The Łojasiewicz exponent over a field of arbitrary characteristic

Let K be an algebraically closed field and let K((XQ)) denote the field of generalized series with coefficients in K. We propose definitions of the local Łojasiewicz exponent of F = ( f1, . . . , fm) ∈ K[[X, Y ]]m as well as of the Łojasiewicz exponent at infinity of F = ( f1, . . . , fm) ∈ K[X, Y ]m, which generalize the familiar case of K = C and F ∈ C{X, Y }m (resp. F ∈ C[X, Y ]m), see Cha˛dzy´nski and Krasi´nski (In: Singularities, 1988; In: Singularities, 1988; Ann Polon Math 67(3):297–301, 1997; Ann Polon Math 67(2):191–197, 1997), and prove some basic properties of such numbers. Namely, we show that in both cases the exponent is attained on a parametrization of a component of F (Theorems 6 and 7), thus being a rational number. To this end, we define the notion of the Łojasiewicz pseudoexponent of F ∈ (K((XQ))[Y ])m for which we give a description of all the generalized series that extract the pseudoexponent, in terms of their jets. In particular, we show that there exist only finitely many jets of generalized series giving the pseudoexponent of F (Theorem 5). The main tool in the proofs is the algebraic version of Newton’s Polygon Method. The results are illustrated with some explicit examples

Edge Partitions of Optimal 22-plane and 33-plane Graphs

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for $k=1$, we focus on optimal $2$-plane and $3$-plane graphs, which are $2$-plane and $3$-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple optimal $2$-plane graph into a $1$-plane graph and a forest, while (ii) an edge partition formed by a $1$-plane graph and two plane forests always exists and can be computed in linear time. (iii) We describe efficient algorithms to partition the edges of a simple optimal $2$-plane graph into a $1$-plane graph and a plane graph with maximum vertex degree $12$, or with maximum vertex degree $8$ if the optimal $2$-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) We exhibit an infinite family of simple optimal $2$-plane graphs such that in any edge partition composed of a $1$-plane graph and a plane graph, the plane graph has maximum vertex degree at least $6$ and the $1$-plane graph has maximum vertex degree at least $12$. (v) We show that every optimal $3$-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a $2$-plane graph and two plane forests

A quotient of the Lubin-Tate tower II

In this article we construct the quotient M_1/P(K) of the infinite-level Lubin-Tate space M_1 by the parabolic subgroup P(K) of GL(n,K) of block form (n-1,1) as a perfectoid space, generalizing results of one of the authors (JL) to arbitrary n and K/Q_p finite. For this we prove some perfectoidness results for certain Harris-Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze's candidate for the mod p Jacquet-Langlands and the mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of M_1/P(K) when n = 2, and shows that M_1/Q(K) is not perfectoid for maximal parabolics Q not conjugate to P.Comment: with an appendix by David Hanse

Local and global structure of connections on nonarchimedean curves

Consider a vector bundle with connection on a p-adic analytic curve in the sense of Berkovich. We collect some improvements and refinements of recent results on the structure of such connections, and on the convergence of local horizontal sections. This builds on work from the author's 2010 book and on subsequent improvements by Baldassarri and by Poineau and Pulita. One key result exclusive to this paper is that the convergence polygon of a connection is locally constant around every type 4 point

On the Witt vector frobenius

We study the kernel and cokernel of the Frobenius map on the p-typical Witt vectors of a commutative ring, not necessarily of characteristic p. We give many equivalent conditions to surjectivity of the Frobenus map on both finite and infinite length Witt vectors. In particular, surjectivity on finite Witt vectors turns out to be stable under certain integral extensions; this provides a clean formulation of a strong generalization of Faltings’s almost purity theorem from p-adic Hodge theory, incorporating recent improvements by Keddlaya and Liu, and by Scholze

Relative p-ADIC hodge theory: Foundations

We describe a new approach to relative p-adic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry in the style of Berkovich. In this paper, we give a thorough development of ?-modules over a relative Robba ring associated to a perfect Banach ring of characteristic p, including the relationship between these objects and étale Zp-local systems and Qp-local systems on the algebraic and analytic spaces associated to the base ring, and the relationship between étale cohomology and φ-cohomology. We also make a critical link to mixed characteristic by exhibiting an equivalence of tensor categories between the finite étale algebras over an arbitrary perfect Banach algebra over a nontrivially normed complete field of characteristic p and the finite étale algebras over a corresponding Banach Qp-algebra. This recovers the homeomorphism between the absolute Galois groups of Fp((π)) and Qp(μp∞)given by the field of norms construction of Fontaine and Wintenberger, as well as generalizations considered by Andreatta, Brinon, Faltings, Gabber, Ramero, Scholl, and most recently Scholze. Applications to the description of étale local systems on nonarchimedean analytic spaces over p-adic fields will be described in subsequent papers

An algebraic sato-tate group and sato-tate conjecture

We make explicit a construction of Serre giving a definition of an algebraic Sato-Tate group associated to an abelian variety over a number field, which is conjecturally linked to the distribution of normalized L-factors as in the usual Sato-Tate conjecture for elliptic curves. The connected part of the algebraic Sato-Tate group is closely related to theMumford-Tate group, but the group of components carries additional arithmetic information. We then check that, in many cases where the Mumford-Tate group is completely determined by the endomorphisms of the abelian variety, the algebraic Sato-Tate group can also be described explicitly in terms of endomorphisms. In particular, we cover all abelian varieties (not necessarily absolutely simple) of dimension at most 3; this result figures prominently in the analysis of Sato-Tate groups for abelian surfaces given recently by Fité, Kedlaya, Rotger, and Sutherland

New methods for (φ,Γ)(\varphi, \Gamma) ( φ , Γ ) -modules

We provide new proofs of two key results of p-adic Hodge theory: the Fontaine-Wintenberger isomorphism between Galois groups in characteristic 0 and characteristic p, and the Cherbonnier-Colmez theorem on decompletion of (phi, Gamma)-modules. These proofs are derived from joint work with Liu on relative p-adic Hodge theory, and are closely related to Scholze's study of perfectoid algebras and spaces