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Sarah Livia Zerbes

Publications and source records attributed to Sarah Livia Zerbes.

46 records · Page 3Linked to original sources
arXiv2010

Wach modules and critical slope p-adic L-functions

We study Kato and Perrin-Riou's critical slope p-adic L-function attached to an ordinary modular form, using the methods of our earlier work with Lei. 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.

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arXiv2010

Wach modules and Iwasawa theory for modular forms

For a crystalline p-adic representation of the absolute Galois group of Qp, we define a family of Coleman maps (linear maps from the Iwasawa cohomology of the representation to the Iwasawa algebra), using the theory of Wach modules. Let f = sum(a_n q^n) be a normalized new modular 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 two Coleman maps with values in the Iwasawa algebra of Zp^* (after extending scalars to some extension of Qp). 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 a_p=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 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.

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arXiv2010

Coleman maps and the p-adic regulator

This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative Hodge-Tate weights. In this paper, we study these Coleman maps using Perrin-Riou's p-adic regulator L_V. Denote by H(Γ) the algebra of Qp-valued distributions on Γ= Gal(Qp(μ(p^\infty) / Qp). Our first result determines the H(Γ)-elementary divisors of the quotient of D_{cris}(V) \otimes H(Γ) by the H(Γ)-submodule generated by (ϕ* N(V))^{ψ= 0}, where N(V) is the Wach module of V. By comparing the determinant of this map with that of L_V (which can be computed via Perrin-Riou's explicit reciprocity law), we obtain a precise description of the images of the Coleman maps. In the case when V arises from a modular form, we get some stronger results about the integral Coleman maps, and we can remove many technical assumptions that were required in our previous work in order to reformulate Kato's main conjecture in terms of cotorsion Selmer groups and bounded p-adic L-functions.

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arXiv2010

Akashi series of Selmer groups

We study the Selmer group of an elliptic curve over an admissible p-adic Lie extension of a number field F . We give a formula for the Akashi series attached to this module, in terms of the corresponding objects for the cyclotomic Zp-extension and certain correction terms. This extends our earlier work [arXiv:math/0404431], in particular since it applies to elliptic curves having split multiplicative reduction at some primes above p, in which case the Akashi series can have additional zeros.

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arXiv2010

Generalised Euler characteristics of Selmer groups

Let E be an elliptic curve defined over a number field F, and let p be a prime >= 5. In this paper we study the structure of the Selmer group of E over p-adic Lie extensions $F_\infty$ of F. In particular, under certain global and local conditions on $F_\infty$ we relate the generalised $Gal(F_\infty / F)$-Euler characteristic of $Sel(E / F_\infty)$ to the generalised Euler characteristic of the Selmer group over the cyclotomic Zp-extension of F. This invariant generalises the classical Euler characteristic to the case when the rank of E(F) is positive. Moreover, we show that the global and local conditions on $F_\infty$ are satisfied for a large class of p-adic Lie extensions of F .

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arXiv2008

Bloch-Kato exponential maps for local fields with imperfect residue fields

In this paper, we generalise the construction of the Bloch-Kato exponential map to complete discrete valuation fields of mixed characteristic (0,p) whose residue fields have a finite p-basis. As an application we prove an explicit reciprocity law, extending a result of Cherbonnier and Colmez in the classical case. This result relies on the calculation of the Galois cohomology of a p-adic representation V in terms of its (phi,G)-module.

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arXiv2007

The higher Hilbert pairing via (phi,G)-modules

We prove the Tate duality for higher dimensional local fields of mixed characteristic (0,p), when p is an odd prime, using the theory of higher fields of norms. Assuming that p is not ramified in the basefield, we then use this construction to define the higher Hilbert pairing. In particular, we show that the Hilbert pairing is non-degenerate, and we re-discover the formulae of Brueckner and Vostokov.

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arXiv2006

Generalized logarithmic derivatives for K_n

We construct (generalized) logarithmic derivatives for general n-dimensional local fields K of mixed characteristics (0,p) in which p is not necessarily a prime element with residue field k such that [k:k^p]=p^{n-1}. For the construction of the logarithmic derivative map, we define n-dimensional rings of overconvergent series and show that - as in the 1-dimensional case - they can be interpreted as functions converging on some annulus of the open unit p-adic disc. Using the generalized logarithmic derivative, we give a new construction of Kato's n-dimensional dual exponential map.

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arXiv2004

Selmer Groups over p-adic Lie Extensions I

Let $E$ be an elliptic curve defined over a number field $F$. In this paper, we study the structure of the $p^\infty$-Selmer group of $E$ over $p$-adic Lie extensions $F_\infty$ of $F$ which are obtained by adjoining to $F$ the $p$-division points of an abelian variety $A$ defined over $F$. The main focus of the paper is the calculation of the $\Gal(F_\infty/F)$-Euler characteristic of the $p^\infty$-Selmer group of $E$. The final section illustrates the main theory with the example of an elliptic curve of conductor 294.

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