Properties of Néron Models

On the complexity group of stable curves

“…Of course this is also a bound on the cardinality of the group of components of the Néron models of the Jacobians of such families. However, notice that this bound is different from the ones found by Lorenzini in [Lor93] (see also [BLR90], Theorem 9 sec.6.9). Indeed, given any strictly henselian discrete valuation ring R, with algebraically closed residue field k and field of fractions K, any regular family of curves f : X −→ spec R such that the general fiber X K is of genus g, and the Jacobian J K has potential good reduction, Lorenzini finds a bound, depending only on g, for the group of components of the Néron model of f .…”

“…Remark 4.3. As already noticed, the geometric meaning of this result is that it gives a bound on the group of connected components of the Néron model of the degree-d Picard variety for families of stable curves ( [BLR90], Theorem 1, sec.9.6), and as well on the number of irreducible components of the fibres of the scheme P d g constructed in [Cap08] and of P d,g of [Cap94]. Of course this is also a bound on the cardinality of the group of components of the Néron models of the Jacobians of such families.…”

“…More precisely, let R be a discrete valuation ring with fraction field K and residue field k and denote by B the spectrum of R. Consider a flat and proper regular curve X −→ B, such that the closed fiber X k is geometrically irreducible. Under some technical assumptions, it is defined the group Φ of connected components of the the Néron model of the Jacobian J K of the generic fiber X K (see [BLR90] sec. 9.6).…”

From the Function-Sheaf Dictionary to Quasicharacters of -Adic Tori

“…When K ′ /K is unramified, the Néron model of T K is a form of the Néron model for G m,K , with the nontrivial automorphism σ ∈ Gal(K ′ /K) mapping (x, n) to (σ(x), −n) for x ∈ R ′× and n ∈ Z specifying the copy of G m,R ′ . This example illustrates the compatibility between Néron models and unramified base change [6,§10.1,Prop. 3].…”

Geometry of Fermat adeles

“…By Theorem 6.9 we may assume the family where (E, θ) is the Jacobian of the curve corresponding to P . By Artin approximation [4], p. 91, it is enough to find an R ∈ Witt p with algebraically closed residue field and P ∈ U (R) such that (6.10) holds. Fix an R ∈ Witt p with algebraically closed residue field k. Since M g ⊗ k is irreducible and the ordinary locus in it is open and dense [20] it follows that, with exception of finitely many characteristics (which we can always discard by enlarging S), the ordinary locus in U ⊗ k is nonempty.…”