In this paper we study division algebras over the function fields of curves over $\Q_p$. The first and main tool is to view these fields as function fields over nonsingular $S$ which are projective of relative dimension 1 over the $p$ adic ring $\Z_p$. A previous paper showed such division algebras had index bounded by $n^2$ assuming the exponent was $n$ and $n$ was prime to $p$. In this paper we consider algebras of degree (and hence exponent) $q \not= p$ and show these algebras are cyclic. We also find a geometric criterion for a Brauer class to have index $q$.
Let $F$ be the function field of a smooth curve over the $p$-adic number field $\Q_p$. We show that for each prime-to-$p$ number $n$ the $n$-torsion subgroup $\H^2(F,μ_n)={}_n\Br(F)$ is generated by $\Z/n$-cyclic classes; in fact the $\Z/n$-length is equal to two. It follows that the Brauer dimension of $F$ is two (first proved in \cite{Sa97}), and any $F$-division algebra of period $n$ and index $n^2$ is decomposable.
Exact bounds for the positions of the branch points for cyclic coverings of the $p$-adic projective line by Mumford curves are calculated in two ways. Firstly, by using Fumiharu Kato's *-trees, and secondly by giving explicit matrix representations of the Schottky groups corresponding to the Mumford curves above the projective line through combinatorial group theory.
We study the ramification of fierce cyclic Galois extensions of a local field $K$ of characteristic zero with a one-dimensional residue field of characteristic $p>0$. Using Kato's theory of the refined Swan conductor, we associate to such an extension a ramification datum, consisting of a sequence of pairs $(δ_i,ω_i)$, where $δ_i$ is a positive rational number and $ω_i$ a differential form on the residue field of $K$. Our main result gives necessary and sufficient conditions on such sequences to occur as a ramification datum of a fierce cyclic extension of $K$.
Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the "algebraic" local moduli in characteristic p of $G_0$, by $G$ the universal deformation of $G_0$ over $S$, and by $U\subset S$ the ordinary locus of $G$. The etale part of $G$ over $U$ gives rise to a monodromy representation $ρ$ of the fundamental group of $U$ on the Tate module of $G$. Motivated by a famous theorem of Igusa, we prove in this article that $ρ$ is surjective if $G_0$ is connected and HW-cyclic. This latter condition is equivale…
We construct indecomposable and noncrossed product division algebras over function fields of smooth curves X over Z_p. This is done by defining an index preserving morphism s:Br(\hat K(X))' -> Br(K(X))' which splits res:Br(K(X)) -> Br(\hat K(X)), where \hat K(X) is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over \hat K(X).
We show that for an associative algebra A and its ideal I such that the I-adic topology on A coincides with the p-adic topology, the relative continuous K-theory pro-spectrum "lim"K(A_i, IA_i), where A_i :=A/p^i A, is naturally isogenous to the cyclic chain pro-complex "lim"CC(A_i) (subject to minor conditions on A). This identification is a continuous version of the classical Goodwillie isomorphism. The work comes from an attempt to understand the article of Bloch, Esnault, and Kerz "p-adic deformations of algebraic cycle classes".
Let F be the function field of a curve over a p-adic field. Let D/F be a central division algebra of prime exponent $\ell$ which is different from p. Assume that F contains a primitive ${\ell}^2$-th root of unity. Then the group $SK_1(D):=SL_1(D)/[D^*, D^*]$ is trivial.
We establish a valuative version of Grothendieck's section conjecture for curves over p-adic local fields. The image of every section is contained in the decomposition subgroup of a valuation which prolongs the p-adic valuation to the function field of the curve.
Deligne and Kato proved a formula computing the dimension of the nearby cycles complex of an l-adic sheaf on a relative curve over an excellent strictly henselian trait. In this article, we reprove this formula using Abbes-Saito's ramification theory.
These are the notes accompanying 13 lectures given by the authors at the Clay Mathematics Institute Summer School 2014 in Madrid. The notes give an introduction into the theory of $\ell$-adic sheaves with emphasis on their ramification theory on curves.
We prove that a category of degree zero vector bundles with "potentially strongly semistable reduction" on a p-adic curve is a neutral Tannakian category. We also make a first study of the corresponding affine group scheme. In particular, we determine its group of connected components using a theorem of Weil.
In this paper, we prove a `cut-by-curves criterion' for the overconvergence of integrable connections on certain rigid analytic spaces and certain varieties over $p$-adic fields.
I prove the surjectivity of the $p$-adic regulator from Quillen's $K_2$ of Tate curve to the $p$-adic etale cohomology group when the base field is contained in a cyclotomic extension of $Q_p$. This implies the finiteness of torsion part of $K_1$ of Tate curves thanks to Suslin's exact sequence.
Let E be an elliptic curve with additive reduction over the p-adic numbers, and let G be the group of p-adic points on E that have good reduction. This paper gives necessary and sufficient conditions for G to contain non-trivial p-torsion.
Let K be a p-adic field and F the function field of a curve over K. Let G be a connected linear algebraic group over F of classical type. Suppose the prime p is a good prime for G. Then we prove that projective homogeneous spaces under G over F satisfy a local global principle for rational points with respect to discrete valuations of F . If G is a semisimple simply connected group over F , then we also prove that principal homogeneous spaces under G over F satisfy a local global principle for rational points with respect to discrete valuations of F.
We define functorial isomorphisms of parallel transport along étale paths for a class of principal $G$-bundles on a $p$-adic curve. Here $G$ is a connected reductive algebraic group of finite presentation and the considered principal bundles are just those with potentially strongly semistable reduction of degree zero. The constructed isomorphisms yield continous functors from the étale fundamental groupoid of the given curve to the category of topological spaces with a simply transitive continous right $G(\mathbb{C}_{p})$-action. This generalizes a construction in the case of vector bundles on…
We extend our previous theory of etale parallel transport to a larger class of slope zero vector bundles on p-adic curves. The new class is stable under pullback by ramified coverings. We also construct p-adic representations of a central extension of the fundamental group for certain bundles of non-zero slope.
In [DW05] and [DW07], C. Deninger and A. Werner developed a partial p-adic analogue of the classical Narasimhan-Seshadri correspondence between vector bundles and representations of the fundamental group. We will investigate the various monodromy groups that occur in this theory, that is the image of these representations and their Zariski closure as well as the Tannaka dual group of these vector bundles.
We show how non-vanishing of p-adic L functions controls the dimensions of Selmer varieties associated to the complement of the origin in an elliptic curve with CM. As a corollary, one obtains a π_1-proof of the theorem of Siegel for such curves.
Let D be a quaternion division algebra over a totally real number field F which splits exactly at one infinite place. We assume that there is a p-adic place where D doesn't split. Then the associated Shimura curve has a Cherednik unifomization by the p-adic upper half plane. We define an integral model of the Shimura curve an over the integers O of this p-adic place. We prove an uniformization theorem for the completion of the model over the formal spectrum Spf O.
Given a finite morphism $φ:Y\to X$ of quasi-smooth Berkovich curves over a complete, algebraically closed field $k$ of characteristic $0$, we prove a Riemann-Hurwitz formula relating their Euler-Poincaré characteristics (calculated using De Rham cohomology of their overconvergent structure sheaf). The main tools are $p$-adic Runge's theorem together with valuation polygons of analytic functions. Using the results obtained, we provide another point of view on Riemann-Hurwitz formula for finite morphisms of curves over algebraically closed fields of positive characteristic.
We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) étale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric étale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.
In this paper, we seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics, i.e. a characterization, in terms of geometry mod p, of curves in positive characteristics which are reduction of Shimura curves over the complex field. Specifically, we study the liftablity of a curve in moduli space of principally polarized abelian varieties over k, char k=p. We show that some conditions on the l-adic monodromy over such a curve imply that this curve can be lifted to a Shimura curve.
For a vector bundle E on a model of a smooth projective curve over a p-adic number field a p-adic representation of the geometric fundamental group of X has been defined in work with Annette Werner if the reduction of E is strongly semistable of degree zero. In the present note we calculate the reduction of this representation using the theory of Nori's fundamental group scheme.
We describe an algorithm to compute the local component at p of the Coleman-Gross p-adic height pairing on divisors on hyperelliptic curves. As the height pairing is given in terms of a Coleman integral, we also provide new techniques to evaluate Coleman integrals of meromorphic differentials and present our algorithms as implemented in Sage.
In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic analogue of the result of Beilinson and Levin expressing the complex elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. Our result is valid even if the elliptic curve has supersingular…
We establish a relation between intersection numbers of special cycles on a Shimura curve and special values of derivatives of metaplectic Eisenstein series at a place of bad reduction where p-adic uniformization in the sense of Cherednik and Drinfeld holds. The result extends the one established by one of us (S. Kudla: Ann. of Math. 146 (1997)) for the archimedean place and for the non-archimedean places of good reduction. The bulk of the paper is concerned with the corresponding problem on the Drinfeld upper half plane (the formal scheme version).
We introduce stable tropical curves and use these to count covers of the $p$-adic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the covers.
Moduli spaces for Galois covers of p-adic Mumford curves by Mumford curves are constructed using Herrlich's Teichmuller spaces, Andre's orbifold fundamental groups, and Kato's graphs of groups encoding ramification data of charts for Mumford orbifolds.
In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of k-th invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid-analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, th…
We prove that the geometric etale fundamental group of a (geometrically connected) rigid smooth $p$-adic affinoid curve is a semi-direct factor of a certain profinite free group. We also prove that the maximal pro-$p$ (resp. maximal prime-to-$p$) quotient of this geometric étale fundamental group is pro-$p$ free of infinite rank (resp. (pro-)prime-to-$p$ free of finite computable rank).
Let $A$ be an abelian variety over ${\bf C}_p$ ($p$ a prime number) and $V\hookrightarrow A$ a closed subvariety. The conjecture of Tate-Voloch predicts that the $p$-adic distance from a torsion point $T\not\in V({\bf C}_p)$ to the variety $V$ is bounded below by a strictly positive constant. This conjecture is proven by Hrushovski and Scanlon, when $A$ has a model over $\bar{\bf C}_p$. We give an explicit formula for this constant, in the case where $V$ is a curve embedded into its Jacobian and $V$ has a model over a number field. This explicit formula involves analytic and arakelovian invari…
Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer type is computed.
This paper deals with connections on $p$-adic analytic curves, in the sense of Berkovich. The curves must be compact but the connections are allowed to have a finite number of meromorphic singularities on them. For any choice of a semistable formal model of the curve, we define an intrinsic notion of normalized radius of convergence as a function on the curve, with values in $(0,1]$. For a sufficiently refined choice of the semistable model, we prove continuity and logarithmic concavity of that function. We characterize \emph{Robba connections}, that is connections whose sheaf of solutions is …
We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell-Weil group tensored with the rationals.
The tempered fundamental group of a p-adic analytic space classifies coverings that are dominated by a topological covering (for the Berkovich topology) of a finite etale covering of the space. Here we construct cospecialization homomorphisms between (p') versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log reduction. In particular, we will have to study invariance of the geometric log fundamental group of…
Let $k$ be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve $X_1$ defined over $k$, there exists a lifting $X$ of the curve to the ring $W(k)$ of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over $X/W$. As a consequence, it gives rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group $π_1(X_K)$ of the generic fiber of $X$. This curve $X$ and its associated representation is in close relation wit…
We consider a finite étale morphism $f:Y \to X$ of quasi-smooth Berkovich curves over a complete nonarchimedean non-trivially valued field $k$, assumed algebraically closed and of characteristic 0, and a skeleton $Γ_f=(Γ_Y,Γ_X)$ of the morphism $f$. We prove that $Γ_f$ radializes $f$ if and only if $Γ_X$ controls the pushforward of the constant $p$-adic differential equation $f_*(\mathcal{O}_Y,d_Y)$. Furthermore, when $f$ is a finite étale morphism of open unit discs, we prove that $f$ is radial if and only if the number of preimages of a point $x\in X$, counted without multiplicity, only depe…
In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser fo…
In this note, we introduce the notion of an unramified strongly cyclic covering for a cyclic curve, a class that has similar properties to, and contains, unramified double covers of hyperelliptic curves. We determine several of their basic properties, extending the theorems in \cite{MR2217998} to this larger class. In particular, we will write down equations for smooth affine models, determine when they are isomorphic, and discuss the curves that they are ramified cyclic covers of.
An origami (also known as square-tiled surface) is a Riemann surface covering a torus with at most one branch point. Lifting two generators of the fundamental group of the punctured torus decomposes the surface into finitely many unit squares. By varying the complex structure of the torus one obtains easily accessible examples of Teichmüller curves in the moduli space of Riemann surfaces. The p-adic analogues of Riemann surfaces are Mumford curves. A p-adic origami is defined as a covering of Mumford curves with at most one branch point, where the bottom curve has genus one. A classification o…
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.
We construct a semi-stable formal model of a wide open rigid curve with a semi-stable covering, and study the l-adic cohomology of the rigid curve. We describe the l-adic cohomology of the rigid curve using the l-adic cohomology of the irreducible components of a semi-stable reduction, and homology and cohomology of some graphs. We also prove the functoriality of the description for a finite flat morphism that is compatible with semi-stable coverings of wide open rigid curves.
We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorize by the retraction through a locally finite subgraph of the curve.
Using topological cyclic homology, we give a refinement of Beilinson's $p$-adic Goodwillie isomorphism between relative continuous $K$-theory and cyclic homology. As a result, we generalize results of Bloch-Esnault-Kerz and Beilinson on the $p$-adic deformations of $K$-theory classes. Furthermore, we prove structural results for the Bhatt-Morrow-Scholze filtration on $TC$ and identify the graded pieces with the syntomic cohomology of Fontaine-Messing.
Deninger and Werner developed an analogue for p-adic curves of the classical correspondence of Narasimhan and Seshadri between stable bundles of degree zero and unitary representations of the topological fundamental group for a complex smooth proper curve. Using parallel transport, they associated functorially to every vector bundle on a p-adic curve whose reduction is strongly semi-stable of degree 0 a p-adic representation of the fundamental group of the curve. They asked several questions: whether their functor is fully faithful; whether the cohomology of the local systems produced by this …
