The Néron model over the Igusa curves

Abstract

We analyze the geometry of rational p-division points in degenerating families of elliptic curves in characteristic p. We classify the possible Kodaira symbols and determine for the Igusa moduli problem the reduction type of the universal curve. Special attention is paid to characteristic 2 and 3, where wild ramification and stacky phenomena show up.

Purely additive reduction of abelian varieties with torsion

“…Let E/K be an elliptic curve with a K-rational point of order p. The reduction types that can occur in the case where K is the fraction field of a discrete valuation ring of characteristic p ≥ 5 have already been studied by Liedtke and Schröer in [18]. More precisely, among other results in [18], they prove the following theorem (see Theorem 4.3 of [18] and the Theorem in page 2158 of [18]).…”

The $μ$-invariant change for abelian varieties over finite $p$-extensions of global fields

“…Recall that the group scheme C p := ker F is the maximal connected subgroup scheme of A p whereas E (p) p := ker V is the maximal étale subgroup scheme of A (p) p . These groups are Cartier dual to each other and satisfy the exact sequence (see [LSc10,§3])…”

Torsion and twists of abelian varieties

“…Let 𝐸∕𝐾 be an elliptic curve with good reduction and let ∕𝑅 be the Néron Model of 𝐸∕𝐾. Assume that the Hasse invariant of ∕𝑅 has vanishing order 𝑝−1 2 (see [17,Section 5] for information on the Hasse invariant of ∕𝑅). Let 𝐸 (𝑝) ∕𝐾 be the Frobenius pullback of the curve 𝐸∕𝐾, which still has good reduction by [17,Proposition 7.2].…”