Adic Local Field Research Articles

The Semi-absolute Anabelian Geometry of Geometrically Pro-$p$ Arithmetic Fundamental Groups of Associated Low-Dimensional Configuration Spaces

Let $p$ be a prime number. In the present paper, we study geometrically pro-$p$ arithmetic fundamental groups of low-dimensional configuration spaces associated to a given hyperbolic curve over an arithmetic field such as a number field or a $p$-adic local field. Our main results concern the group-theoretic reconstruction of the function field of certain tripods (i.e., copies of the projective line minus three points) that lie inside such a configuration space from the associated geometrically pro-$p$ arithmetic fundamental group, equipped with the auxiliary data constituted by the collection of decomposition groups determined by the closed points of the associated compactified configuration space.

The test function conjecture for local models of Weil-restricted groups

We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached toallconnected reductive groups over$p$-adic local fields with$p\geqslant 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.

Faltings extension and Hodge-Tate filtration for abelian varieties over p-adic local fields with imperfect residue fields

AbstractLet K be a complete discrete valuation field of characteristic $0$ , with not necessarily perfect residue field of characteristic $p>0$ . We define a Faltings extension of $\mathcal {O}_K$ over $\mathbb {Z}_p$ , and we construct a Hodge-Tate filtration for abelian varieties over K by generalizing Fontaine’s construction [Fon82] where he treated the perfect residue field case.

WEIERSTRASS-TYPE FUNCTIONS IN p-ADIC LOCAL FIELDS

The Weierstrass nowhere differentiable function has been studied often as example of functions whose graphs are fractals in [Formula: see text]. This paper investigates the Weierstrass-type function in the [Formula: see text]-adic local field [Formula: see text] whose graph is a repelling set of a discrete dynamical system, and proves that there exists a linear connection between the orders of the [Formula: see text]-adic calculus and the dimensions of the corresponding graphs.

Breuil-Mézard conjectures for central division algebras

We formulate an analogue of the Breuil-Mezard conjecture for the group of units of a central division algebra over a $p$-adic local field, and we prove that it follows from the conjecture for $\mathrm{GL}_n$. To do so we construct a transfer of inertial types and Serre weights between the maximal compact subgroups of these two groups, in terms of Deligne-Lusztig theory, and we prove its compatibility with mod $p$ reduction, via the inertial Jacquet-Langlands correspondence and certain explicit character formulas. We also prove analogous statements for $\ell$-adic coefficients.

On the analytic properties of intertwining operators II: Local degree bounds and limit multiplicities

In this paper we continue to study the degrees of matrix coefficients of intertwining operators associated to reductive groups over $p$-adic local fields. Together with previous analysis of global normalizing factors we can control the analytic properties of global intertwining operators for a large class of reductive groups over number fields, in particular for inner forms of $GL(n)$ and $SL(n)$ and quasi-split classical groups. This has a direct application to the limit multiplicity problem for these groups.

Higher Chow cycles on Abelian surfaces and a non-Archimedean analogue of the Hodge--conjecture

AbstractWe construct new indecomposable elements in the higher Chow group $CH^2(A,1)$ of a principally polarized Abelian surface over a $p$-adic local field, which generalize an element constructed by Collino [Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393–415]. These elements are constructed using a generalization, due to Birkenhake and Wilhelm [Humbert surfaces and the Kummer plane, Trans. Amer. Math. Soc. 355 (2003), 1819–1841 (electronic)], of a classical construction of Humbert. They can be used to prove a non-Archimedean analogue of the Hodge-${\mathcal{D}}$-conjecture – namely, the surjectivity of the boundary map in the localization sequence – in the case where the Abelian surface has good and ordinary reduction.

Crystalline and semi-stable representations in the imperfect residue field case

Let $K$ be a $p$-adic local field with residue field $k$ such that $[k : k^p] = p^e \lt \infty$ and $V$ be a $p$-adic representation of $\mathrm{Gal}(\overline{K} / K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a potentially crystalline (resp. potentially semi-stable) representation of $\mathrm{Gal}(\overline{K} / K)$ if and only if $V$ is a potentially crystalline (resp. potentially semi-stable) representation of $\mathrm{Gal}(\overline{K^\mathrm{pf}} / K^\mathrm{pf})$ where $K^\mathrm{pf} / K$ is a certain $p$-adic local field whose residue field is the smallest perfect field $k^\mathrm{pf}$ containing $k$. As an application, we prove the $p$-adic monodromy theorem of Fontaine in the imperfect residue field case.

On the existence of non-geometric sections of arithmetic fundamental groups

We show the existence of group-theoretic sections of the “étale-by-geometrically abelian” quotient of the arithmetic fundamental group of hyperbolic curves over \(p\)-adic local fields relative to a proper and flat model which are non-geometric, i.e., which do not arise from rational points.

The logarithmic growth of element of Robba ring which satisfies Frobenius equation over bounded Robba ring

We study the logarithmic growth of an element of the Robba ring which satisfies a Frobenius equation over the bounded Robba ring. Chiarellotto and Tsuzuki computed the logarithmic growth of analytic functions on the open unit disc with coefficients in a $p$-adic local field which satisfy Frobenius equations over bounded functions of rank 2. We extend their result by replacing those functions by elements of the Robba ring which satisfy Frobenius equations over the bounded Robba ring. Moreover, we will see, in special cases, the zeros of these functions have some cyclicity and the logarithmic growth can be computed by the zeros of these function.

A finiteness theorem for zero-cycles overp-adic fields

In this paper we prove a finiteness result concerning the Chow group of zero-cycles for varieties over $p$-adic local fields. In this final version, there are several corrections concerning mathematical symbols and reference to related known results.

On the period-index problem in light of the section conjecture

Period and index of a curve $X/K$ over a $p$-adic local field $K$ such that the fundamental group $\pi_1(X/K)$ admits a splitting are shown to be powers of $p$. As a consequence, examples of curves over number fields are constructed where having sections is obstructed locally at a $p$-adic place. Hence the section conjecture holds for these curves as there are neither sections nor rational points.

An explicit matching theorem for level zero discrete series of unit groups ofp-adic simple algebras

For A | F a central simple algebra over a p -adic local field the group of units A × ≅ GL m ( D d ) is a general linear group over a central division algebra D d | F of index d . The product n = dm being fixed, the Abstract Matching Theorem (AMT) implies the existence of bijective maps ℐℒ between the sets of discrete series representations of the groups A × such that a character relation is preserved. In this paper we construct maximal level zero extended type components for every level zero discrete series representation of A × . Its maximal level zero extended type determines the discrete series representation uniquely (without any twist ambiguities as for the usual types) and, conversely, every level zero discrete series representation ∏ contains a maximal level zero extended type component ## add figure 'crll.2005.585.173_01.gif' ## which is unique up to conjugacy. In order to determine how ℐℒ matches the extended types we find certain regular elliptic elements where the characters of ## add figure 'crll.2005.585.173_01.gif' ## and ∏ are the same and we compute the character values at these elements by using a version of Shintani descent which we develop in Appendix B. Surprisingly, we find that AMT also implies explicit Shintani descent for irreducible characters of finite general linear groups which have cuspidal descent.