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Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point. (English) Zbl 0805.14018

Let \(K\) be a number field, \(S\) a finite set of primes (i.e. discrete valuations) and \({\mathfrak O}_ S\) the ring of \(S\)-integers in \(K\). The authors consider complete, smooth irreducible curves of genus 2 over \(K\). Such a curve \(C\) has exactly six Weierstrass points, and these are precisely the ramification points of the associated canonical double covering \(C \to \mathbb{P}^ 1_ K\). A curve is said to have good reduction at a prime \(v \notin S\) if there exists a relatively proper, smooth scheme \({\mathcal C}\) of dimension one over \(\text{Spec} ({\mathfrak O}_ v)\) such that \({\mathcal C} \otimes_{{\mathfrak O}_ v} K \cong C\). It can be shown that if \(C\) has good reduction at \(v\) then so does its Jacobian variety \(J_ C\). We regard two curves as equivalent if their corresponding Jacobian varieties are isogenous. This may in principle be verified by computing the traces of Frobenius at all primes yielding good reduction for the two Jacobian varieties and checking agreement up to an effective bound. In general, though, this bound is too large for practical computation. A courser equivalence relation is introduced in this paper: two curves of genus 2 are \(N\)-equivalent if, for all rational prime \(p\) with \(2 \leq p \leq N\) and \(p \notin S\), the traces of Frobenius at \(p\) for the Jacobians of the two curves agree. In this paper the authors choose the convenient but somewhat arbitrary bound of 89. The main result of the paper is that there are ninety-five 89-equivalence classes of curves of genus 2 defined over \(\mathbb{Q}\) having a rational Weierstrass point and good reduction away from 2. A consequence is that there are at least 95 isogeny classes of Jacobian varieties of such curves.

MSC:

14H55 Riemann surfaces; Weierstrass points; gap sequences
14H45 Special algebraic curves and curves of low genus
14H40 Jacobians, Prym varieties
14K02 Isogeny
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