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, 2013

"... I do research in number theory and arithmetic geometry, and particularly in the area related to Galois representations and deformation of abelian varieties. It is a celebrated theorem of Serre and Tate that the deformation theory of an abelian variety A over a field of characteristic p> 0 is equi ..."

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I do research in number theory and arithmetic geometry, and particularly in the area related to Galois representations and deformation of abelian varieties. It is a celebrated theorem of Serre and Tate that the deformation theory of an abelian variety A over a field of characteristic p> 0 is equivalent that of its Barsotti-Tate group. When the abelian variety A is ordinary, there is a structure of formal group on the infinitesimal deformation space of A/k, and any deformation of A is determined by the Serre-Tate coordinates on the deformation space ([5]). Moreover, the Serre-Tate deformation theory can be applied to determine the local structure of Shimura varieties of Hodge type at ordinary locus ([11],[14]). My research has been forcused on applying the Serre-Tate theory to two types of questions in arithmetic geometry: the first is the study of the local behavior of Galois representations attached to nearly ordinary Hilbert modular forms; the second is the study of special cases of the Mumford-Tate conjecture. I will give a summary of my work ([19],[20]) toward these two directions below. 1 Local behavior of Hilbert modular Galois representations Let F be a totally real field inside an algebraic closure Q ̄ of Q and let FA be the adele ring of F. Fix a rational prime p and a prime p of F over p. Let m be an integral ideal of F and Sk(m,C) be the space of Hilbert modular cusp forms of parallel weight k and level m. For each prime q of F and fractional ideal n of F prime to m, let T (q) and 〈n 〉 be the Hecke operator acting on Sk(m,C). Suppose that we have a Hilbert modular form f ∈ Sk(m,C) such that f is an eigenvector for all the above Hecke operators, i.e. there exist c(q, f), d(n, f) such that T (q)(f) = c(q, f)f and 〈n〉(f) = d(n, f)f. Let Kf be the field generated over Q by these eigenvalues c(q, f), d(n, f), which is known to be a number field. Let λ be a prime of Kf over the rational prime p and let Kf,λ be the completion of Kf at λ. Then it is well known that there is a continuous representation ρf,λ: Gal(Q̄/F) → GL2(Kf,λ) attached to f. Moreover, if f is nearly p-ordinary in the sense that c(p, f) ∈ Kf ⊂ Kf,λ is a p-adic unit, then the restriction of ρf,λ to the decomposition group Dp of Gal(Q̄/F) at p is reducible and of the following shape: ρf,λ|Dp ∼