Thursday, 11 January 2007

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

As Minhyong suggests, a curve C (with C(mathbbQp)neq0) which has bad reduction but whose jacobian J has good reduction would do the affair. This works because the cohomology of C is essentially the same as that of J, and because an abelian mathbbQp-variety has good reduction if and only if its l-adic étale cohomology is unramified for some (and hence for every) prime lneqp (Néron-Ogg-Shafarevich) or its p-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(ainmathbbZptimes)
when pneq2,3, and y2=(x3+x+1)(x3+a34x+b36), with a,binmathbbZ3times, for p=3.



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



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

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