

I am working on arithmetic algebraic geometry, and especially interested in the padic or characteristic p aspects of Shimura varieties and their applications to arithmetic problems. In Langlands program, Shimura varieties are usually used as a bridge between automorphic forms and Galois representations, and their padic geometry can provide interesting information on the congruence of modular forms.
In an earlier joint work with Liang Xiao, we obtained an explicit description of the global geometry of GorenOort stratification of some quaternionic Shimura varieties (including Hilbert modular varieties), namely each GorenOort stratum is a bundle of products of projective lines over another quaternionic Shimurva varieties. Using this description, we gave an explicit optimal slopeweight bound for the classicality of overconvergent Hilbert modular forms. Another application is about the Tate conjecture on the special fiber of Hilbert modular varieties at an inert prime. Using iterations of GorenOort divisors, we gave an explicit construction of generic Tate cycles on Hilbert modular varieties at an inert prime. Later on, we found that such a phenomena always appears in the setup of some unitary Shimura varieties.
In a recent joint work with Yifeng Liu, we obtain an explicit description of the supersingular locus of Hilbert modular varieties at an inert prime. As an arithmetic application, we proved a generalization of Ribet’s classical theorem on the level raising of modular forms to the Hilbert case. Such a result is then used to give an upper bound of the triple product BlochKato Selmer group attached to an elliptic curve over a totally real cubic field, under the assumption that certain diagonal cycles are cohomologically nontrivial. This can be viewed as an triple product analogue of Kolyvagin’s work on ShafarevichTate groups of elliptic curves under the nontorsion assumption of Heegner points. In the future, we are trying to generalize this approach to the case of unitary Shimura varieties, and we hope that this can give new evidence for BlochKato conjecture in this case.


Research Area DE
I am recently working on the Tate conjectures for the characteristic p fibers of Hilbert modular varieties, and also some unitary Shimura varieties. I showed the generic Tate classes on such varieties are generated by the cohomology classes of irreducible components of the supersingular locus. I obtained this result by computing explicitly the intersection matrix of the supersingular locus. This computation of intersection matrix also leads to an upper bound for BlochKato's Selmer groups for the twisted triple product motive attached to an elliptic curve over a cubic totally real field. 


[ 1] Yichao Tian
Canonical subgroups of BarsottiTate groups Ann. of Math. (2) , 172: (2): 955988 2010[ 2] Yichao Tian
padic monodromy of the universal deformation of a HWcyclic BarsottiTate group Doc. Math. , 14: : 397440 2009[ 3] Yichao Tian
Classicality of overconvergent Hilbert eigenforms: case of quadratic residue degrees Rend. Semin. Mat. Univ. Padova , 132: : 133229 2014[ 4] Payman L. Kassaei, Shu Sasaki, Yichao Tian
Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case Forum Math. Sigma , 2: : e18, 58 2014[ 5] Yichao Tian, Liang Xiao
On GorenOort stratification for quaternionic Shimura varieties Compos. Math. , 152: (10): 21342220 2016[ 6] Yichao Tian, Liang Xiao
padic cohomology and classicality of overconvergent Hilbert modular forms Astérisque (382): 73162 2016 ISBN: 9782856298435[7] Yichao Tian, Yifeng Liu
Number Theory (math.NT); Algebraic Geometry (math.AG) arXiv preprint arXiv:1710.11492 2017 [8] David Helm, Yichao Tian, Liang Xiao
On Tate conjecture for the special fibers of some unitary Shimura varieties arXiv preprint arXiv:1410.2343 2014



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