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 my future project joint with Yifeng Liu, I want to apply our current geometric techniques to study the level raising of automorphic forms. In the case of classical modular forms, it was a theorem due to Ribet, and was used by Bertonili and Darmon to study the anticyclotomic Iwasawa main conjecture. In our project, we want to use the information on the supersingular locus of Hilbert modular varieties, which is obtained by previous techniques with GorenOort stratification, to prove a generalization of Ribetâ€™s theorem in this context. Such a result is related to Selmer group of certain motive attached to some Hilbert cuspidal eigenforms,and may provide new evident to the BlochKato conjecture. This geometric approach to level raising can be adapted to other Shimura varieties, we hope to study the similar problems for Siegel threefold, and some unitary Shimura varieties of group.


[ 1] Yichao Tian
Canonical subgroups of BarsottiTate groups Ann. of Math. (2) , 172: (2): 955988 2010 DOI: 10.4007/annals.2010.172.955[ 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 DOI: 10.4171/RSMUP/13210[ 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 DOI: 10.1112/S0010437X16007326[ 6] Yichao Tian, Liang Xiao
padic cohomology and classicality of overconvergent Hilbert modular forms AstÃ©risque (382): 73162 2016 ISBN: 9782856298435[7] Yichao Tian, Liang Xiao
Tate cycles on some quaternionic Shimura varieties mod p arXiv preprint arXiv:1410.2321 2014 [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

