Let $Ω$ be an algebraic closure of ${\mathbb Q}_p$ and let $F$ be a finite extension of ${\mathbb Q}_p$ contained in $Ω$. Given positive integers $f$ and $e$, the number of extensions $K/F$ contained in $Ω$ with residue degree $f$ and ramification index $e$ was computed by Krasner. This paper is concerned with the number ${\mathfrak I}(F,f,e)$ of $F$-isomorphism classes of such extensions. We determine ${\mathfrak I}(F,f,e)$ completely when $p^2\nmid e$ and get partial results when $p^2\parallel e$. When $s$ is large, ${\mathfrak I}({\mathbb Q}_p,f,e)$ is equal to the number of isomorphism classes of finite commutative chain rings with residue field ${\mathbb F}_{p^f}$, ramification index $e$, and length $s$.
We give a new proof of Ohta's Lambda-adic Eichler-Shimura isomorphism using p-adic Hodge theory and the results of Bloch-Kato and Hyodo on p-adic etale cohomology.
We deduce a formula enumerating the isomorphism classes of extensions of a $\kp$-adic field $K$ with given ramification $e$ and inertia $f$. The formula follows from a simple group-theoretic lemma, plus the Krasner formula and an elementary class field theory computation. It shows that the number of classes only depends on the ramification and inertia of the extensions $K/\Q_p$, and $K(ζ_{p^m})/K$ obtained adding the $p^m$-th roots of 1, for all $p^m$ dividing $e$.
We discuss the formalism of Iwasawa theory descent in the setting of the localized K_1-groups of Fukaya and Kato. We then prove interpolation formulas for the `leading terms' of the global Zeta isomorphisms that are associated to certain Tate motives and of the p-adic L-functions that are associated to certain critical motives.
In this article, we prove (many parts of) the rank two case of the Kato's local epsilon-conjecture using the Colmez's p-adic local Langlands correspondence for GL_2(Q_p). We show that a Colmez's pairing defined in his study of locally algebraic vectors gives us the conjectural epsilon-isomorphisms for (almost) all the families of p-adic representations of Gal(overline{Q}_p/Q_p) of rank two, which satisfy the desired interpolation property for the de Rham and trianguline case. For the de Rham and non trianguline case, we also show this interpolation property for the "critical" range of Hodge-Ta…
In this paper we consider the problem of classifying the isomorphism classes of extensions of degree pk of a p-adic field, restricting to the case of extensions without intermediate fields. We establish a correspondence between the isomorphism classes of these extensions and some Kummer extensions of a suitable field F containing K. We then describe such classes in terms of the representations of Gal(F/K). Finally, for k = 2 and for each possible Galois group G, we count the number of isomorphism classes of the extensions whose normal closure has a Galois group isomorphic to G. As a byproduct,…
In this paper we define a $p$-adic analogue of the Borel regulator for the $K$-theory of $p$-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this $p$-adic regulator to the Bloch-Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups.
We show that the Deligne formal model of the Drinfeld p-adic halfplane relative to a non-archimedean local field F represents a moduli problem of polarized O_F-modules with an action of the ring of integers O_E in a quadratic extension E of F. The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of SL_2(F) and SU(C)(F) for a two-dimensional split hermitian space C for E/F.
This article is the first one of a series aiming to construct an isomorphism between the p-adic Lubin-Tate and Drinfeld towers, describe this isomorphism and give applications. We construct a p-adic equivariant integral model of the Lubin-Tate tower with infinite level. This formal scheme will be later compared to another one associated to the Drinfeld tower.
Laurent Berger attached a p-adic differential equation N_rig(M) with a Frobenius structure to an arbitrary de Rham (phi, Gamma)-module over a Robba ring. In this article, we compare the local epsilon conjecture for the cyclotomic deformation of M with that of N_rig(M). We first define an isomorphism between the fundamental lines of their cyclotomic deformations using the second author's results on the big exponential map. As a main result of the article, we show that this isomorphism enables us to reduce the local epsilon conjecture for the cyclotomic deformation of M to that of N_rig(M).
Given a $p$-adic field $K$ and a prime number $\ell$, we count the total number of the isomorphism classes of $p^\ell$-extensions of $K$ having no intermediate fields. Moreover for each group that can appear as Galois group of the normal closure of such an extension, we count the number of isomorphism classes that contain extensions whose normal closure has Galois group isomorphic to the given group. Finally we determine the ramification groups and the discriminant of the composite of all $p^\ell$-extensions of K with no intermediate fields. The principal tool is a result, proved at the beginn…
Suppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported, K-biequivariant functions f: G(F) \to End V. These Hecke algebras were first considered by Barthel-Livne for GL_2. They play a role in the recent mod p and p-adic Langlands correspondences for GL_2(Q_p), in generalisations of Serre's conjecture on the modularity of mod p …
In this paper, we prove the "local epsilon-isomorphism conjecture" of Fukaya and Kato for a particular class of Galois modules obtained by tensoring a Zp-lattice in a crystalline representation of the Galois group of Qp with a representation of an abelian quotient of the Galois group with values in a suitable p-adic local ring. This can be regarded as a local analogue of the Iwasawa main conjecture for abelian p-adic Lie extensions of Qp, extending earlier work of Benois and Berger for the cyclotomic extension. We show that such an epsilon-isomorphism can be constructed using the Perrin-Riou r…
We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of two-sided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2.
This paper is a continuation of a previous paper in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dime…
Let $K$ be a finite extension of the $p$-adic numbers $\mathbb Q_p$ with ring of integers $\mathcal O_K$, $\mathcal X$ a regular scheme, proper, flat, and geometrically irreducible over $\mathcal O_K$ of dimension $d$, and $\mathcal X_K$ its generic fiber. We show, under some assumptions on $\mathcal X_K$, that there is a reciprocity isomorphism of locally compact groups $H_{ar}^{2d-1}(\mathcal X_K, \mathbb Z(d)) \simeq π_1^{ab}(\mathcal X_K)_{W}$ from a new cohomology theory to an integral model $π_1^{ab}(\mathcal X_K)_{W}$ of the abelianized geometric fundamental groups $π_1^{ab}(\mathcal X_…
We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-etale site, which makes all constructions completely functorial.
In this paper the new techniques and results concerning the structure theory of modules over non-commutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions K of number fields k "up to pseudo-isomorphism". In particular, a close relationship is revealed between the Selmer group of abelian varieties, the Galois group of the maximal abelian unramified p-extension of K as well as the Galois group of the maximal abelian outside S unramified p-extension where S is a finite set of certain places of k. Moreover, we determine the Galois module structure …
We compute the universal deformations of cuspidal representations $π$ of $\GL_2(F)$ over an algebraically closed field of characteristic $l$, where $F$ is a local field of residue characteristic $p$ not equal to $l$. When $π$ is supercuspidal there is an irreducible, two-dimensional representation $ρ$ of $G_F$ that corresponds to $π$ by the mod $l$ local Langlands correspondence of Vign{é}ras; we show there is a natural isomorphism between the universal deformation rings of $π$ and $ρ$ that induces the usual local Langlands correspondence on characteristic zero points. Our work establishes cer…
Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\geq 3$ and $a_p=0$. We generalise the definition of Kobayashi's plus/minus Selmer groups over $\mathbb{Q}(μ_{p^\infty})$ to $p$-adic Lie extensions $K_\infty$ of $\mathbb{Q}$ containing $\mathbb{Q}(μ_{p^\infty})$, using the theory of $(ϕ,Γ)$-modules and Berger's comparison isomorphisms. We show that these Selmer groups can be equally described using the "jumping conditions" of Kobayashi via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer group…
We define functorial isomorphisms of parallel transport along etale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.
In this paper we consider the (affine) Schur algebra introduced by Vignéras as the endomorphism algebra of certain permutation modules for the Iwahori-Matsumoto Hecke algebra. This algebra describes, for a general linear group over a $p$-adic field, a large part of the unipotent block over fields of characteristic different from $p$. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a bypr…
We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$ and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$ and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. We also classify all morphisms in the category character sheaves on $G$. As an application, we study character sheaves on Greenberg transforms of loc…
A superspecial curve is a (non-singular) curve over a field of positive characteristic whose Jacobian variety is isomorphic to a product of supersingular elliptic curves over the algebraic closure. It is known that for given genus and characteristic, there exist only finitely many superspecial curves, up to isomorphism over an algebraically closed field. In this article, we give a brief survey on results of counting isomorphism classes of superspecial curves. In particular, this article summarizes some recent results in the case of genera four and five, obtained by the author and S.\ Harashita…
Let K be a non-archimedean local field. This paper gives an explicit isomorphism between the dual of the special representation of GL_{n+1}(K)$and the space of harmonic cochains defined on the Bruhat-Tits building of GL_{n+1}(K), the definition of harmonic cochains is in the sense of E. de Shalit [Residues on buildings and de Rham cohomology of p-adic symmetric domains, Duke Math. J., 106, 123-191, (2000)]. We deduce, applying a work of P. Schneider and U. Stuhler, [The cohomology of p-adic symmetric spaces, Inv. Math. 105, 47-122, (1991)], that for K of any characteristic, there exists a GL_{…
