Local invariants of isogenous elliptic curves

Abstract

We investigate how various invariants of elliptic curves, such as the discriminant, Kodaira type, Tamagawa number and real and complex periods, change under an isogeny of prime degree p. For elliptic curves over l-adic fields, the classification is almost complete (the exception is wild potentially supersingular reduction when l = p), and is summarised in a table.

Growth of in towers for isogenous curves

“…(2) If E/K has tame reduction, the reduction stays tame over F. Furthermore, 0 δ, δ ′ , δ F , δ ′ F < 12 by [9] Thm. 3.1.…”

“…Theorems 1.1-1.3 are proved in §8. In §4, §5 we rely on the results of [9] that describe how local invariants of elliptic curves change under isogeny.…”

“…(1) Theorem 5.1 of [9] (or [9] Table 1) describes the change in the discriminant under isogenies of prime degree. If E has potentially good reduction, and either l = p or E has good or potentially ordinary reduction, then δ = δ ′ , δ F = δ ′ F .…”

“…Otherwise, by decomposing φ into endomorphisms and isogenies of prime degree if necessary, we may assume that deg φ = p is prime. Now apply the classification for the quotient c c ′ from [9] Table 1. All the conditions in the table that determine c c ′ (e.g.…”

“…) is 0 unless E has additive potentially supersingular reduction at p, see [9] Table 1. In this exceptional case, µ does not have to be an integer, see Example 1.6.…”