Hecke operators in half-integral weight

Soma Purkait[1]

  • [1] Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 1, page 233-251
  • ISSN: 1246-7405

Abstract

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In [6], Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm’s bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators T p ℓ with p prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators T p ℓ commute with the Shimura correspondence.

How to cite

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Purkait, Soma. "Hecke operators in half-integral weight." Journal de Théorie des Nombres de Bordeaux 26.1 (2014): 233-251. <http://eudml.org/doc/275727>.

@article{Purkait2014,
abstract = {In [6], Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm’s bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators $T_\{p^\ell \}$ with $p$ prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators $T_\{p^\{\ell \}\}$ commute with the Shimura correspondence.},
affiliation = {Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom},
author = {Purkait, Soma},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {4},
number = {1},
pages = {233-251},
publisher = {Société Arithmétique de Bordeaux},
title = {Hecke operators in half-integral weight},
url = {http://eudml.org/doc/275727},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Purkait, Soma
TI - Hecke operators in half-integral weight
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/4//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 1
SP - 233
EP - 251
AB - In [6], Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm’s bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators $T_{p^\ell }$ with $p$ prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators $T_{p^{\ell }}$ commute with the Shimura correspondence.
LA - eng
UR - http://eudml.org/doc/275727
ER -

References

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  1. F. Diamond and J. Shurman, A First Course in Modular Forms, GTM 228, Springer-Verlag, 2005. Zbl1062.11022MR2112196
  2. W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32–72. Zbl0475.10025MR660784
  3. T. Miyake, Modular Forms, Springer-Verlag, 1989. Zbl0701.11014
  4. S. Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Mathematical Journal 56 (1975), 147–161. Zbl0303.10027MR364106
  5. K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q -Series, CBMS 102, American Mathematical Society, 2004. Zbl1119.11026MR2020489
  6. G. Shimura, On Modular Forms of Half Integral Weight, Annals of Mathematics, Second Series, Vol. 97, 3 (1973), pp. 440–481. Zbl0266.10022MR332663
  7. W. Stein, Modular forms, a computational approach, Graduate Studies in Mathematics 79, American Mathematical Society, 2007. Zbl1110.11015MR2289048
  8. J. Sturm, On the Congruence of Modular Forms. Number theory (New York, 1984-1985), Lecture Notes in Math. 1240, Springer, Berlin, (1987), 275–280. Zbl0615.10035MR894516

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