claimroutine

Uniform ramification cut for beta

Gives uniform ramification cut killed by β\beta at small primes
controls wild ramification of beta at small primes
Claim·§4.8 A uniform bound for the modular level·inside lemModular level is polynomially bounded in M

Statement (paper)

Fix a prime qq of SS-type smaller than the fixed constant, a place vv of LL above qq, and put F=LvF=L_v. Then there is a bound uβu_\beta, independent of ss (depending only on the finitely many possible embedded fields F/QqF/\Q_q and on the uniformly bounded Qq\Q_q-unit conductor of χ\chi), such that for every u>uβu>u_\beta the upper ramification group GQquG_{\Q_q}^u lies in GFG_F and is killed by the character β∣GF\beta|_{G_F}; in particular β\beta kills UFb=1+mFbU_F^b=1+\mathfrak m_F^b for a uniform integer bb.

In plain English

For a fixed small prime qq of SS-type and a place vv of LL above it, with F=LvF=L_v, there is a bound uβu_\beta independent of ss such that for all u>uβu>u_\beta the upper ramification group GQquG_{\Q_q}^u lies in GFG_F and is killed by β∣GF\beta|_{G_F}. In particular β\beta kills UFb=1+mFbU_F^b=1+\mathfrak m_F^b for some uniform integer bb.

Why it's here

This controls the wild ramification of the square-root cochain β\beta at small primes uniformly in ss. The bound uses only that finitely many local fields F/QqF/\Q_q occur and that the Qq\Q_q-unit conductor of χ\chi is uniformly bounded (from the sign-splitting lemma), and it feeds the uniform conductor bound at small primes used to get the level bound N≤CMaN\le CM^a.
Classified routine:
Follows from local class field theory and standard facts on upper ramification groups (e.g. Serre, Local Fields): there are finitely many local fields F/Q_q of bounded degree, and the Q_q-unit conductor of chi is uniformly bounded by the sign-splitting lemma, which together give a uniform cutoff.

Uses

lemSign-splitting cochain with square-root character

Used by

claimUniform conductor bound gives N ≤ CM²

Sources

PaperOpen
Section
§4.8 A uniform bound for the modular level
Page
—
Label
—
Line
sections/04-height.tex:1074
Leannot aligned

Phase 2 aligns this node to a Lean declaration.