The formal group of the Jacobian of an algebraic curve
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 \(p\).
formal group of the Jacobian of a complete nonsingular algebraic curve
scientific article; zbMATH DE number 508862