claimstandardinferred_dep

Modular Jacobian dimension and height bounds

Bounds dim⁡J≤CN2\dim J\le CN^2 and hF(J)≤CN12h_{\mathrm F}(J)\le CN^{12}
polynomial control of the modular Jacobian
Claim·§4.9 The modular factor and the final height bound·inside thmHeight bounded by radical of contact primes

Statement (paper)

With g0=dim⁡Jg_0=\dim J, the modular Jacobian JJ satisfies g0≤CN2g_0\le C N^2 and max⁡{0,hF(J)}≤CN12\max\{0,h_{\mathrm F}(J)\}\le C N^{12} for a constant CC.

In plain English

The modular Jacobian JJ has dimension g0≤CN2g_0\le CN^2 and stable Faltings height satisfying max⁡{0,hF(J)}≤CN12\max\{0,h_{\mathrm F}(J)\}\le CN^{12}, for some constant CC, so both are polynomial in the level NN.

Why it's here

Since JJ contains AA as an isogeny factor (the modular-factor lemma), the Faltings height of AA can be controlled through JJ. This claim supplies the polynomial-in-NN bounds on dim⁡J\dim J and hF(J)h_{\mathrm F}(J) that the next claim uses, via an isogeny comparison, to get hF(A)≤CMCh_{\mathrm F}(A)\le CM^C.
Classified standard:
Known bounds for modular Jacobians: the genus of X_1(N) (or X_0(N)) is O(N^2), and the Faltings height of J_1(N) or J_0(N) is polynomially bounded in N, as in the height bounds for modular Jacobians due to Ullmo and Jorgenson–Kramer or the Edixhoven–Couveignes book on computing Galois representations. The exponent 12 is a crude polynomial bound.

Uses

lemModular Jacobian contains A as isogeny factor

Used by

claimFaltings height of A is polynomial in M

Sources

PaperOpen
Section
§4.9 The modular factor and the final height bound
Page
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Label
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sections/04-height.tex:1205
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Modular Jacobian height polynomial in N" rests on hei:modular-factor