Profile
Profile

Prof. Dr. Yichao Tian

Bonn Junior Fellow

E-Mail: tian(at)math.uni-bonn.de
Telefon: +49 228 73 62245
Homepage: http://www.math.uni-bonn.de/people/tian/
Raum: 4.028
Institute: Mathematical Institute
Forschungsbereich: Research Area DE

Research Profile

I am working on arithmetic algebraic geometry, and especially interested in the p-adic 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 p-adic 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 Goren-Oort stratification of some quaternionic Shimura varieties (including Hilbert modular varieties), namely each Goren-Oort stratum is a bundle of products of projective lines over another quaternionic Shimurva varieties. Using this description, we gave an explicit optimal slope-weight 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 Goren-Oort 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 anti-cyclotomic 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 Goren-Oort stratification, to prove a generalization of Ribet’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 Bloch-Kato 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.

Selected Publications

[1] Yichao Tian
Canonical subgroups of Barsotti-Tate groups
Ann. of Math. (2) , 172: (2): 955--988
2010
[2] Yichao Tian
p-adic monodromy of the universal deformation of a HW-cyclic Barsotti-Tate group
Doc. Math. , 14: : 397--440
2009
[3] Yichao Tian
Classicality of overconvergent Hilbert eigenforms: case of quadratic residue degrees
Rend. Semin. Mat. Univ. Padova , 132: : 133--229
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 Goren-Oort stratification for quaternionic Shimura varieties
Compos. Math. , 152: (10): 2134--2220
2016
[6] Yichao Tian, Liang Xiao
p-adic cohomology and classicality of overconvergent Hilbert modular forms
Astérisque (382): 73--162
2016
ISBN: 978-2-85629-843-5
[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

Publication List

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