claimroutineinferred_dep

Abelian surface family with quaternionic multiplication

Builds abelian surface family over

Y

with quaternionic multiplication

family whose fibers carry the Faltings heights

Claim·§4.2 The abelian surface family·inside thmHeight bounded by radical of contact primes

Statement (paper)

There exist a finite level map

Y\to X

, unramified over

X

, and a principally polarized abelian surface scheme

\mathcal A\to Y

such that, with

\pi:Y\to X\to X^*\simeq\mathbf P^1

, the following hold. Every geometric fiber

\mathcal A_y

y\in Y(\overline\Q)

, carries an embedding

B\hookrightarrow\operatorname{End}^0_{\overline\Q}(\mathcal A_y)

given by left multiplication;

\pi

has ramification index two at every point above

\mathcal C

and index one elsewhere; fibers with the same image under

\pi

are geometrically isogenous, so that if

\pi(y)=s\in\mathbf P^1(\Q)

then

\mathcal A_y

is isogenous to each of its Galois conjugates; and the polarized moduli map is nonconstant on every geometric component of

Y

In plain English

After passing to a finite unramified level cover

Y\to X

, there is a principally polarized abelian surface scheme

\mathcal A\to Y

whose fibers all carry a left-multiplication action of

B

. The map

\pi:Y\to X^*\simeq\mathbf P^1

has ramification index two above

\mathcal C

and one elsewhere, fibers over the same point of

X^*

are geometrically isogenous (so each fiber over a rational point is isogenous to its Galois conjugates), and the moduli map is nonconstant on every geometric component.

Why it's here

This turns the Shimura curve quotient with its five rational branch values

\mathcal C

into an actual family of abelian surfaces, with the ramification of

\pi

exactly at

\mathcal C

, which is what ties bad reduction behavior to the radical condition in the height bound. The isogeny-to-conjugates property and the nonconstant moduli map are what the later Faltings-height arguments for the CM and non-CM cases use.

Classified routine:

Standard construction of a fake elliptic curve (QM abelian surface) family over a level cover of a quaternionic Shimura curve, using the moduli interpretation of Shimura curves (Shimura; Buzzard; Milne) together with the Atkin–Lehner quotient computed in the previous claim.

Uses

claimQuaternionic Shimura curve quotient has five rational branch values

Used by

claimCM fibers have uniformly bounded Faltings heightclaimNon-CM case: quasi-isogenies and sign cocycle

Sources

PaperOpen

Section: §4.2 The abelian surface family
Page: —
Label: —
Line: sections/04-height.tex:177

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Abelian surface family with B-action and isogenous conjugates" rests on claim-hei-height-1