Lukas Melninkas

Local root and Tamagawa numbers of hyperelliptic curves

Given a p-adic field K we consider a hyperelliptic curve of genus (p-1)/2 defined over K. Under the hypothesis that the associated l-adic Galois representation is wildly ramified, we give a formula for the number of rational components of the Néron model of the Jacobian, generalizing Tate's algorithm. As an application for genera 1 and 2, we assume further that the Galois representation has the maximal possible inertia image and give geometric formulae for the root numbers.