### Abstract

In this paper we determine the explicit structure of the semisimple part of the Hecke algebra that acts on Drinfeld modular forms of full level modulo T. We show that modulo T the Hecke algebra has a non-zero semisimple part. In contrast, a well-known theorem of Serre asserts that for classical modular forms the action of Tℓ for any odd prime ℓ is nilpotent modulo 2. After proving the result for Drinfeld modular forms modulo T, we use computations of the Hecke action modulo T to show that certain powers of the Drinfeld modular form h cannot be eigenforms. Finally, we pose a question a positive answer to which will mean that the Hecke algebra that acts on Drinfeld modular forms of full level is not smooth for large weights, which again contrasts the classical situation.

Original language | English (US) |
---|---|

Pages (from-to) | 186-200 |

Number of pages | 15 |

Journal | Journal of Number Theory |

Volume | 137 |

DOIs | |

State | Published - Apr 2014 |

### Fingerprint

### Keywords

- Drinfeld modular forms
- Reduction of Drinfeld modular forms modulo T

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*137*, 186-200. https://doi.org/10.1016/j.jnt.2013.11.010

**On the action of Hecke operators on Drinfeld modular forms.** / Joshi, Kirti N; Petrov, Aleksandar.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 137, pp. 186-200. https://doi.org/10.1016/j.jnt.2013.11.010

}

TY - JOUR

T1 - On the action of Hecke operators on Drinfeld modular forms

AU - Joshi, Kirti N

AU - Petrov, Aleksandar

PY - 2014/4

Y1 - 2014/4

N2 - In this paper we determine the explicit structure of the semisimple part of the Hecke algebra that acts on Drinfeld modular forms of full level modulo T. We show that modulo T the Hecke algebra has a non-zero semisimple part. In contrast, a well-known theorem of Serre asserts that for classical modular forms the action of Tℓ for any odd prime ℓ is nilpotent modulo 2. After proving the result for Drinfeld modular forms modulo T, we use computations of the Hecke action modulo T to show that certain powers of the Drinfeld modular form h cannot be eigenforms. Finally, we pose a question a positive answer to which will mean that the Hecke algebra that acts on Drinfeld modular forms of full level is not smooth for large weights, which again contrasts the classical situation.

AB - In this paper we determine the explicit structure of the semisimple part of the Hecke algebra that acts on Drinfeld modular forms of full level modulo T. We show that modulo T the Hecke algebra has a non-zero semisimple part. In contrast, a well-known theorem of Serre asserts that for classical modular forms the action of Tℓ for any odd prime ℓ is nilpotent modulo 2. After proving the result for Drinfeld modular forms modulo T, we use computations of the Hecke action modulo T to show that certain powers of the Drinfeld modular form h cannot be eigenforms. Finally, we pose a question a positive answer to which will mean that the Hecke algebra that acts on Drinfeld modular forms of full level is not smooth for large weights, which again contrasts the classical situation.

KW - Drinfeld modular forms

KW - Reduction of Drinfeld modular forms modulo T

UR - http://www.scopus.com/inward/record.url?scp=84892522799&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892522799&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2013.11.010

DO - 10.1016/j.jnt.2013.11.010

M3 - Article

VL - 137

SP - 186

EP - 200

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -