Thursday, 11 January 2007

nt.number theory - A nice variety without a smooth model

As Minhyong suggests, a curve CC (with C(mathbbQp)neq0C(mathbbQp)neq0) which has bad reduction but whose jacobian JJ has good reduction would do the affair. This works because the cohomology of CC is essentially the same as that of JJ, and because an abelian mathbbQpmathbbQp-variety has good reduction if and only if its ll-adic étale cohomology is unramified for some (and hence for every) prime lneqplneqp (Néron-Ogg-Shafarevich) or its pp-adic étale cohomology is crystalline (Fontaine-Coleman-Iovita).



I asked Qing Liu for explicit examples. He suggested the curve
y2=(x3+1)(x3+ap6)qquad(ainmathbbZtpimes)y2=(x3+1)(x3+ap6)qquad(ainmathbbZtpimes)
when pneq2,3pneq2,3, and y2=(x3+x+1)(x3+a34x+b36)y2=(x3+x+1)(x3+a34x+b36), with a,binmathbbZt3imesa,binmathbbZt3imes, for p=3p=3.



He refers to Proposition 10.3.44 in his book for computing the stable reduction of these CC, and to Bosch-Lütkebohmert-Raynaud,
Néron models, Chapter 9, for showing that JJ has good reduction.



I "accept" this answer as coming from Minhyong Kim and Qing Liu.

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