A condition for the rationality of certain elliptic modular forms over primes dividing the level

Andrea Mori

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1991)

  • Volume: 2, Issue: 2, page 103-109
  • ISSN: 1120-6330

Abstract

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Let f be a weight k holomorphic automorphic form with respect to Γ 0 ⁢ N . We prove a sufficient condition for the integrality of f over primes dividing N . This condition is expressed in terms of the values at particular C ⁢ M curves of the forms obtained by iterated application of the weight k Maaß operator to f and extends previous results of the Author.

How to cite

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Mori, Andrea. "A condition for the rationality of certain elliptic modular forms over primes dividing the level." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 2.2 (1991): 103-109. <http://eudml.org/doc/244261>.

@article{Mori1991,
abstract = {Let \( f \) be a weight \( k \) holomorphic automorphic form with respect to \( \Gamma\_\{0\} (N) \). We prove a sufficient condition for the integrality of \( f \) over primes dividing \( N \). This condition is expressed in terms of the values at particular \( CM \) curves of the forms obtained by iterated application of the weight \( k \) Maaß operator to \( f \) and extends previous results of the Author.},
author = {Mori, Andrea},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Modular forms; Modular curves; Complex multiplications; complex multiplication; modular curve; integrality; Maass operator; automorphic forms; values at CM curves},
language = {eng},
month = {6},
number = {2},
pages = {103-109},
publisher = {Accademia Nazionale dei Lincei},
title = {A condition for the rationality of certain elliptic modular forms over primes dividing the level},
url = {http://eudml.org/doc/244261},
volume = {2},
year = {1991},
}

TY - JOUR
AU - Mori, Andrea
TI - A condition for the rationality of certain elliptic modular forms over primes dividing the level
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1991/6//
PB - Accademia Nazionale dei Lincei
VL - 2
IS - 2
SP - 103
EP - 109
AB - Let \( f \) be a weight \( k \) holomorphic automorphic form with respect to \( \Gamma_{0} (N) \). We prove a sufficient condition for the integrality of \( f \) over primes dividing \( N \). This condition is expressed in terms of the values at particular \( CM \) curves of the forms obtained by iterated application of the weight \( k \) Maaß operator to \( f \) and extends previous results of the Author.
LA - eng
KW - Modular forms; Modular curves; Complex multiplications; complex multiplication; modular curve; integrality; Maass operator; automorphic forms; values at CM curves
UR - http://eudml.org/doc/244261
ER -

References

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  1. DELIGNE, P. - RAPOPORT, M., Les schémas de modules de courbes elliptiques. In: Modular functions of one variable. II. Lecture Notes in Math., vol. 349, Springer Verlag, 1973, 143-316. Zbl0281.14010MR337993
  2. DRINFELD, V. G., Elliptic modules. Math. USSR Sbornik, 23 (4), 1973. Zbl0386.20022
  3. HARRIS, M., Special values of zeta functions attached to Siegel modular forms. Ann. Sc. Ec. Norm. Sup., s. IV, vol. 14, 1981, 77-120. Zbl0465.10022MR618732
  4. KATZ, N., p-adic properties of modular schemes and modular forms. In: Modular functions of one variable. III. Lecture Notes in Math., vol. 350, Springer Verlag, 1973, 70-189. Zbl0271.10033MR447119
  5. KATZ, N., Serre-Tate local moduli. In: Surfaces algebraiques. Lecture Notes in Math., vol. 868, Springer Verlag, 1981, 138-202. Zbl0477.14007MR638600
  6. KATZ, N. - MAZUR, B., Arithmetic moduli of elliptic curves. Annals of Math. Studies, vol. 108, Princeton Univ. Press, 1985. Zbl0576.14026MR772569
  7. MORI, A., Integrality of elliptic modular forms via Maaß operators. Ph. D. thesis, Brandeis University, 1989. MR2637460
  8. MORI, A., An integrality criterion for elliptic modular forms. Rend. Mat. Acc. Lincei, s. 9, vol. 1, 1990, 3-9. Zbl0702.11025MR1081819
  9. MORI, A., A characterization of integral elliptic automorphic forms. To appear. Zbl0822.11033MR1081819
  10. SHIMURA, G., Introduction to the arithmetic theory of automorphic functions. Iwanami Shoten and Princeton Univ. Press, 1971. Zbl0872.11023MR314766

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