The formal group of the Jacobian of an algebraic curve
type: Item
instance of: scholarly article
title: The formal group of the Jacobian of an algebraic curve
zbMATH Open document ID: 0804.14020
author: Margaret N. Freije
published in: Pacific Journal of Mathematics
publication date: 1995-01-22T00:00:00Z
review text: After some of the basic facts about higher dimensional formal groups are reviewed, an explicit construction of the formal group of the Jacobian of a complete nonsingular algebraic curve \(C\) over a field of characteristic zero is given using a basis for the holomorphic differentials on \(C\) at a rational non-Weierstrass point. Let \(l\) be an odd positive prime and \[ \Gamma_ 0(l) = \left \{ {a\;b \choose c \;d} \in \text{SL}_ 2 (\mathbb{Z}) \mid c \equiv 0 \pmod l \right\}. \] Let \(X_ 0 (l)\) be the modular curve associated to \(\Gamma_ 0(l)\). As an example of the above construction, the formal group of the Jacobian of \(X_ 0(l)\) is constructed. The connection between the differentials on \(X_ 0(l)\) and the Fourier expansions of cusp forms of weight 2 on \(\Gamma_ 0(l)\) is reviewed. Using this and a result of \textit{T. Honda}, the author proves that this formal group is \(p\)-integral for all but finitely many primes \(p\).
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14F10
zbMATH DE Number: 508862
zbMATH Keywords: formal group of the Jacobian of a complete nonsingular algebraic curve
zbMATH Keywords: holomorphic differentials
zbMATH Keywords: rational non-Weierstrass point
zbMATH Keywords: modular curve
zbMATH Keywords: cusp forms
reviewed by: Ryutaro Horiuchi
MaRDI profile type: Publication
full work available at URL: pjm.1993.157.241
OpenAlex ID: W2041065668
DOI: 10.2140/PJM.1993.157.241
description: scientific article; zbMATH DE number 508862