### Abstract

In a previous paper [1], we defined the space of toric forms script T sign(l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ_{1}(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) ≠ 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.

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

Pages (from-to) | 149-165 |

Number of pages | 17 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 539 |

State | Published - Dec 1 2001 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal fur die Reine und Angewandte Mathematik*, (539), 149-165.

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*Journal fur die Reine und Angewandte Mathematik*, no. 539, pp. 149-165.

**Toric modular forms and nonvanishing of L-functions.** / Borisov, Lev; Gunnells, Paul E.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Toric modular forms and nonvanishing of L-functions

AU - Borisov, Lev

AU - Gunnells, Paul E.

PY - 2001/12/1

Y1 - 2001/12/1

N2 - In a previous paper [1], we defined the space of toric forms script T sign(l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) ≠ 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.

AB - In a previous paper [1], we defined the space of toric forms script T sign(l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) ≠ 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.

UR - http://www.scopus.com/inward/record.url?scp=23044530693&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23044530693&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:23044530693

SP - 149

EP - 165

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 539

ER -