Curves of Genus 2 with (N, N) Decomposable Jacobians

✍ Scribed by T. Shaska

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 344 KB
• Volume: 31
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1006/jsco.2001.0439

✦ Synopsis

Let C be a curve of genus 2 and ψ 1 : C -→ E 1 a map of degree n, from C to an elliptic curve E 1 , both curves defined over C. This map induces a degree n map φ 1 : P 1 -→ P 1 which we call a Frey-Kani covering. We determine all possible ramifications for φ 1 . If ψ 1 : C -→ E 1 is maximal then there exists a maximal map ψ 2 : C -→ E 2 , of degree n, to some elliptic curve E 2 such that there is an isogeny of degree n 2 from the Jacobian J C to E 1 × E 2 . We say that J C is (n, n)-decomposable. If the degree n is odd the pair (ψ 2 , E 2 ) is canonically determined. For n = 3, 5, and 7, we give arithmetic examples of curves whose Jacobians are (n, n)-decomposable.