TY - CHAP

T1 - Semistable types of hyperelliptic curves

AU - Maistret, Celine

AU - Dokchitser, Tim

AU - Dokchitser, Vladimir

AU - Morgan, Adam

PY - 2019/1/22

Y1 - 2019/1/22

N2 - In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining equation (`cluster pictures'). We construct explicit combinatorial one-to-one correspondences between the three, which furthermore respect automorphisms and allow to keep track of the monodromy pairing and the Tamagawa group of the Jacobian. We introduce a classification scheme and a naming convention for semistable types of hyperelliptic curves and types with a Frobenius action. This is the higher genus analogue of the distinction between good, split and non-split multiplicative reduction for elliptic curves. Our motivation is to understand L-factors, Galois representations, conductors, Tamagawa numbers and other local invariants of hyperelliptic curves and their Jacobians.

AB - In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining equation (`cluster pictures'). We construct explicit combinatorial one-to-one correspondences between the three, which furthermore respect automorphisms and allow to keep track of the monodromy pairing and the Tamagawa group of the Jacobian. We introduce a classification scheme and a naming convention for semistable types of hyperelliptic curves and types with a Frobenius action. This is the higher genus analogue of the distinction between good, split and non-split multiplicative reduction for elliptic curves. Our motivation is to understand L-factors, Galois representations, conductors, Tamagawa numbers and other local invariants of hyperelliptic curves and their Jacobians.

UR - https://arxiv.org/abs/1704.08338

UR - https://bookstore.ams.org/conm-724/

M3 - Chapter in a book

SN - 9781470442477

T3 - Contemporary Mathematics

SP - 73

EP - 136

BT - Algebraic Curves and Their Applications

A2 - Beshaj, Lubjana

A2 - Shaska, Tony

PB - American Mathematical Society

ER -