（一）当前研究方向:

（1）算术基本群；
（2）岩泽理论；
（3）自守表示与L-函数。

（二）2010年开始，每年主攻一个专题。

2019年研究专题：“arithmetic fundamental groups and Iwasawa theory”（“算术基本群与岩泽理论”）.

2018年研究专题：“arithmetic fundamental groups and Noether's problem on rationality”（“算术基本群与诺特有理化问题”）.

 
诺特在1910年代，研究了这样一个问题：纯超越扩张在有限群作用下的不变子域，是否仍是纯超越扩张。这就是现在所称的“诺特有理化问题”。一百年来，人们只获得了一些零星的结果。现在仍是公开问题。详见：
https://en.wikipedia.org/wiki/Rational_variety

 

很偶然地发现，我们在“算术基本群”方面的一些结果，可以用来解释诺特有理化问题中的一些现象。

2010年-2017年研究专题：
“arithmetic fundamental groups”（“算术基本群”）；
“motivic Galois groups”（“动形Galois群”）；
“Iwasawa theory”（“岩泽理论”）；
“L-functions of motives”（“动形L-函数”）；
“automorphic representations and L-functions”（“自守表示与L-函数”）.

（三）早期工作的预印本

Here are my previous preprints,

available from 

https://arxiv.org/search/math?query=feng-wen+an&searchtype=author&abstracts=show&order=-announced_date_first&size=50

 

[1] On the arithmetic Galois covers of higher relative dimensions. arXiv:1009.3213
 
Abstract: In this paper we will give the calculus, the criterion, and the existence of the arithmetic Galois covers of higher relative dimensions.

MSC Class: 11G35; 14H30; 14H37; 14J50

[2] On the unramified extension of an arithmetic function field in several variables. arXiv:1006.5143
 
Abstract: In this paper we will give a scheme-theoretic discussion on the unramified extensions of an arithmetic function field in several variables. The notion of unramified discussed here is parallel to that in algebraic number theory and for the case of classical varieties, coincides with that in Lang's theory of unramified class fields of a function field in several variables.

MSC Class: Primary 11G35; Secondary 14F35; 14G25

[3] On the transcendental Galois extensions. arXiv:1004.5036
 
Abstract: In this paper the transcendental Galois extensions of a field will be introduced as counterparts to algebraic Galois ones. There exist several types of transcendental Galois extensions of a given field, from the weakest one to the strongest one, such as Galois, tame Galois, strong Galois, and absolute Galois. The four Galois extensions are distinct from each other in general …

MSC Class: 11J81; 12F20; 14H25

[4] Algebraic Anabelian Functors. arXiv:0912.4017
 
Abstract: In this paper we will prove that there exists a covariant functor, called algebraic anabelian functor, from the category of algebraic schemes over a given field to the category of outer homomorphism sets of groups. The algebraic anabelian functor, given in a canonical manner, is full and faithful. It reformulates the anabelian geometry over a field. As an application of the anabelian functor, we…

MSC Class: 14F35; 11G35

[5] On the section conjecture of Grothendieck. arXiv:0911.1523
 
Abstract: For a given arithmetic scheme, in this paper we will introduce and discuss the monodromy action on a universal cover of the étale fundamental group and the monodromy action on an \emph{sp}-completion constructed by the graph functor, respectively; then by these results we will give a proof of the section conjecture of Grothendieck for arithmetic schemes.

MSC Class: 14F35 (Primary); 11G35 (Secondary)

[6] Notes on the quasi-galois closed schemes. arXiv:0911.1073

Abstract: Let
f:X→Y be a surjective morphism of integral schemes. Then X is said to be quasi-galois closed over
Y by f if X has a unique conjugate over Y in an algebraically closed field. Such a notion has been applied to the computation of étale fundamental groups. In this paper we will use affine coverings with values in a fixed field to discuss quasi-galois closed and then give a suffi…

MSC Class: 14H30

[7] On the algebraic fundamental groups. arXiv:0910.4691

Abstract: Passing from arithmetic schemes to algebraic schemes, in a similar manner we will have the computation of the étale fundamental group of an algebraic scheme and then will define and discuss the qc fundamental group of an algebraic scheme in this paper. The qc fundamental group will also give a prior estimate of the étale fundamental group.

MSC Class: 14F35; 11G35

[8] On the etale fundamental groups of arithmetic schemes, revised. arXiv:0910.4646

Abstract: In this paper we will give a computation of the étale fundamental group of an integral arithmetic scheme. For such a scheme, we will prove that the étale fundamental group is naturally isomorphic to the Galois group of the maximal formally unramified extension over the function field. It consists of the main theorem of the paper. Here, formally unramified will be proved to be arithmetically unrami…

MSC Class: 14F35; 11G35

[9] On the arithmetic fundamental groups. arXiv:0910.0605
 
Abstract: In this paper we will define a qc fundamental group for an arithmetic scheme by quasi-galois closed covers. Then we will give a computation for such a group and will prove that the etale fundamental group of an arithmetic scheme is a normal subgroup in our qc fundamental group, which make up the main theorem of the paper. Hence, our group gives us a prior estimate of the etale fundamental group.…

MSC Class: 14F35; 11G35

[10] On the etale fundamental groups of arithmetic schemes. arXiv:0910.0157
 
Abstract: In this paper we will give the computation of the etale fundamental group of an arithmetic scheme.

MSC Class: 14F35 (Primary); 11G35 (Secondary)

[11] On the existence of geometric models for function fields in several variables. arXiv:0909.1993
 
Abstract: In this paper we will give an explicit construction of the geometric model for a prescribed extension of a function field in several variables over a number field. As a by-product, we will also prove the existence of quasi-galois closed covers of arithmetic schemes (in eprint arXiv:0907.0842).

MSC Class: 11G35 (Primary); 14J50 (Secondary)

[12] A short note on quasi-galois closed and pseudo-galois covers. arXiv:0909.0240

Abstract: There is a big difference between "quasi-galois" in the eprint (arXiv:0907.0842) and "pseudo-galois" in the sense of Suslin-Voevodsky.

MSC Class: 14J50; 14G40

[13] Automorphism Groups of Quasi-galois Closed Arithmetic Schemes. arXiv:0907.0842
 
Abstract: Assume that X and Y are arithmetic schemes, i.e., integral schemes of finite types over
Spec(Z). Then X is said to be quasi-galois closed over Y if X has a unique conjugate over Y in some certain algebraically closed field, where the conjugate of X over Y is defined in an evident manner. Now suppose that
φ:X→Y is a surjective morphism of finite type such that …

MSC Class: 14J50; 14G40; 14G45; 14H30; 14H37; 12F10

[14] The Combinatorial Norm of a Morphism of Schemes. arXiv:0801.2609
 
Abstract: In this paper we will prove that there exists a covariant functor from the category of schemes to the category of graphs. This functor provides a combination between algebraic varieties and combinatorial graphs so that the invariants defined on graphs can be introduced to algebraic varieties in a natural manner. By the functor, we will define the combinatorial norm of a morphism of schemes. Then…

MSC Class: 14A15 (Primary); 05C30 (Secondary); 13E99; 14M05; 55M10

[15] Affine Structures on a Ringed Space and Schemes. arXiv:0706.0579

Abstract: In this paper we will first introduce the notion of affine structures on a ringed space and then obtain several properties. Affine structures on a ringed space, arising mainly from complex analytical spaces of algebraic schemes over number fields, behave like differential structures on a smooth manifold. As one does for differential manifolds, we will use pseudogroups of affine transformations …

MSC Class: 14A15; 14A25; 57R55

[16] The Conjugates of Algebraic Schemes. arXiv:math/0702493
 
Abstract: Fixed an algebraic scheme Y. We suggest a definition for the conjugate of an algebraic scheme X over Y in an evident manner; then X is said to be Galois closed over Y if X has a unique conjugate over Y. Now let X and Y both be integral and let X be Galois closed over Y by a surjective morphism φ of finite type. Then φ?(k(Y)) is a subfield of k(X) by φ. …

MSC Class: 14J50; 11R37

[17] The Specializations in a Scheme. arXiv:math/0509587
 
Abstract: In this paper we will obtain some further properties for specializations in a scheme. Using these results, we will take a picture for a scheme and a picture for a morphism of schemes. In particular, we will prove that every morphism of schemes is specialization-preserving and of norm not greater than one (under some condition); a necessary and sufficient condition will be given for an injective…

MSC Class: 14A15;14A25; 14C99; 14M05