claimroutineinferred_dep

CM fibers have uniformly bounded Faltings height

Bounds Faltings height uniformly for CM fibers
CM case of the height bound
Claim·§4.6 The endomorphism dichotomy and descent up to isogeny·inside thmHeight bounded by radical of contact primes

Statement (paper)

If CB=kC_B=k is an imaginary quadratic field, then A∼LE2A\sim_L E^2 for a CM elliptic curve E/LE/L, and the stable Faltings height hF(A)h_{\mathrm F}(A) is bounded by a constant that is uniform over all such fibers (so Theorem  holds for these fibers, since M≥1M\ge1).

In plain English

If the endomorphism field CBC_B equals the imaginary quadratic field kk, then AA is isogenous over LL to E2E^2 for a CM elliptic curve E/LE/L, and its stable Faltings height hF(A)h_{\mathrm F}(A) is bounded by a constant uniform over all such fibers. Hence the height bound of Theorem hei:height holds for these fibers, since M≥1M\ge1.

Why it's here

This settles the CM case of the endomorphism dichotomy: the fiber splits up to isogeny as a square of a CM curve, and CM elliptic curves have Faltings height bounded uniformly in terms of the bounded field LL from hei-hom-field. Combined with hei-base-height, which bounds the base height by the Faltings height, this gives a bound on h(s)\hgt(s) by a constant, which is at most HMS(s)cH M_S(s)^c because M≥1M\ge1.
Classified routine:
Follows from standard facts: a QM abelian surface with CM splits up to isogeny as E2E^2 with EE CM, and CM elliptic curves over a field of bounded degree have only finitely many jj-invariants (class number bounds), hence bounded Faltings height. This is combined with the bounded field from hei-hom-field.

Uses

lemBounded field for all geometric homomorphismslemBase height bounded by Faltings heightclaimAbelian surface family with quaternionic multiplication

Used by

Nothing in the paper

Sources

PaperOpen
Section
§4.6 The endomorphism dichotomy and descent up to isogeny
Page
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Label
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Line
sections/04-height.tex:622
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "CM fibers have uniformly bounded Faltings height" rests on hei:hom-field
  • info
    inferred_dep
    outline: "CM fibers have uniformly bounded Faltings height" rests on hei:base-height
  • info
    inferred_dep
    outline: "CM fibers have uniformly bounded Faltings height" rests on claim-hei-height-2