claimroutineinferred_dep

Order and conductor bounds for chi

Bounds order of χ\chi by CMCM and shows tame bounded conductors
quantitative control of chi
Claim·§4.6 The endomorphism dichotomy and descent up to isogeny·inside lemSign-splitting cochain with square-root character

Statement (paper)

The Dirichlet character χ\chi has order mm satisfying m≤2∏q∈T, q≠2(q−1)≤CMm\le 2\prod_{q\in T,\ q\ne 2}(q-1)\le C M. Its conductor at each odd prime q∈Tq\in T divides qq, so χ\chi is tame there, its conductor at 22 divides 44, and χ\chi is unramified at all other primes; in particular the local conductor of χ\chi at any fixed prime is bounded independently of ss.

In plain English

The character χ\chi has order m≤2∏q∈T, q≠2(q−1)≤CMm \le 2\prod_{q\in T,\,q\ne 2}(q-1)\le CM. It has conductor dividing qq at each odd q∈Tq\in T (so it is tame there), conductor dividing 44 at 22, and is unramified elsewhere, so its local conductors are bounded independently of ss.

Why it's here

This claim gives the quantitative part of the sign-splitting lemma: the order bound m≤CMm\le CM and the tame, bounded conductors. These follow because the character from the previous claim is unramified outside TT, so it is a product of local characters at primes of TT whose size is controlled by the prime.
Classified routine:
Direct bookkeeping from the structure of Dirichlet characters unramified outside T: such a character factors through (Z/4·∏q)^×, so its order divides the exponent of that group and its conductors are tame at odd q. This is standard class field theory for Q applied to the character from the previous claim.

Uses

claimGlobal character matching the obstruction class

Used by

Nothing in the paper

Sources

PaperOpen
Section
§4.6 The endomorphism dichotomy and descent up to isogeny
Page
—
Label
—
Line
sections/04-height.tex:815
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Order and conductor bounds for chi" rests on claim-hei-sign-splitting-2