claimstandard
Level exponents equal Artin conductors
Level exponent at equals Artin conductor exponent of
reduces the level bound to local conductors
Claim·§4.8 A uniform bound for the modular level·inside lemModular level is polynomially bounded in M
Statement (paper)
For every prime , the exponent of in the primitive level of the weight-two form equals the Artin-conductor exponent at of its -adic representation (for , using at all primes other than and at ), including the Weil--Deligne monodromy contribution. Moreover, this exponent is zero for every prime .
In plain English
For each prime , the exponent of in the primitive level of the weight-two newform equals the Artin-conductor exponent at of , with the Weil–Deligne monodromy term included. Here is used at primes other than 2 and at . This exponent is zero for every prime .
Why it's here
This turns the bound on the level into a bound on local Artin conductors of Galois representations. The compatibility of and with the same newform lets one use whichever representation is coprime to at each prime. Later claims then only need to bound conductors at the primes in , with the small primes handled by uniform ramification cuts.
Classified standard:
This is local-global compatibility of the Langlands correspondence for weight-two newforms: the level exponent equals the conductor of the Weil–Deligne representation at q≠ℓ (Carayol 1986, building on Deligne, Langlands). Vanishing outside T follows because the representation is unramified there.
Uses
Used by
Sources
PaperOpen
- Section
- §4.8 A uniform bound for the modular level
- Page
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- Label
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sections/04-height.tex:1051
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