lemmaroutineinferred_dep

3-adic and 2-adic representations are compatible

Shows r3r_3 comes from the same newform as r2r_2
reads the level from conductors at several primes
Lemma·hei:compatibility·§4.7 Two-dimensional representations and modularity

Statement (paper)

The representation r3r_3 is the 33-adic representation of the same algebraic conjugate of ff that realizes r2r_2.

In plain English

The 3-adic representation r3r_3 is the 3-adic representation attached to the same algebraic conjugate of the newform ff that gives r2r_2.

Why it's here

Modularity was shown only for the 2-adic representation r2r_2, producing the newform ff. This lemma ties r3r_3 to that same form, so both r2r_2 and r3r_3 are members of one compatible system. This lets the level of ff be read off from the Artin conductors of rℓr_\ell for several ℓ\ell, which the level-bound lemma uses.
Classified routine:
Follows by a standard compatible-systems argument: the characteristic polynomials of Frobenius of r_2 and r_3 agree (e.g. both come from the same motive/abelian variety), so by Chebotarev density and Brauer–Nesbitt r_3 is isomorphic to the 3-adic representation attached to the conjugate of f realizing r_2.

Uses

claimThe 2-adic representation r_2 is modular

Used by

lemModular level is polynomially bounded in MclaimLevel exponents equal Artin conductors

Sources

PaperOpen
Section
§4.7 Two-dimensional representations and modularity
Page
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Label
hei:compatibility
Line
sections/04-height.tex:986
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    critique: Lemma hei:compatibility compares r_3 with "the same algebraic conjugate of f that realizes r_2" (04-height.tex:987-988). This presupposes the modularity identification made in claim-hei-height-6 (lines 952-979).