27.
Slopes of modular forms and geometry of eigencurves, (with Ruochuan Liu, Nha Truong and Bin Zhao), arXiv:2302.07697.
|
| Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil-Buzzard-Emerton on the crystalline slopes of Kisin's crystabelian deformation spaces, (b) Gouvea's (kâ1)/(p+1)-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurve. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and (f) a refined version of Coleman's spectral halo conjecture. |
26.
A local analogue of the ghost conjecture of Bergdall-Pollack, (with Ruochuan Liu, Nha Truong and Bin Zhao), arXiv:2206.15372.
|
| We formulate a local analogue of the ghost conjecture of Bergdall and Pollack, which essentially relies purely on the representation theory of GL2(Qp). We further study the combinatorial properties of the ghost series as well as its Newton polygon, in particular, giving a characterization of the vertices of the Newton polygon and proving an integrality result of the slopes. In a forthcoming sequel, we will prove this local ghost conjecture under some mild hypothesis and give arithmetic applications. |
25.
Deformation of rigid conjugate self-dual Galois representations, (with Yifeng Liu, Yichao Tian, Wei Zhang, and Xinwen Zhu), arXiv:2108.06998.
|
| This is an appendix the following joint work. We study deformations of conjugate self-dual Galois representations. The study has two folds. First, we prove an R=T type theorem for a conjugate self-dual Galois representation with coefficients in a finite field, satisfying a certain property called rigid. Second, we study the rigidity property for the family of residue Galois representations attached to a symmetric power of an elliptic curve, as well as to a regular algebraic conjugate self-dual cuspidal representation. |
|
|
| In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg L-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin-Selberg L-function, then the Bloch-Kato Selmer group is of rank one. |
23*.
Cycles on Shimura varieties via geometric Satake, Example, (with Xinwen Zhu), pdf
|
| This is a part of the forthcoming update of the article below, which aims to provide some examples and explicit computations of cycles. It will be subsumed into the new version of the article once it was ready. |
23.
Cycles on Shimura varieties via geometric Satake, (with Xinwen Zhu), arXiv:1707.05700
|
| 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. |
22.
On the parity conjecture in finite-slope families, (with Jonathan Pottharst), arXiv:1410.5050.
|
| We generalize to the finite-slope setting several techniques due to Nekovář concerning the parity conjecture for self-dual motives. In particular we show that, for an irreducible p-adic analytic family 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 Nekovář for Hilbert modular forms from the ordinary case to the finite-slope case. |
21.
Tate cycles on quaternionic Shimura varieties over finite fields, (with Yichao Tian), Duke Mathematical Journal, 168 (2019), 1551--1639. arXiv:1410.2321.
|
| Let F be a totally real field in which a prime number p>2 is inert. We continue the study of the
(generalized) Goren-Oort strata on quaternionic Shimura varieties over finite
extensions of Fp. We prove that, when the dimension of the quaternionic Shimura
variety is even, the Tate conjecture for the special fiber of the quaternionic
Shimura variety holds for the cuspidal π-isotypical component, as long as
the two unramified Satake-parameters at p are not differed by a root of
unity. |
20.
On vector-valued twisted conjugate invariant functions on a group, (with Xinwen Zhu), in Representation of Reductive Groups, Proceedings of Symposia in Pure Mathematics 101, 361--425. arXiv:1802.05299.
|
| We study the space of vector-valued (twisted) conjugate invariant functions on a connected reductive group.
|
19.
Unramfiedness of Galois representations arising from Hilbert modular surfaces, (with Matthew Emerton and Davide Reduzzi), Forum of Mathematics Sigma, 5 (2017), E29. arXiv:1410.6203 Journal, erratum.
|
| Let p be a prime number and F a totally real number field. For each prime P of F above p we construct a Hecke operator TP acting on (mod pm) Katz Hilbert modular classes which agrees with the classical Hecke operator at 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 1 are unramified at p when [F:Q]=2. Some partial and some conjectural results are obtained when [F:Q]>2. |
18.
Tate cycles on some unitary Shimura varieties mod p, (with David Helm and Yichao Tian), Algebraic Number Theory 11 (2017), 2213--2288. arXiv:1410.2343 Journal.
|
| Let F be a totally real field in which a fixed prime p is inert, and let
E be a CM extension of F in which p splits. We fix two positive integers
r and s. We investigate the Tate conjecture on the special fiber of
G(U(r,s)×U(s,r))-Shimura variety. We construct cycles which we
conjecture to generate the Tate classes and verify our conjecture in the case
of G(U(1,s)×U(s,1)). We also discuss the general conjecture regarding
special cycles on the special fibers of unitary Shimura varieties. |
17.
Slopes for higher rank Artin-Schreier-Witt Towers, (with Rufei Ren, Daqing Wan, and Myungjun Yu), Trans. Amer. Math. Soc. 370 (2018), 6411-6432. Journal arXiv:1605.02254.
|
| We fix a monic polynomial f(x)∈Fq[x] over a finite field of characteristic p, and consider the
Zpl-Artin-Schreier-Witt tower defined by f(x); this
is a tower of curves … → Cm → Cm-1 → … → C0=A1, whose Galois group is canonically isomorphic to
Zpl, the degree l unramified extension of
Zp, which is abstractly isomorphic to (Zp)l 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 Zpl-Artin-Schreier-Witt tower. This
extends the main result in [DWX] from rank one case l=1 to the higher rank
case l≥1.
|
16.
The eigencurve over the boundary of weight space, (with Ruochuan Liu and Daqing Wan), Duke Mathematical Journal 166 (2017), 1739-1787. arXiv:1412.2584, Journal.
|
| We prove that the eigencurve associated to a definite quaternion algebra over Q satisfies the following properties, as conjectured by Coleman-Mazur and Buzzard-Kilford: (a) over the boundary annuli of the weight space, the eigencurve is a disjoint union of (countably) infinitely many connected components each finite and flat over the weight annuli, (b) the Up-slopes of points on each fixed connected component are proportional to the p-adic valuations of the parameter on the 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 moving on an irreducible connected component of the eigencurve towards the boundary, the slope converges to zero.
|
15.
Slopes of eigencurves over boundary disks, (with Daqing Wan and Jun Zhang), Mathematische Annalen 369 (2017), 487-537. arXiv:1407.0279, Journal.
|
Let p be a prime number. We study the slopes of Up-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 Up-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. |
14.
Partial Hasse invariants on splitting models of Hilbert modular varieties, (with Davide Reduzzi), Annales Scientifiques
de l'ENS 50 (2017), 579-607. arXiv:1405.6349 Journal, erratum.
|
| 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 mod pm, extending a previous result of M. Emerton and the authors which required p to be unramified in F. |
13.
Galois representations and torsion in the coherent cohomology of Hilbert modular varieties, (with Matthew Emerton and Davide Reduzzi) Journal für die reine und angewandte Mathematik 726 (2017), 93-127, arXiv:1307.8003, Journal.
|
| Let F be a totally real number field and let p be a prime unramified in F. We prove the existence of Galois pseudo-representations attached to mod pm Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties for F of level prime to p. |
12.
On Goren-Oort stratification for quaternionic Shimura varieties, (with Yichao Tian), Compositio Mathematica 152 (2016), 2134--2220. arXiv:1308.0790 Journal.
|
| 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 $(P1)N$-bundle over other quaternionic Shimura varieties (for some appropriate N). |
11.
p-adic cohomology and classicality of overconvergent Hilbert modular forms, (with Yichao Tian), in Astérisque 382 (2016), 73-162. arXiv:1308.0779.
|
| 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 Up-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. |
10.
Newton slopes for Artin-Schreier-Witt towers, (with Christopher Davis and Daqing Wan), Mathematische Annalen 364 (2016), Issue 3, 1451-1468. arXiv:1310.5311, Journal, erratum.
|
| We fix a monic polynomial over a finite field and consider the associated Artin-Schreier-Witt tower of curves. 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. |
9.
Tensor being crystalline implies each factor being crystalline up to twists, (Appendix to "On automorphy of certain Galois representations of GO4-type", by Tong Liu and Jiu-Kang Yu, Jounal of Number Theory (special issue for Prof. Winnie Li) 161 (2016) 49-74, pdf.
|
| We give a simple proof of the fact that if the tensor product of two p-adic representations of GQp is crystalline, then the two representations are crystalline themselves up to twisting by a character. |
8.
Cleanliness and log-characteristic cycles of vector bundles with flat connections, Mathematische Annalen 362 (2015), Issue 1-2, 579-627. arXiv:1104.1224, Journal
|
| 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 cleanness 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 logarithmic 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. |
7.
Gauss-Manin connections for p-adic families of nearly overconvergent modular forms, (with Robert Harron) Annales de l'institut Fourier 64 (2014), no.6, 2449-2464. arXiv:1308.1732, Journal.
|
| 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. |
6.
Ramification of higher local fields,
approaches and questions, (with Igor Zhukov), Algebra i Analiz, 26 (2014), issue 5, 1-40. reprinted in Proceedings of the 2nd International Conference on Valuation Theory. pdf
|
| This is a survey paper, in which we try to organize facts, ideas and problems concerning ramification in finite extensions of complete discrete valuation fields with arbitrary residue fields. |
5.
Cohomology of arithmetic families of (φ, Γ)-modules, (with Kiran S. Kedlaya and Jonathan Pottharst), Journal of American Mathematical Society 27 (2014), 1043-1115. arXiv:1203.5718, Journal, erratum.
|
| We prove the finiteness of the (φ, Γ)-cohomology and the Iwasawa cohomology of arithmetic families of (φ, Γ)-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 classical points. |
4.
On refined ramification filtrations in the equal characteristic case, Algebra and Number Theory 6 (2012), no. 8, 1579-1667.
arXiv:0911.1802, Journal
|
| Let k be a complete discretely valued field of equal characteristic p > 0. We introduce the notion of differential refined Artin and Swan conductors for a representation of the absolute Galois group Gk with finite local monodromy. We prove that the differential refined Swan conductors coincide with the arithmetic Swan conductors defined by Saito. Also, we study its relation with the toroidal variation of Swan conductors. |
3.
On ramification filtrations and p-adic differential equations, II: mixed characteristic case, Compositio Mathematica 148 (2012), no.2, 415-463.
arXiv:0811.3792, Journal
|
Let K be a complete discrete valuation field of mixed characteristic (0, p) with possibly imperfect residue field. We prove a Hasse-Arf theorem for the arithmetic ramification ramifications on GK, 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 at group schemes over OK.
I wrote a summary on this and the previous paper, with more intuitive explanation. |
2.
On ramification filtrations and p-adic differential equations, I: equal characterisitc case, Algebra and Number Theory 4 (2010), no.8, 969-1027.
arXiv:0801.4962,
Journal
|
| Let k be a complete discretely valued field of equal characteristic p > 0 with possibly imperfect residue field, and let Gk be its Galois group. We prove that the conductors computed by the arithmetic ramification ramifications on Gk coincide with the differential Artin and Swan conductors of Galois representations of Gk defined by Kedlaya. As a consequence, we obtain 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. |
1.
Differential modules on p-adic polyannuli, (with Kiran S. Kedlaya), Journal de l'Institut de Mathematiques de Jussieu 9 (2010), 155-201. arXiv:0804.1495
Journal, Erratum
|
| We consider variational properties of some numerical invariants, measuring convergence of local horizontal sections, associated to differential modules on polyannuli over a nonarchimedean field of characteristic zero. This extends prior work in the one-dimensional case of Christol, Dwork, Robba, Young, et al, as well as the result on variation of generic and subsidiary generic radii of convergence, from Kedlaya's course on p-adic differential equations. Our results do not require positive residue characteristic; thus besides their relevance to the study of Swan conductors for isocrystals, they are germane to the formal classification of flat meromorphic connections on complex manifolds. |