## 所員 - 譚福成（TAN, Fucheng）-

名前

**譚福成（TAN, Fucheng）**
職
講師

E-Mail
ftan （emailアドレスには＠kurims.kyoto-u.ac.jp をつけてください）

研究内容
Arithmetic Geometry

紹 介

My research interests lie in Arithmetic Geometry and Number Theory. I currently focus on the study of p-adic Hodge theory, anabelian geometry, and modularity of Galois representations.

In number theory, especially in Langlands Program, a central question is: Which Galois representations come from algebraic geometry? It is conjectured by Fontaine and Mazur that the key condition is ``potentially log-crystalline" (also called potentially semi-stable). About twenty years ago, a highly non-trivial case of this conjecture was proved by Wiles, namely the Taniyama-Shimura conjecture. Today, the Fontaine-Mazur conjecture in dimension two for the rational field is almost settled, as a result of various works in the past decades, including our work [2].

In fact, the condition ``log-crystalline" was rooted in the study of comparison between p-adic etale cohomology and crystalline cohomology, the so-called comparison theorem in p-adic Hodge theory, initially known as Grothendieck's mysterious functor, which was proved in various generalities. In [4], we have adapted Scholze's approach of pro-etale site to prove the comparison for etale cohomology with non-trivial coefficients, in the relative setting, i.e. for morphisms between formal schemes.

It has been known that p-adic Hodge theory plays an essential role in anabelian geometry, for instance, in Mochizuki's proof of Grothendieck's Anabelian Conjecture. Recently, I have been studying Mochizuki's theory of Inter-universal Teichmuller theory, which is in certain sense a global comparison theorem. I am investigating further arithmetic applications of these two theories. In particular, motivated by the recent study of p-adic Teichmuller theory via p-adic Simpson correspondence, I am led to expect that the p-adic comparison theorems will help with the study of p-adic Simpson correspondence, as well as that of Inter-universal Teichmuller theory.

P-adic Hodge theory also has applications to (families of) automorphic forms. In [5] I obtain a construction of eigenvarieties in dimension two over arbitrary number fields via p-adic Hodge theory. In [3], we have managed to construct pieces of eigenvarieties in the Siegel-Hilbert setting. In [1] the framework of Kummer logarithmic adic spaces and Kummer pro-etale site were developed for the study of overconvergent Eichler-Shimura morphisms.

In number theory, especially in Langlands Program, a central question is: Which Galois representations come from algebraic geometry? It is conjectured by Fontaine and Mazur that the key condition is ``potentially log-crystalline" (also called potentially semi-stable). About twenty years ago, a highly non-trivial case of this conjecture was proved by Wiles, namely the Taniyama-Shimura conjecture. Today, the Fontaine-Mazur conjecture in dimension two for the rational field is almost settled, as a result of various works in the past decades, including our work [2].

In fact, the condition ``log-crystalline" was rooted in the study of comparison between p-adic etale cohomology and crystalline cohomology, the so-called comparison theorem in p-adic Hodge theory, initially known as Grothendieck's mysterious functor, which was proved in various generalities. In [4], we have adapted Scholze's approach of pro-etale site to prove the comparison for etale cohomology with non-trivial coefficients, in the relative setting, i.e. for morphisms between formal schemes.

It has been known that p-adic Hodge theory plays an essential role in anabelian geometry, for instance, in Mochizuki's proof of Grothendieck's Anabelian Conjecture. Recently, I have been studying Mochizuki's theory of Inter-universal Teichmuller theory, which is in certain sense a global comparison theorem. I am investigating further arithmetic applications of these two theories. In particular, motivated by the recent study of p-adic Teichmuller theory via p-adic Simpson correspondence, I am led to expect that the p-adic comparison theorems will help with the study of p-adic Simpson correspondence, as well as that of Inter-universal Teichmuller theory.

P-adic Hodge theory also has applications to (families of) automorphic forms. In [5] I obtain a construction of eigenvarieties in dimension two over arbitrary number fields via p-adic Hodge theory. In [3], we have managed to construct pieces of eigenvarieties in the Siegel-Hilbert setting. In [1] the framework of Kummer logarithmic adic spaces and Kummer pro-etale site were developed for the study of overconvergent Eichler-Shimura morphisms.

- H. Diao and F. Tan, The overconvergent Eichler-Shimura morphisms for modular curves, preprint.
- Y. Hu and F. Tan, The Breuil-Mezard conjecture for non-scalar split residual representations, Annales Scientifiques de l'Ecole Normale Superieure 48, 2015 (4), 1381-1419.
- C.-P. Mok and F. Tan, Overconvergent family of Siegel-Hilbert modular forms, Canadian Journal of Mathematics 67, 2015 (4), 893-922.
- F. Tan and J. Tong, Crystalline comparison isomorphisms in p-adic Hodge theory: the absolutely unramified case, submitted.
- F. Tan, Families of p-adic Galois representations. MIT thesis, 2011.