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clarify "over Qp" as being over the Robba ring of Qp
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Rob Harron
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Hey Rob! You may want to take a look at Colmez's "Représentations triangulines de dimension 2" (here). He's working with (φ, Γ)-modules over Qp (by which I mean the Robba ring of Qp) not Fp((T)) though. But he shows that every one-dimensional (φ, Γ)-module over Qp is given by a p-adic character δ of $\mathbf{Q}_p^\times$ with the action of φ on a basis vector given by δ(p) and the action of $\gamma\in\Gamma$ given by $\delta(\chi(\gamma))$ (with $\chi$ the $p$-adic cyclotomic character). See section 2 and proposition 3.1 in particular of Colmez's paper (in which he also describes the $H^1$'s of these one-dimensional $(\phi,\Gamma)$-modules).

EDIT: looking at Colmez's paper, it looks like the way that he knows that this is all one-dimensional $(\phi,\Gamma)$-modules is by using the equivalence of catgeories with Galois representations, so I guess this doesn't really answer your question as it doesn't address the $(\phi,\Gamma)$-modules intrinsically.

Hey Rob! You may want to take a look at Colmez's "Représentations triangulines de dimension 2" (here). He's working with (φ, Γ)-modules over Qp not Fp though. But he shows that every one-dimensional (φ, Γ)-module over Qp is given by a p-adic character δ of $\mathbf{Q}_p^\times$ with the action of φ on a basis vector given by δ(p) and the action of $\gamma\in\Gamma$ given by $\delta(\chi(\gamma))$ (with $\chi$ the $p$-adic cyclotomic character). See section 2 and proposition 3.1 in particular of Colmez's paper (in which he also describes the $H^1$'s of these one-dimensional $(\phi,\Gamma)$-modules).

EDIT: looking at Colmez's paper, it looks like the way that he knows that this is all one-dimensional $(\phi,\Gamma)$-modules is by using the equivalence of catgeories with Galois representations, so I guess this doesn't really answer your question as it doesn't address the $(\phi,\Gamma)$-modules intrinsically.

Hey Rob! You may want to take a look at Colmez's "Représentations triangulines de dimension 2" (here). He's working with (φ, Γ)-modules over Qp (by which I mean the Robba ring of Qp) not Fp((T)) though. But he shows that every one-dimensional (φ, Γ)-module over Qp is given by a p-adic character δ of $\mathbf{Q}_p^\times$ with the action of φ on a basis vector given by δ(p) and the action of $\gamma\in\Gamma$ given by $\delta(\chi(\gamma))$ (with $\chi$ the $p$-adic cyclotomic character). See section 2 and proposition 3.1 in particular of Colmez's paper (in which he also describes the $H^1$'s of these one-dimensional $(\phi,\Gamma)$-modules).

EDIT: looking at Colmez's paper, it looks like the way that he knows that this is all one-dimensional $(\phi,\Gamma)$-modules is by using the equivalence of catgeories with Galois representations, so I guess this doesn't really answer your question as it doesn't address the $(\phi,\Gamma)$-modules intrinsically.

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Rob Harron
  • 4.8k
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  • 35

Hey Rob! You may want to take a look at Colmez's "Représentations triangulines de dimension 2" (here). He's working with (φ, Γ)-modules over Qp not Fp though. But he shows that every one-dimensional (φ, Γ)-module over Qp is given by a p-adic character δ of $\mathbf{Q}_p^\times$ with the action of φ on a basis vector given by δ(p) and the action of $\gamma\in\Gamma$ given by $\delta(\chi(\gamma))$ (with $\chi$ the $p$-adic cyclotomic character). See section 2 and proposition 3.1 in particular of Colmez's paper (in which he also describes the $H^1$'s of these one-dimensional $(\phi,\Gamma)$-modules).

EDIT: looking at Colmez's paper, it looks like the way that he knows that this is all one-dimensional $(\phi,\Gamma)$-modules is by using the equivalence of catgeories with Galois representations, so I guess this doesn't really answer your question as it doesn't address the $(\phi,\Gamma)$-modules intrinsically.

Hey Rob! You may want to take a look at Colmez's "Représentations triangulines de dimension 2" (here). He's working with (φ, Γ)-modules over Qp not Fp though. But he shows that every one-dimensional (φ, Γ)-module over Qp is given by a p-adic character δ of $\mathbf{Q}_p^\times$ with the action of φ on a basis vector given by δ(p) and the action of $\gamma\in\Gamma$ given by $\delta(\chi(\gamma))$ (with $\chi$ the $p$-adic cyclotomic character). See section 2 and proposition 3.1 in particular of Colmez's paper (in which he also describes the $H^1$'s of these one-dimensional $(\phi,\Gamma)$-modules).

Hey Rob! You may want to take a look at Colmez's "Représentations triangulines de dimension 2" (here). He's working with (φ, Γ)-modules over Qp not Fp though. But he shows that every one-dimensional (φ, Γ)-module over Qp is given by a p-adic character δ of $\mathbf{Q}_p^\times$ with the action of φ on a basis vector given by δ(p) and the action of $\gamma\in\Gamma$ given by $\delta(\chi(\gamma))$ (with $\chi$ the $p$-adic cyclotomic character). See section 2 and proposition 3.1 in particular of Colmez's paper (in which he also describes the $H^1$'s of these one-dimensional $(\phi,\Gamma)$-modules).

EDIT: looking at Colmez's paper, it looks like the way that he knows that this is all one-dimensional $(\phi,\Gamma)$-modules is by using the equivalence of catgeories with Galois representations, so I guess this doesn't really answer your question as it doesn't address the $(\phi,\Gamma)$-modules intrinsically.

Source Link
Rob Harron
  • 4.8k
  • 2
  • 25
  • 35

Hey Rob! You may want to take a look at Colmez's "Représentations triangulines de dimension 2" (here). He's working with (φ, Γ)-modules over Qp not Fp though. But he shows that every one-dimensional (φ, Γ)-module over Qp is given by a p-adic character δ of $\mathbf{Q}_p^\times$ with the action of φ on a basis vector given by δ(p) and the action of $\gamma\in\Gamma$ given by $\delta(\chi(\gamma))$ (with $\chi$ the $p$-adic cyclotomic character). See section 2 and proposition 3.1 in particular of Colmez's paper (in which he also describes the $H^1$'s of these one-dimensional $(\phi,\Gamma)$-modules).