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Rheinische Friedrich-Wilhelms-Universität Bonn
Université des Sciences et Technologies de Lille (Lille-1)
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Papers by Sarah Zerbes
Asian Journal of Mathematics, 2010
We define a family of Coleman maps for positive crystalline p-adic representations of the absolut... more We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f = anq n be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f , we define Coleman maps Col i for i = 1, 2 with values in Q p ⊗ Zp Λ, where Λ is the Iwasawa algebra of Z × p . Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.
We define a family of Coleman maps for positive crystalline p-adic representations of the absolut... more We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f = anq n be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f , we define Coleman maps Col i for i = 1, 2 with values in Q p ⊗ Zp Λ, where Λ is the Iwasawa algebra of Z × p . Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.
We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomo... more We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of Qp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2.
In this paper, we study the Coleman maps for a crystalline representation V with non-negative Hod... more In this paper, we study the Coleman maps for a crystalline representation V with non-negative Hodge-Tate weights via Perrin-Riou's p-adic regulator L V . Denote by H(Γ) the algebra of Qp-valued distributions on Γ = Gal(Qp(µ p ∞ )/Qp). Our first result determines the H(Γ)-elementary divisors of the quotient
We study Kato and Perrin-Riou's critical slope p-adic L-function attached to an ordinary modular ... more We study Kato and Perrin-Riou's critical slope p-adic L-function attached to an ordinary modular form using the methods of . We show that it may be decomposed as a sum of two bounded measures multiplied by explicit distributions depending only on the local properties of the modular form at p. We use this decomposition to prove results on the zeros of the p-adic L-function, and we show that our results match the behaviour observed in examples calculated by Pollack and Stevens in [PS08].
Asian Journal of Mathematics, 2010
We define a family of Coleman maps for positive crystalline p-adic representations of the absolut... more We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f = anq n be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f , we define Coleman maps Col i for i = 1, 2 with values in Q p ⊗ Zp Λ, where Λ is the Iwasawa algebra of Z × p . Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.
We define a family of Coleman maps for positive crystalline p-adic representations of the absolut... more We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f = anq n be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f , we define Coleman maps Col i for i = 1, 2 with values in Q p ⊗ Zp Λ, where Λ is the Iwasawa algebra of Z × p . Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.
We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomo... more We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of Qp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2.
In this paper, we study the Coleman maps for a crystalline representation V with non-negative Hod... more In this paper, we study the Coleman maps for a crystalline representation V with non-negative Hodge-Tate weights via Perrin-Riou's p-adic regulator L V . Denote by H(Γ) the algebra of Qp-valued distributions on Γ = Gal(Qp(µ p ∞ )/Qp). Our first result determines the H(Γ)-elementary divisors of the quotient
We study Kato and Perrin-Riou's critical slope p-adic L-function attached to an ordinary modular ... more We study Kato and Perrin-Riou's critical slope p-adic L-function attached to an ordinary modular form using the methods of . We show that it may be decomposed as a sum of two bounded measures multiplied by explicit distributions depending only on the local properties of the modular form at p. We use this decomposition to prove results on the zeros of the p-adic L-function, and we show that our results match the behaviour observed in examples calculated by Pollack and Stevens in [PS08].