Yesterday Adrian Iovita gave an extremely interesting talk at the local Quebec-Vermont Number Theory Seminar (QVNTS) on a -adic criterion for good reduction of curves.
If is an abelian variety over some -adic local field , then it is an old result that has good reduction if and only if its Tate module is unramified for one (and hence all) primes . Fontaine reformulated this criterion for : in this case, one has that has good reduction if and only if is crystalline.
Is there an analogue for curves, say expressed in terms of etale ? It turns out that there isnt. The is an abelian invariant and doesn’t capture enough information about the curve. However, one can formulate conditions analogous to the above if one works with the etale fundamental group of the curve, rather than its abelianization!
In the 1990’s, Takayuki Oda proved the following: for let denote the largest pro- quotient of , where is a geometric point of . Then there is a natural homomorphism (representation) of the absolute Galois group of . If we define a filtration on by setting and otherwise letting denote the closure of the commutator subgroup , and we write , then the representation induces representations of into for all . With this notation, Oda proved that the curve has good reduction if and only if for one (and hence all) primes , the representation is unramified (that is, it vanishes on inertia). In fact, one only has to verify that the first few are unramified (say for ; I’m not sure if one can get by with less).
Now Andreatta-Iovita-Kim have essentially completed the story for curves: in the case , it turns out that the quotients are the -points of a finite group scheme over , such that the corresponding Hopf algebra is endowed with a natural action of . Say that is crystalline if this -adic representation is crystalline. Then has good reduction if and only if is crystalline for all . As above, it suffices to check this for the first few . This result hasn’t appeared yet, but I look forward to their paper!
Edit: Fixed the alphabetical ordering of the attribution above.