Monodromy of Airy and Kloosterman sheaves
A useful tool to study a family of curves is the geometric monodromy group \(G\). Using Fourier transforms the author essentially proves the following three theorems: For any \(g\geq 3\), the geometric monodromy group of hyperelliptic curves of genus \(g\) and \(2\)-rank zero over \(\overline{\mathbb{F}_2}\) is \(Sp(2g)\). If \(p>2\), and \(\mathcal F\) is a Lie irreducible Airy sheaf over \(A^1\otimes\overline{\mathbb{F}_p}\), then the Lie algebra of \(G\) is \(sl\) or \(sp\) in its standard representation. Let \(\mathcal F\) be a Lie irreducible Kloosterman sheaf on \(A^1\) over \(\overline{\mathbb{F}_p}\). If \(p>\sqrt{\text{rank }{\mathcal F}+1}\), then the Lie algebra of \(G\) is either \(sl\) or \(sp\) or \(so\) in its standard representation.
scientific article; zbMATH DE number 1491011
scientific article