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David Loeffler
David Loeffler
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It is difficult to even define the conductor at $\ell$. The problem is that rho is infinitely wildly ramified at ell$\ell$ (it must be, since its determinant is a power of the ell$\ell$-adic cyclotomic character); so the naive definition of the conductor would be infinity.

The morally right way to define the conductor at p of a p-adic Galois representation is to use Fontaine's p-adic Hodge theory. In your case the representation is crystalline at ell$\ell$, so the conductor is 1, matching the level of the modular form.

It is difficult to even define the conductor at $\ell$. The problem is that rho is infinitely wildly ramified at ell (it must be, since its determinant is a power of the ell-adic cyclotomic character); so the naive definition of the conductor would be infinity.

The morally right way to define the conductor at p of a p-adic Galois representation is to use Fontaine's p-adic Hodge theory. In your case the representation is crystalline at ell, so the conductor is 1, matching the level of the modular form.

It is difficult to even define the conductor at $\ell$. The problem is that rho is infinitely wildly ramified at $\ell$ (it must be, since its determinant is a power of the $\ell$-adic cyclotomic character); so the naive definition of the conductor would be infinity.

The morally right way to define the conductor at p of a p-adic Galois representation is to use Fontaine's p-adic Hodge theory. In your case the representation is crystalline at $\ell$, so the conductor is 1, matching the level of the modular form.

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David Loeffler
David Loeffler
  • 37k
  • 3
  • 89
  • 194

It is difficult to even define the conductor at $\ell$. The problem is that rho is infinitely wildly ramified at ell (it must be, since its determinant is a power of the ell-adic cyclotomic character); so the naive definition of the conductor would be infinity.

The morally right way to define the conductor at p of a p-adic Galois representation is to use Fontaine's p-adic Hodge theory. In your case the representation is crystalline at ell, so the conductor is 1, matching the level of the modular form.