A user's guide to the local arithmetic of hyperelliptic curves

Abstract

A new approach has been recently developed to study the arithmetic of hyperelliptic curves 𝑦 2 = 𝑓(𝑥) over local fields of odd residue characteristic via combinatorial data associated to the roots of 𝑓. Since its introduction, numerous papers have used this machinery of 'cluster pictures' to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.

2‐Selmer parity for hyperelliptic curves in quadratic extensions

“…Remark Instead of appealing to Proposition 12.18, an alternative approach to proving Proposition 13.20 might be to draw on work of Betts [3, Section 3] (see also [2, Section 10]), which gives a description in terms of clusters for the individual Tamagawa numbers $$c(J/L)$$ and $$c(J/K)$$. From this, one might then hope to prove the result by computing explicitly the quotient $$c(J/L)/c(J/K)$$ and appealing to ().…”

“…We caution that [20] contains some minor errors. These are discussed and corrected in the PhD thesis of Nowell [45] (see also [2, Section 9]). Remark For an alternative, but related, approach to constructing regular models of hyperelliptic curves over non‐archimedean local fields of odd residue characteristic, see the works of Srinivasan [59] and Obus–Srinivasan [46].…”

“…With the relevant invariants being described by Lemma 14.7, that the minimal regular SNC model takes the desired form now follows from specialising [20, Theorems 7.12 and 7.18] to the case in hand, save for the fact that the statements of these results contain some minor errors. A corrected version of the relevant results appears in the PhD thesis of Nowell [45, Theorems 9.23, 9.31 and 9.32] (see also [2, Section 9]), from which one obtains the claimed description of the minimal regular SNC model.$$\Box$$…”

The 2-parity conjecture for elliptic curves with isomorphic 2-torsion

“…To keep track of the arithmetic invariants needed to compute λf,K as in lemma 2.6, we use the machinery of ‘clusters’ [17,18]. Clusters allow us to extract arithmetic invariants of hyperelliptic curves over p-adic fields with p odd from simple combinatorial data (see example 4.4 below for a worked out example).…”

Root number of the Jacobian of y 2 =x p +a

“…Now suppose that = p is an odd prime. We will use the theory of clusters to compute the corresponding inertia representation and refer the reader to the user's guide [1] for the relevant definitions and theorem statements. The set of roots is or p respectively by [1,Theorem 13.4].…”