Endomorphism Algebras of Jacobians

✍ Scribed by Jordan S. Ellenberg

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 228 KB
• Volume: 162
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2001.1994

✦ Synopsis

Very little is known about the existence of curves, and families of curves, whose Jacobians are acted on by large rings of endomorphisms. In this paper, we show the existence of curves X with an injection K / Ä Hom(Jac(X ), Jac(X )) Z Q, where K is a subfield of even index at most 10 in a primitive cyclotomic field Q(p), or a subfield of index 2 in Q(p q ) or Q(`p:). This result generalizes previous work of Brumer, Mestre, and Tautz, Top, and Verberkmoes. Our curves are constructed as branched covers of the projective line, and the endomorphisms arise as quotients of double coset algebras of the Galois groups of these coverings. In certain cases, we show the existence of continuous families of Jacobians admitting the desired endomorphism algebras. At the end, we raise some questions about upper bounds for endomorphism algebras of the type described here and ask about a ``non-Galois version of Hurwitz's theorem. '' 2001 Academic Press