Cusp forms of weight 1 associated to Fermat curves
type: Item
instance of: scholarly article
title: Cusp forms of weight 1 associated to Fermat curves
zbMATH Open document ID: 0860.11021
DOI: 10.1215/S0012-7094-96-08306-4
author: Tonghai Yang
published in: Duke Mathematical Journal
publication date: 1996-11-10T00:00:00Z
review text: Let \(\Delta\) be the free subgroup of the principal congruence subgroup \(\Gamma (2)\) generated by \(A= \left(\begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)\) and \(B = \left (\begin{smallmatrix} 1 & 0 \\ 2 & 1 \end{smallmatrix} \right)\) (one has \(\Gamma(2) = \{\pm I\} \Delta)\), and for \(N\in\mathbb{N}\) let \(\Phi(N)\) be the subgroup of \(\Delta\) generated by \(A^N\) and \(B^N\) and the commutator \([\Delta,\Delta]\). Then the associated modular curve \(X(\Phi (N))\) is isomorphic to the Fermat curve \(X^N + Y^N=Z^N\). As is well-known, \(\Phi(N)\) is not a congruence subgroup unless \(N=1,2,4,8\). Using the above isomorphism the author constructs a canonical basis for the space of modular forms and cusp forms of weight 1 for \(\Phi(N)\), and, as a consequence, obtains explicit dimension formulas for these spaces. One should note that the classical Riemann-Roch theorem in general only gives explicit dimension formulas for spaces of cusp forms of weight \(k\geq 2\).
reviewed by: Winfried Kohnen
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14C40
zbMATH DE Number: 912123
zbMATH Keywords: canonical basis for the space of cusp forms of weight 1
zbMATH Keywords: principal congruence subgroup
zbMATH Keywords: modular curve
zbMATH Keywords: Fermat curve
zbMATH Keywords: canonical basis for the space of modular forms
zbMATH Keywords: dimension formulas
MaRDI profile type: Publication
cites work: Formes modulaires de poids $1$
cites work: Q4842213
cites work: Q3931656
cites work: Some results on the Mordell-Weil group of the Jacobian of the Fermat curve
cites work: Q3887686
cites work: Q4693061
cites work: Q1838549
cites work: The spectrum of Fermat curves
cites work: Q5666618
cites work: Points at infinity on the Fermat curves
cites work: Q4141905
cites work: Artin’s conjecture for representations of octahedral type
full work available at URL: s0012-7094-96-08306-4
OpenAlex ID: W2027232072
description: scientific article; zbMATH DE number 912123