claimroutineinferred_dep

Faltings height of A is polynomial in M

Bounds Faltings height hF(A)≤CMCh_{\mathrm F}(A)\le CM^C
non-CM case of the height bound
Claim·§4.9 The modular factor and the final height bound·inside thmHeight bounded by radical of contact primes

Statement (paper)

There is a constant CC, independent of ss, such that hF(A)≤CMCh_{\mathrm F}(A)\le C M^C. This follows by choosing a principally polarized complement DZD_Z of A8A^8 with C1=J8∼L′C2=A8×DZC_1=J^8\sim_{L'}C_2=A^8\times D_Z, an isogeny ϕ:C1→C2\phi:C_1\to C_2 satisfying log⁡deg⁡ϕ≤CMC\log\deg\phi\le C M^C, and the inequality 8hF(A)+hF(DZ)≤hF(C1)+12log⁡deg⁡ϕ≤CMC8h_{\mathrm F}(A)+h_{\mathrm F}(D_Z)\le h_{\mathrm F}(C_1)+\tfrac12\log\deg\phi\le C M^C together with hF(DZ)≥−12(dim⁡DZ)log⁡(2π2)h_{\mathrm F}(D_Z)\ge-\tfrac12(\dim D_Z)\log(2\pi^2).

In plain English

There is a constant CC, independent of ss, such that the Faltings height of the abelian variety AA satisfies hF(A)≤CMCh_{\mathrm F}(A)\le C M^C.

Why it's here

The modular Jacobian JJ has dimension and Faltings height polynomial in the level NN, and NN is polynomial in MM. Since JJ is isogenous to A×D0A\times D_0, one builds a principally polarized complement DZD_Z of A8A^8 and an isogeny J8→A8×DZJ^8\to A^8\times D_Z of degree log⁡deg⁡ϕ≤CMC\log\deg\phi\le CM^C. The isogeny height inequality bounds 8hF(A)+hF(DZ)8h_{\mathrm F}(A)+h_{\mathrm F}(D_Z) above, and the universal lower bound on hF(DZ)h_{\mathrm F}(D_Z) then isolates hF(A)h_{\mathrm F}(A).
Classified routine:
Follows from the preceding Jacobian dimension/height bounds and level bound by the standard Faltings isogeny height inequality, Zarhin's trick (A^8 times its dual-type complement is principally polarized), and the Bost-type universal lower bound on Faltings height; this is standard bookkeeping, not a new idea.

Uses

claimModular Jacobian dimension and height boundslemModular level is polynomially bounded in MlemModular Jacobian contains A as isogeny factorclaimCorrected Galois representation on Tate module

Used by

Nothing in the paper

Sources

PaperOpen
Section
§4.9 The modular factor and the final height bound
Page
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sections/04-height.tex:1232
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Faltings height of A polynomial in M via isogeny" rests on claim-hei-height-7
  • info
    inferred_dep
    outline: "Faltings height of A polynomial in M via isogeny" rests on hei:level-bound
  • info
    inferred_dep
    outline: "Faltings height of A polynomial in M via isogeny" rests on hei:modular-factor
  • info
    inferred_dep
    outline: "Faltings height of A polynomial in M via isogeny" rests on claim-hei-height-5