This primarily expository article collects together some facts from the literature about the monodromy of differential equations on a p-adic (rigid analytic) annulus, though often with simpler proofs. These include Matsuda's classification of quasi-unipotent nabla-modules, the Christol-Mebkhout construction of the ramification filtration, and the Christol-Dwork Frobenius antecedent theorem. We also briefly discuss the p-adic local monodromy theorem without proof.
We provide an effective version of Katz' criterion for finiteness of the monodromy group of a lisse, pure of weight zero, $\ell$-adic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf
We establish a generalization of the p-adic local monodromy theorem (of Andre, Mebkhout, and the author) in which differential equations on rigid analytic annuli are replaced by differential equations on so-called fake annuli. The latter correspond loosely to completions of a Laurent polynomial ring with respect to a monomial valuation. The result represents a step towards a higher-dimensional version of the p-adic local monodromy theorem (the "problem of semistable reduction"); it can also be viewed as a slightly novel presentation of the original p-adic local monodromy theorem.
Let $X$ be a smooth connected algebraic curve over an algebraically closed field $k$. We study the deformation of $\ell$-adic Galois representations of the function field of $X$ while keeping the local Galois representations at all places undeformed.
In this paper we prove a Gross-Zagier type formula for the anticyclotomic p-adic L-function of an elliptic modular form f of higher weight and of multiplicative type at p. For such f we also decribe explicitely the local Galois representation attached to it at p and compare the Fontaine-Mazur and Teitelbaum L-invariants.
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.
We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a p-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable etale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morph…
For any simple algebraic group $G$ of exceptional type, we construct geometric $\ell$-adic Galois representations with algebraic monodromy group equal to $G$, in particular producing the first such examples in types $\mathrm{F}_4$ and $\mathrm{E}_6$. To do this, we extend to general reductive groups Ravi Ramakrishna's techniques for lifting odd two-dimensional Galois representations to geometric $\ell$-adic representations.
The weight-monodromy conjecture claims the coincidence of the weight filtration and the monodromy filtration, up to shift, on the $l$-adic étale cohomology of a proper smooth variety over a complete discrete valuation field. Although it has been proved in some cases, the case of dimension $\geq 3$ in mixed characteristic is still open so far. The aim of this paper is to give a proof of the weight-monodromy conjecture for a threefold which has a projective strictly semistable model such that, for each irreducible component of the special fiber, the Picard number is equal to the second $l$-adic …
We prove an analogue of the Tate isogeny conjecture and the semi-simplicity conjecture for overconvergent crystalline Dieudonné modules of abelian varieties defined over global function fields of characteristic $p$. As a corollary we deduce that monodromy groups of such overconvergent crystalline Dieudonné modules are reductive, and after a finite base change of coefficients their connected components are the same as the connected components of monodromy groups of Galois representations on the corresponding $l$-adic Tate modules, for $l$ different from $p$. We also show such a result for gener…
Let K be a complete discrete valuation field of mixed characteristic (0,p) and G_K the absolute Galois group of K. In this paper, we will prove the p-adic monodromy theorem for p-adic representations of G_K without any assumption on the residue field of K, for example the finiteness of a p-basis of the residue field of K. The main point of the proof is a construction of (phi,G_K)-module Nrig^+(V) for a de Rham representation V, which is a generalization of Pierre Colmez' Nrig^+(V). In particular, our proof is essentially different from Kazuma Morita's proof in the case when the residue field a…
Let $G$ be a connected reductive group over a $p$-adic local field $F$. We propose and study the notions of $G$-$φ$-modules and $G$-$(φ,\nabla)$-modules over the Robba ring, which are exact faithful $F$-linear tensor functors from the category of $G$-representations on finite-dimensional $F$-vector spaces to the categories of $φ$-modules and $(φ,\nabla)$-modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlaya's slope filtration theorem in this context, and show that $G$-$(φ,\nabla)$-modules over the Robba ring are "$G$-quasi-unipotent", which i…
Slopes and weights of $\ell$-adic cohomology of rigid spaces
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）；University of the Chinese Academy of Sciences（中国科学院大学）
For a family of varieties, we prove that the alternating sum of the traces of "local" monodromy acting on the $\ell$-adic étale cohomology groups of the generic fiber is an integer which is independent of $\ell$.
Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. Following the one we initiated previously, we present an overview of a new monograph developing new features of the p-adic Simpson correspondence, inspired by our construction of the relative Hodge-Tate spectral sequence. First, we address the connection to Hodge-Tate local systems. Second, we establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, we expand the scope of our…
In this paper, we give explicit descriptions of Hyodo and Kato's Frobenius and Monodromy operators on the first $p$-adic de Rham cohomology groups of curves and Abelian varieties with semi-stable reduction over local fields of mixed characteristic. This paper was motivated by the first author's paper "A $p$-adic Shimura isomorphism and periods of modular forms," where conjectural definitions of these operators for curves with semi-stable reduction were given.
In this paper we prove that for any connected reductive algebraic group G and a large enough prime $l$, there are continuous homomorphisms $$\mathrm{Gal}(\bar\mathbb Q/\mathbb Q) \to G(\bar\mathbb Q_l)$$ with Zariski-dense image, in particular we produce the first such examples for $SL_n, Sp_{2n}, Spin_n, E_6^{sc}$ and $E_7^{sc}$. To do this, we start with a mod-$l$ representation of $\mathrm{Gal}(\bar\mathbb Q/\mathbb Q)$ related to the Weyl group of $G$ and use a variation of Stefan Patrikis' generalization of a method of Ravi Ramakrishna to deform it to characteristic zero.
In this paper, we prove a $p$-adic analogous of the Kulikov-Persson-Pinkham classification theorem [Persson:1981wp] for the central fiber of a degeneration of $K3$-surfaces in terms of the nilpotency degree of the monodromy of the family. Namely, let $X_K$ be a be a smooth, projective $K3$-surface which has a minimal semi-stable model $X$ over $\mathcal O_K$. If we let $N_{st}$ be the monodromy operator on $D_{st}(H^2_{et}}(X_{\overline K},\mathbb Q_p))$, then we prove that the degree of nilpotency of $N_{st}$ determines the type of the special fiber of $X$. As a consequence we give a criterio…
The purpose of this article is to prove a "Newton over Hodge" result for finite characters on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $ρ:π_1^{et}(V) \to \mathbb{C}^\times$. In this article, we prove a lower bound on the Newton polygon of the $L$-function $L(ρ,s)$. The estimate depends on monodromy invariants of $ρ$: the Swan conductor and the local exponents. Under certain nondegeneracy assumptions this lower bound agrees with the irregular Hodge filt…
Let $C$ be a smooth curve over a finite field in characteristic $p$ and let $M$ be an overconvergent $F$-isocrystal over $C$. After replacing $C$ with a dense open subset $M$ obtains a slope filtration, whose steps interpolate the Frobenius eigenvalues of $M$ with bounded slope. This is a purely $p$-adic phenomenon; there is no counterpart in the theory of lisse $\ell$-adic sheaves. The graded pieces of this slope filtration correspond to lisse $p$-adic sheaves, which we call geometric. Geometric lisse $p$-adic sheaves are mysterious. While they fit together to build an overconvergent $F$-isoc…
We the study of the monodromy of local systems with bounded ramification on a punctured disc defined over a non-archimedean valued field of characteristic zero. First, we construct the local Fourier transforms and we establish their main properties. Also, we show that the Fourier transform of a "perverse sheaf" with bounded ramification on the affine line, is again a "perverse sheaf" with bounded ramification (we put that in quotes, because actually the language of perversity is not used). These foundations are then used to exhibit a natural break decomposition for local systems with bounded r…
We study the relation of the middle convolution to the $\ell$-adic Fourier transformation in the étale context. Using Katz' work and Laumon's theory of local Fourier transformations we obtain a detailed description of the local monodromy and the determinant of Katz' middle convolution functor $\MC_χ$ in the tame case. The theory of local $ε$-constants then implies that the property of an étale sheaf of having an at most quadratic determinant is often preserved under $\MC_χ$ if $χ$ is quadratic.
We study the topology of the punctured disc defined over a non-archimedean field of characteristic zero. Chapter two includes a new proof of the so-called p-adic Riemann existence theorem. This release completes the study of breaks and break decompositions of the monodromy representation of a sheaf around the origin of the punctured disc.
We calculate the local Fourier transformations for a class of $\bar{\mathbb Q}_\ell$-sheaves. In particular, we verify a conjecture of Laumon and Malgrange. As an application, we calculate the local monodromy of $\ell$-adic hypergeometric sheaves introduced by Katz. We also discuss the characteristic $p$ analogue of the Turrittin-Levelt Theorem for $D$-modules. The method used in this paper can be used to show a conjecture of Ramero which states that the Fourier transformation of an analytic sheaf with meromorphic ramification still has meromorphic ramification.
We compute the density of the set of ordinary primes of an abelian surface over a number field in terms of the l-adic monodromy group. Using the classification of l-adic monodromy groups of abelian surfaces by Fite, Kedlaya, Rotger, and Sutherland, we show the density is 1, 1/2, or 1/4.
The aim of this paper is to study certain properties of the weight spectral sequences of Rapoport-Zink by a specialization argument. By reducing to the case over finite fields previously treated by Deligne, we prove that the weight filtration and the monodromy filtration defined on the $l$-adic étale cohomology coincide, up to shift, for proper smooth varieties over equal characteristic local fields. We also prove that the weight spectral sequences degenerate at $E_2$ in any characteristic without using log geometry. Moreover, as an application, we give a modulo $p>0$ reduction proof of a Hodg…
We define a numerical invariant, the differential Swan conductor, for certain differential modules on a rigid analytic annulus over a p-adic field. This gives a definition of a conductor for p-adic Galois representations with finite local monodromy over an equal characteristic discretely valued field, which agrees with the usual Swan conductor when the residue field is perfect. We also establish analogues of some key properties of the usual Swan conductor, such as integrality (the Hasse-Arf theorem), and the fact that the graded pieces of the associated ramification filtration on Galois groups…
We investigate the action of the Weil group on the compactly supported l-adic cohomology groups of rigid spaces over local fields. We prove that every eigenvalue of the action is a Weil number when either a rigid space is smooth or the characteristic of the base field is equal to 0. Since a smooth rigid space is locally isomorphic to the Raynaud generic fiber of an algebraizable formal scheme, we can use the techniques of alterations and monodromy spectral sequences in the smooth case. In the general case, we use the continuity theorem of Huber, which requires the restriction on the characteri…
We strengthen the compatibility between local and global Langlands correspondences for GL_{n} when n is even and l=p. Let L be a CM field and Π a cuspidal automorphic representation of GL_{n}(\mathbb{A}_{L}) which is conjugate self-dual and regular algebraic. In this case, there is an l-adic Galois representation associated to Π, which is known to be compatible with local Langlands in almost all cases when l=p by recent work of Barnet-Lamb, Gee, Geraghty and Taylor. The compatibility was proved only up to semisimplification unless Π has Shin-regular weight. We extend the compatibility to Frobe…
We present a detailed overview of the construction of the A_inf-cohomology theory from the preprint "Integral p-adic Hodge theory", joint with B. Bhatt and P. Scholze. We focus particularly on the p-adic analogue of the Cartier isomorphism via relative de Rham--Witt complexes.
The purpose of this article is to prove a ``Newton over Hodge'' result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. For a regular function $\overline{f}$ on $V$, we may form the $L$-function $L(\overline{f},V,s)$ associated to the exponential sums of $\overline{f}$. In this article, we prove a lower estimate on the Newton polygon of $L(\overline{f},V,s)$. The estimate depends on the local monodromy of $f$ around each point $x \in X-V$. This confirms a hope of Deligne tha…
We investigate local-global compatibility for cuspidal automorphic representations $π$ for GL(2) over CM fields that are regular algebraic of weight $0$. We prove that for a Dirichlet density one set of primes $l$ and any $ι: \overline{\mathbf{Q}}_l \cong \mathbf{C}$, the $l$-adic Galois representation attached to $π$ and $ι$ has nontrivial monodromy at any $v \nmid l$ in $F$ at which $π$ is special.
We resolve the local semistable reduction problem for overconvergent F-isocrystals at monomial valuations (Abhyankar valuations of height 1 and residue transcendence degree 0). We first introduce a higher-dimensional analogue of the generic radius of convergence for a p-adic differential module, which obeys a convexity property. We then combine this convexity property with a form of the p-adic local monodromy theorem for so-called fake annuli.
We give an alternative proof of Faltings's theorem (Mordell's conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings's original proof, but is in spirit closer to the methods of Chabauty and Kim: we replace the use of abelian varieties by a more detailed analysis of the variation of $p$-adic Galois representations in a family of algebraic varieties. The key inputs into this analysis are the comparison theorems of $p$-adic Hodge theory, and explicit topological computations of monodromy. By the same methods w…
We prove that the monodromy of an irreducible cohomologically complex rigid local system with finite determinant and quasi-unipotent local monodromies at infinity on a smooth quasiprojective complex variety $X$ is integral. This answers positively a special case of a conjecture by Carlos Simpson. On a smooth projective variety, the argument relies on Drinfeld's theorem on the existence of $\ell$-adic companions over a finite field. When the variety is quasiprojective, one has in addition to control the weights and the monodromy at infinity.
Let $X$ be a complete smooth variety defined over number field $K$ and $i$ an integer. The absolute Galois group of $K$ acts on the $i$th $l$-adic etale cohomology of $X$ for all $l$, producing a system of $l$-adic representations $\{Φ_l\}$. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of $Φ_\ell$ admits a common reductive $Q$-form for all $l$ if $X$ is projective. Denote by $Γ_l$ and $G_l$ respectively the monodromy group and the algebraic monodromy group of $Φ_l^{ss}$, the semisimplification of $Φ_\ell$. Assuming…
We strengthen the local-global compatibility of Langlands correspondences for $GL_{n}$ in the case when $n$ is even and $l\not=p$. Let $L$ be a CM field and $Π$ be a cuspidal automorphic representation of $GL_{n}(\mathbb{A}_{L})$ which is conjugate self-dual. Assume that $Π_{\infty}$ is cohomological and not "slightly regular", as defined by Shin. In this case, Chenevier and Harris constructed an $l$-adic Galois representation $R_{l}(Π)$ and proved the local-global compatibility up to semisimplification at primes $v$ not dividing $l$. We extend this compatibility by showing that the Frobenius …
Let K be a p-adic local field with residue field k such that [k:k^p]=p^e<\infty and V be a p-adic representation of Gal(\bar{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 Gal(\bar{K}/K) if and only if V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(\bar{K^{pf}}/K^{pf}) where K^{pf}/K is a certain p-adic local field whose residue field is the smallest perfect field k^{pf} containing k. As an application, we prove the p-adic monodromy theorem of Fo…
