I am working on number theory or arithmetic geometry, especially on Iwasawa theory. Iwasawa theory is a research field in number theory where one pursuits very deep and mysterious relations between two rather different objects; one is an algebraic object concerning the Selmer group (or more generally the Selmer complexes) which is obtained from a Galois representation of an algebraic number field via cohomology theory, and the other is an analytic object called the p-adic zeta function which p-adically interpolates special values of a certain zeta function (or an L-function). Many research fields (for instance, commutative algebra, cohomology theory like p-adic Hodge theory, theory on modular forms and automorphic representations, p-adic analysis et cetera) intersect at the study of Iwasawa theory and weave up a complicated but graceful tapestry, which has been fascinating researchers of Iwasawa theory. I have especially got interested in non-commutative Iwasawa theory which is a rather new research field, and one of my goals is to reveal characteristic phenomena appearing in number theory for non-commutative Galois extensions by using Iwasawa theoretical methods.
At the same time I am wholly interested in the theory like the arithmetic topology which is developed beyond existing borders of research fields.
I would like to introduce here topics which I have been or am being concerned in for a bit more details.
Iwasawa theory has its origin in historic works of Kenkichi Iwasawa who studied the cyclotomic Zp-extensions of number fields in great detail. Non-commutative Iwasawa theory started from a grandious trial to construct Iwasawa theory on non-commutative extensions of number fields by replacing the cyclotomic Zp-extensions with p-adic Lie extensions (of course not necessarily commutative!!). Even for such non-commutative extensions, basic concepts in Iwasawa theory like characteristic elements and p-adic zeta functions have been introduced and the Iwasawa main conjecture has been formulated after John Henry Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha, Otmar Venjakob and so on. The (non-commutative) Iwasawa main conjecture is closely related to the (non-commutative) equivariant Tamagawa number conjecture, and thus it is also very important from the viepoint of the study on special values of zeta functions.
I have mainly studied the Iwasawa main conjecture for totally real number fields, and have succeeded to prove the main conjecture for p-adic Lie extensions of certain special types (for details, see Papers). At the present, the main conjecture for totally real number fields is verified (under certain assumption on the vanishing of μ-invariants) for general p-adic Lie extensions by Jürgen Ritter, Alfred Weiss and Mahesh Ramesh Kakde. For other motives, however, very little is known up to the present.
In non-commutative Iwasawa theory, a lot of phenomena occur which have never emerged in classical theory; for instance, mysterious congruences between special values of zeta functions like torsion congruences and Möbius-Wall congruences appear. It is one of the most important topics in this research field to construct new mothods which will enable us to study such mysterious phenomena more deeply.
Recently the study of the Iwasawa main conjecture for elliptic cuspforms without complex multiplication has made outstanding developments after results of Kazuya Kato, Christopher McLean Skinner, Eric Urban and so on. So how about on the study of the main conjecture for modular forms with complex multiplication? Basically a cuspform with complex multiplication can be obtained as the theta lifting of a größencharacter of type (A0) on a CM number field, and thus it is not difficult to expect that the main conjecture for modular forms with complex multiplication is closely related to the main conjecture for CM number fields (via Shapiro's lemma). Indeed, Karl Rubin and Kazuya Kato have attacked to the main conjecture for elliptic cuspforms with complex multiplication from such a viewpoint.
Under the obsevation above, Tadashi Ochiai and Kartik Prasanna have precisely described the relation between the two-variable main conjecture for imaginary quadratic fields and the main conjecture for nearly ordinary deformations of elliptic cuspforms with complex multiplication. I am now trying with Tadashi Ochiai to obtain such a detailed description for the relation between the main conjecture for CM number fields and the main conjecture for Hilbert cuspforms with complex multiplication. Meanwhile the Iwasawa main conjecture for CM number fields have been thoroughly studied by Haruzo Hida, Jacques Tilouine, Fabio Mainardi, Ming-Lun Hsieh and so on, and it is also very interesting to search the relation between their results and the main conjecture for Hilbert cuspforms with complex multiplication.
When the fundamental group of a compact, orientable and irreducible 3-manifold has a decomposition as an amalgamated sum (or more generally, has a decomposition associated with a graph of groups), we may construct submanifolds of codimension one in the 3-manifold called essential surfaces which correspond to the decomposition of the fundamental group. Meanwhile it is known that all the graphs of groups correspondiong to decompositions of the fundamental group can be obtained from non-trivial actions of the fundamental group on trees due to the theory of Hyman Bass and Jean-Pierre Serre; hence if one wants to construct essential surfaces in a 3-manifold, it suffices to construct a non-trivial action of its fundamental group on a tree.
Marc Culler and Peter B. Shalen succeeded to obtain non-trivial actions of the fundamental group on trees by focusing on curve components of the SL(2)-character variety associated to the fundamental group and considering its canonical actions on the Bruhat-Tits trees constructed from ideal points of such components. Therefore combining with the observation above, we can systematically obtain essential surfaces each of which corresponds to an ideal point of the SL(2)-character variety. It is remarkable that such highly algebro-geometric methods are effectively used for the purely topological problem to construct essential surfaces.
Now recall that the Bruhat-Tits tree can be regarded as the Bruhat-Tits building associated to the algebraic group SL(2). This simple fact enables us to expect that if we utilese generalities of the theory on Bruhat-Tits buildings, we could develop arguments analogous to Culler-Shalen theory for higher dimensional representations and associated character varieties. Indeed, in my joint work with Takahiro Kitayama, I showed that one could construct certain branched surfaces contained in a 3-manifold by developing an analogue of Culler-Shalen theory for SL(n)-representations of the fundamental group. This result is expected to throw a new light on the many topics in the low-dimensional topology such as the study of non-Haken manifolds. Moreover, since Culler-Shalen theory is highly algebro-geometric theory, there seem to be many possible applications from the viewpoint of the arithmetic topology, which has been developed by Masanori Morishita and so on.