Searcharxiv⌕ Search

arXiv subjects

Liang Xiao

Publications and source records attributed to Liang Xiao.

At least 109 records · Page 6Linked to original sources
arXiv2017

Unramifiedness of Galois representations arising from Hilbert modular surfaces

Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_\mathfrak{p}$ acting on $(\mathrm{mod}\, p^m)$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of F. Calegari and D. Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight ${\bf 1}$ are unramified at $p$ when $[F:\mathbb Q]=2$. Some partial and some conjectural results are obtained when $[F:\mathbb Q]>2$.

math.NT↗
arXiv2017

Cycles on Shimura varieties via geometric Satake

We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in the case of correspondences from a Shimura set to a Shimura variety, we obtain a description of the basic Newton stratum of the latter, and show that the irreducible components of the basic Newton stratum generate all the Tate classes in the middle cohomology of the Shimura variety, under a certain genericity condition. Along the way, we also determine the set of irreducible components of the affine Deligne-Lusztig variety associated to an unramified twisted conjugacy class.

math.AG↗
arXiv2017

Slopes for higher rank Artin-Schreier-Witt Towers

We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$ over a finite field of characteristic $p$, and consider the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower defined by $\bar f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified extension of $\mathbb{Z}_p$, which is abstractly isomorphic to $(\mathbb{Z}_p)^\ell$ as a topological group. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function asymptotically form a finite union of arithmetic progressions. As a corollary, we prove the spectral halo property of the spectral variety associated to the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower. This extends the main result in [DWX] from rank one case $\ell=1$ to the higher rank case $\ell\geq 1$.

math.NT↗
arXiv2016

The eigencurve over the boundary of weight space

We prove that the eigencurve associated to a definite quaternion algebra over $\QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and Buzzard--Kilford: (a) over the boundary annuli of weight space, the eigencurve is a disjoint union of (countably) infinitely many connected components each finite and flat over the weight annuli, (b) the $U_p$-slopes of points on each fixed connected component are proportional to the $p$-adic valuations of the parameter on weight space, and (c) the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves towards the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.

math.NT↗
arXiv2016

Slopes of eigencurves over boundary disks

Let $p$ be a prime number. We study the slopes of $U_p$-eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra. We give a sharp lower bound of the corresponding Newton polygon. The computation happens over a definite quaternion algebra by Jacquet-Langlands correspondence; it generalizes a prior work of Daniel Jacobs who treated the case of $p=3$ with a particular level. In case when the modular forms have a finite character of conductor highly divisible by $p$, we improve the lower bound to show that the slopes of $U_p$-eigenvalues grow roughly like arithmetic progressions as the weight $k$ increases. This is the first very positive evidence for Buzzard-Kilford's conjecture on the behavior of the eigencurve near the boundary of the weight space, that is proved for arbitrary $p$ and general level. We give the exact formula of a fraction of the slope sequence.

math.NT↗
arXiv2016

On Goren-Oort stratification for quaternionic Shimura varieties

Let $F$ be a totally real field in which $p$ is unramified. We study the Goren-Oort stratification of the special fibers of quaternionic Shimura varieties over a place above $p$. We show that each stratum is a $(\mathbb{P}^1)^N$-bundle over other quaternionic Shimura varieties (for some appropriate $N$).

math.AG↗
arXiv2016

p-adic Cohomology and classicality of overconvergent Hilbert modular forms

Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke modules.

math.NT↗
arXiv2016

Partial Hasse invariants on splitting models of Hilbert modular varieties

Let $F$ be a totally real field of degree $g$, and let $p$ be a prime number. We construct $g$ partial Hasse invariants on the characteristic $p$ fiber of the Pappas-Rapoport splitting model of the Hilbert modular variety for $F$ with level prime to $p$, extending the usual partial Hasse invariants defined over the Rapoport locus. In particular, when $p$ ramifies in $F$, we solve the problem of lack of partial Hasse invariants. Using the stratification induced by these generalized partial Hasse invariants on the splitting model, we prove in complete generality the existence of Galois pseudo-representations attached to Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties $\mathrm{mod}$ $p^m$, extending a previous result of M. Emerton and the authors which required $p$ to be unramified in $F$.

math.NT↗
arXiv2016

Newton slopes for Artin-Schreier-Witt towers

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field and consider the Artin-Schreier-Witt tower defined by $f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb A^1$, with total Galois group $\mathbb Z_p$. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function form arithmetic progressions which are independent of the conductor of the character. As a corollary, we obtain a result on the behavior of the slopes of the eigencurve associated to the Artin-Schreier-Witt tower, analogous to the result of Buzzard and Kilford.

math.NT↗
arXiv2015

On the parity conjecture in finite-slope families

We generalize to the finite-slope setting several techniques due to Nekovar concerning the parity conjecture for self-dual motives. In particular we show that, for a $p$-adic analytic family, with irreducible base, of symplectic self-dual global Galois representations whose $(φ,Γ)$-modules at places lying over $p$ satisfy a Panchishkin condition, the validity of the parity conjecture is constant among all specializations that are pure. As an application, we extend some other results of Nekovar for Hilbert modular forms from the ordinary case to the finite-slope case.

math.NT↗
arXiv2014

Cleanliness and log-characteristic cycles for vector bundles with flat connections

Let $X$ be a proper smooth algebraic variety over a field $k$ of characteristic zero and let $D$ be a divisor with simple normal crossings. Let $M$ be a vector bundle over $X-D$ equipped with a flat connection with possible irregular singularities along $D$. We define a cleanliness condition which roughly says that the singularities of the connection are controlled by the singularities at the generic points of $D$. When this condition is satisfied, we compute explicitly the associated log-characteristic cycle, and relate it to the so-called refined irregularities. As a corollary of a log-variant of Kashiwara-Dubson formula, we obtain the Euler characteristic of the de Rham cohomology of the vector bundle, under a mild technical hypothesis on $M$.

math.AG↗
arXiv2014

Gauss-Manin connections for p-adic families of nearly overconvergent modular forms

We interpolate the Gauss-Manin connection in p-adic families of nearly overconvergent modular forms. This gives a family of Maass-Shimura type differential operators from the space of nearly overconvergent modular forms of type r to the space of nearly overconvergent modular forms of type r + 1 with p-adic weight shifted by 2. Our construction is purely geometric, using Andreatta-Iovita-Stevens and Pilloni's geometric construction of eigencurves, and should thus generalize to higher rank groups.

math.NT↗
arXiv2013

Cohomology of arithmetic families of (phi,Gamma)-modules

We prove the finiteness and compatibility with base change of the (phi,Gamma)-cohomology and the Iwasawa cohomology of arithmetic families of (phi,Gamma)-modules. Using this finiteness theorem, we show that a family of Galois representations that is densely pointwise refined in the sense of Mazur is actually trianguline as a family over a large subspace. In the case of the Coleman-Mazur eigencurve, we determine the behavior at all points.

math.NT↗
arXiv2011

On refined ramification filtrations in the equal characteristic case

Let k be a complete discrete valuation field of equal characteristic p>0. Using the tools of p-adic differential modules, we define refined Artin and Swan conductors for a representation of the absolute Galois group $G_k$ with finite local monodromy; this leads to a description of the subquotients of the ramification filtration on $G_k$. We prove that our definition of the refined Swan conductors coincide with that is given by Saito, which uses etale cohomology. We also study its relation with the toroidal variation of the Swan conductors.

math.NT↗
arXiv2011

On Ramification Filtrations and p-adic Differential Equations, II: mixed characteristic case

Let K be a complete discretely valued field of mixed characteristic (0, p) with possibly imperfect residue field. We prove a Hasse-Arf theorem for the arithmetic ramification filtrations on G_K, except possibly in the absolutely unramified and non-logarithmic case, or p=2 and logarithmic case. As an application, we obtain a Hasse-Arf theorem for filtrations on finite flat group schemes over O_K.

math.NT↗
arXiv2010

On ramification filtrations and $p$-adic differential modules, I: equal characteristic case

Let $k$ be a complete discretely valued field of equal characteristic $p > 0$ with possibly imperfect residue field and let $G_k$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on $G_k$ coincide with the differential Artin conductors and Swan conductors of Galois representations of $G_k$. As a consequence, we give a Hasse-Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse-Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors and Borger's conductors.

math.NT↗
arXiv2009

Fingerprints in the Ether: Using the Physical Layer for Wireless Authentication

The wireless medium contains domain-specific information that can be used to complement and enhance traditional security mechanisms. In this paper we propose ways to exploit the fact that, in a typically rich scattering environment, the radio channel response decorrelates quite rapidly in space. Specifically, we describe a physical-layer algorithm that combines channel probing (M complex frequency response samples over a bandwidth W) with hypothesis testing to determine whether current and prior communication attempts are made by the same user (same channel response). In this way, legitimate users can be reliably authenticated and false users can be reliably detected. To evaluate the feasibility of our algorithm, we simulate spatially variable channel responses in real environments using the WiSE ray-tracing tool; and we analyze the ability of a receiver to discriminate between transmitters (users) based on their channel frequency responses in a given office environment. For several rooms in the extremities of the building we considered, we have confirmed the efficacy of our approach under static channel conditions. For example, measuring five frequency response samples over a bandwidth of 100 MHz and using a transmit power of 100 mW, valid users can be verified with 99% confidence while rejecting false users with greater than 95% confidence.

cs.CR↗