038
  18. [18] H. MAAß, Siegel's modular forms and Dirichlet series, Lecture Notes in Math. 216, Berlin-Heidelberg-New York, Springer, 1971. Zbl0224.10028MR49 #8938
  19. [19] Sh.-I. MIZUMOTO, Poles and residues of standard L-functions attached to Siegel modular forms, Math. Ann., 289 (1991), 589-612. Zbl0713.11035MR93b:11058
  20. [20] A.A. PANCHISHKIN, Non-Archimedean L-Functions of Siegel and Hilbert Modular Forms. Springer Lecture Notes in Math. 1471, Berlin-Heidelberg-New York, Springer, 1991. Zbl0732.11026MR93a:11044
  21. [21] A.A. PANCHISHKIN, Admissible Non-Archimedean Standard Zeta Functions associated with Siegel Modular Forms, Proc. Symp. Pure Math., 55, Part 2, 251-292. Zbl0837.11029MR95j:11043
  22. [22] I. PIATETSKI-SHAPIRO, S. RALLIS, A new way to get Euler products, J. reine angew. Math., 392 (1988), 110-124. Zbl0651.10021MR90c:11032
  23. [23] C.G. SCHMIDT, P-adic measures attached to automorphic representations of G1(3), Invent. Math., 92 (1988), 597-631. Zbl0656.10023MR90f:11032
  24. [24] G. SHIMURA, On the Fourier coefficients of modular forms of several variables, Nachr. Akad. Wiss. Göttingen, 1975, 261-268. Zbl0332.32024MR58 #5528
  25. [25] G. SHIMURA, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., XXIX (1976), 783-804. Zbl0348.10015MR55 #7925
  26. [26] G. SHIMURA, Arithmetic of differential operators on symmetric domains, Duke Math. J., 48 (1981), 813-843. Zbl0487.10021MR86m:11032
  27. [27] G. SHIMURA—, Confluent hypergeometric functions on tube domains, Math. Ann., 269 (1982), 269-302. Zbl0502.10013MR84f:32040
  28. [28] G. SHIMURA, On Eisenstein Series, Duke Math. J., 50 (1983), 417-476. Zbl0519.10019MR84k:10019
  29. [29] G. SHIMURA, On differential operators attached to certain representations of classical groups, Invent. Math., 77 (1984), 463-488. Zbl0558.10023MR86c:11034
  30. [30] G. SHIMURA, Differential operators and the singular values of Eisenstein series, Duke Math. J., 51 (1984), 261-329. Zbl0546.10025MR85h:11031
  31. [31] G. SHIMURA, On Eisenstein series of half-integral weight, Duke Math. J., 52 (1985), 281-314. Zbl0577.10025MR87g:11053
  32. [32] G. SHIMURA, On a class of nearly holomorphic automorphic forms, Annals of Math., 123 (1986), 347-406. Zbl0593.10022MR88b:11025a
  33. [33] G. SHIMURA, Nearly holomorphic functions on hermitian symmetric spaces, Math. Ann., 278 (1987), 1-28. Zbl0636.10023MR89b:32044
  34. [34] G. SHIMURA, Invariant differential operators on hermitian symmetric spaces, Annals of Math., 132 (1990), 237-272. Zbl0718.11020MR91i:22015
  35. [35] G. SHIMURA, Differential Operators, Holomorphic Projection, and Singular Forms, Duke Math. J., 76 (1994), 141-173. Zbl0829.11029MR95k:11072
  36. [36] J. STURM, The critical values of zeta functions associated to the symplectic group, Duke Math. J., 48 (1981), 327-350. Zbl0483.10026MR83c:10035
  37. [37] T. TAMAGAWA, On the zeta functions of a division algebra, Annals of Math., 77 (1963), 387-405. Zbl0222.12018MR26 #2468
  38. [38] L. WASHINGTON, Introduction to Cyclotomic Fields, Graduate Texts in Math. 83, Berlin-Heidelberg-New York, Springer, 1982. Zbl0484.12001MR85g:11001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.