Takashi Hara:
On p-adic Artin L-functions for CM fields,
Niigata Algebra Seminar,
Niigata University, 31th/May/2024
Abstract: In 1983, Ralph Greenberg constructed p-adic Artin L-functions for totally real number fields by appropriately "gluing" p-adic Hecke L-functions for various totally real fields via Brauer induction. We will construct in this talk p-adic Artin L-functions for CM fields by also "gluing" p-adic Hecke L-functions for various CM fields. We would especially like to focus on characteristic phenomena and difficulties in the CM field case, which did not appear in Greenberg's work. This is joint work with Tadashi Ochiai (Tokyo Institute of Technology).
Takashi Hara:
Measure-valued modular symbols and explicit description of indefinite integral,
Shinshu Number Theory Mini Workshop,
Shinshu University, Department of Education, 30th/March/2024
Abstract: The cohomological definition of the indefinite integral in Talk 6 is not suitable for actual computation at all. In this talk, we give an explicit description of the indefinite integral, using the measure-valued modular symbols introduced in [BD07] (essentially the same as Λ-adic modular symbols which Greenberg and Stevens introduced in [GS93] to construct their two variable p-adic L-functions).
Takashi Hara:
Definition and properties of (additive and multiplicative) indefinite integral,
Shinshu Number Theory Mini Workshop,
Shinshu University, Department of Education, 29th/March/2024
Abstract: As is explained in Ohshita's talk, the ratio of the two cohomology classes logp (cf,τ) and ordp (cf,τ) coincides with the L-invariant logp(qE) / ordp(qE) [Dar01,Theorem 4]. As a consequence of this theorem, we define the indefinite integral of the double integral introduced in Atsuta's talk (with respect to the p-adic variable τ). By taking an appropriate branch of the p-adic logarithm, we first define the additive indefinite integral cohomologically, and verify its fundamental properties. In particular, we mention that the indefinite integral is uniquely characterised by these properties. We next explain that, by taking a suitable lattice Λp in Cp×, we can consider the multiplicative refinement of the indefinite integral as a Cp× / Λp-valued function. The multiplicative indefinite integral will be used in Sakugawa's talk to construct Stark–Heegner points.
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Abstract: In the first half of this summer school, we studied the formulation of the non-commutative Iwasawa main conjecture for totally number fields following [CFKSV05], and then overviewed the proof of the non-commutative Iwasawa main conjecture due to Mahesh Ramesh Kakde. The proof of the non-commutative Iwasawa main conjecture for totally real number fields was, however, also announced in the arXiv preprint of Jürgen Ritter and Alfred Weiss in advance of Kakde's proof. Ritter and Weiss's proof is based on a strategy rather different from Kakde's, and thus it is very important and interesting to compare their methods. But Ritter and Weiss's proof is written in the language of their equivariant Iwasawa theory (which has already appeared in Nomura's second lecture), and unfortunately it seems to be difficult to compare them (or even to understand both proofs simultaneously) due to this language barrier.
Under these circumstances, I will first compare briefly the formulation in the style of [CFKSV05] to the formulation in the style of Ritter-Weiss's equivariant Iwasawa theory, and then observe that these two formulations essentially coincide. Then I will try to introduce Ritter-Weiss's strategy toward the proof of the non-commutative Iwasawa main conjecture (which, I think, has not been introduced in Japan so seriously) as much as possible, focusing on the difference from the proof of Kakde.
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Abstract: One may prove the non-commutative Iwasawa main conjecture for a p-adic Lie extension of a totally real number field by “patching” the classical Iwasawa main conjectures (verified by Andrew Wiles) for abelian p-adic Lie extensions contained in the p-adic Lie extension under consideration. As an introduction of the following lectures given by Kitajima and Ohshita, I will explain in this lecture the “patching” argument of Iwasawa main conjectures which was first proposed by David Burns and then elaborated by Kazuya Kato.
I will try to give an easy and clear explanation postponing
all the complicated arguments and calculations to Kitajima and Ohshita's lectures, for the audience to understand the heart of Burns and Kato's outstanding ideas.
Takashi Hara: On an extension of Culler-Shalen theory utilising Bruhat-Tits theory,
Kagawa Seminar,
Kagawa University, 12th/July/2014.
Takashi
Hara:
Concerning actions of 3-manifold groups: from topological and arithmetic viewpoints,
Intelligence of Low dimensional topology,
Research Institute for Mathematical Sciences (RIMS), Kyoto University, 22nd/May/2014.
résuméAbstract: Somewhat mysterious as it might seem,
there is a close relation between the (topological) notion
of essential surfaces and (algebraic) actions
of 3-manifold groups on trees. Such a rather classical
observation implies that it should be worth introducing
sophisticated algebraic and arithmetic methods
even to the study of topological objects
such as essential surfaces.
As a practical example, we first present an extension of
Marc Culler and Peter B. Shalen's construction of
essential surfaces based upon the theory of
Bruhat-Tits' buildings (this is a joint work with
Takahiro Kitayama, Tokyo Institute of Technology).
If time permits,
we shall discuss certain perspectives and problems
concerning essential surfaces and actions
of 3-manifold groups from an arithmetic viewpoint.
Takashi
Hara:
On almost divisibility of the Selmer groups and cyclotomic specialisation of the multi-variable Iwasawa main conjecture for CM number fields,
Number Theory Seminar at Waseda University,
Waseda University, 22nd/November/2013.
résumé (Japanese)
Takashi
Hara:
On the cyclotomic Iwasawa main conjecture for Hilbert modular cuspforms with complex multiplication,
The 58th Symposium on Algebra,
Hiroshima University, 29th/August/2013.
résumé (Japanese)
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Abstract: Culler-Shalen theory is a classical theory on construction
of essential surfaces contained in a topological 3-manifold,
which has provided a number of important results in low-dimensional topology.
It is of great interest from the arithmetic viewpoint as well
since a lot of algebraic methods play remarkable roles there:
geometry of character varieties, Bass-Serre theory on trees and so on.
After brief review on classical Culler-Shalen theory,
we discuss in this talk its extension to higher-dimensional
representations of fundamental groups using Bruhat-Tits buildings
of higher rank. We then present certain (trials towards)
applications to arithmetic topology
à la Barry Mazur et Masanori Morishita if time permits.
This is a joint work with Takahiro Kitayama (the University of Tokyo).
Takashi
Hara:
On an extension of Culler-Shalen theory for higher dimensional representations,
Danwakai (
Seminar),
Tokyo University of Science, 12th/July/2013.
résumé (Japanese)
Abstract: Culler-Shalen theory is one of the most important
classical theories in low-dimensional topology which concerns the systematic
constraction of what we call essential surfaces contained in a three
dimensional manifold. It is also very interesting from an algebraic point
of view, for one utilises there many highly algebraic or
algebro-geometric methods such as geometry of moduli
spaces of SL(2)-representations
of fundamental groups, Bass-Serre theory concerning actions of groups on trees, et cetera.
After a brief review on classical Culler-Shalen theory
(especially on algebraic arguments), we shall discuss in this talk its extension
for higher dimensional representations via Bruhat-Tits buildings associated
to the higher dimensional special linear groups SL(n).
We would like to mention several problems raised from a viewpoint of
arithmetic topology if time permits. This is a joint work
with Takahiro Kitayama (the University of Tokyo).
Takashi Hara: On the cyclotomic Iwasawa main conjecture for modular forms with complex multiplication,
HAG (Homotopical Algebraic Geometry) Seminar, (informal talk)
Research Institute for Mathematical Sciences (RIMS), Kyoto University, 21st/June/2013.
Abstract: We discuss technical difficulties on the “specialisation”
of the multivariable Iwasawa main conjecture, adopting
the cyclotomic main conjecture for cuspforms
with complex multiplication as material.
We basically deal with the cases of elliptic cuspforms,
and explain the relation between the cyclotomic
Iwasawa main conjecture for elliptic cuspforms with
complex multiplication and the (two-variable) Iwasawa main
conjecture for imaginary quadratic fields.
If time permits, we would like to mention our results
on the cyclotomic Iwasawa main conjecture for Hilbert
cuspforms with complex multiplication (joint work with
Tadashi Ochiai, Osaka University).
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Abstract: In topology ---a research field where one
pursuits “shapes” of objects---, it goes without saying that
the procedure ‘to decompose manifolds into simpler ones’
is indispensable. For 3-manifolds, in particular, the decomposition
along essential surfaces plays an important role. Marc Culler and Peter Shalen
established in 1983 a method to construct non-trivial essential surfaces
contained in 3-manifolds in a systematic manner.
There they effectively utilised highly algebraic devices;
for instance, geometry of character varieties (moduli of
2-dimensional representations of fundamental groups),
theory of trees established by Hyman Bass and Jean-Pierre Serre
and so on.
After brief review on classical Culler-Shalen theory,
we present in this talk an extension of their theory
to higher dimensional character varieties via Bruhat-Tits theory
and (a trial of) an application to arithmetic topology
à la Barry Mazur et Masanori Morishita.
This is a joint work with Takahiro Kitayama (the University of Tokyo).
Takashi Hara: On the Iwasawa main conjecture for totally real number fields and its non-abelianisation,
HAG (Homotopical Algebraic Geometry) Seminar, (informal talk)
Nagoya University, 23rd/February/2013.
Abstract: This is a survey talk on the non-commutative Iwasawa main conjecture
for totally real number fields especially for non-experts
in this research area. I will first introduce the formulation of the classical
Iwasawa main conjecture for the rational number field, which
is the foundation of Iwasawa theory. Then we will discuss
what kinds of difficulties emerge when we try to
non-abelianise the formulation of the main conjecture, and
how to avoid such problems.
If time permits, I would like to
introduce the «patching arguments» proposed by David Burns and
Kazuya Kato, and explain the strategy towards the proof of the main conjecture
for non-commutative p-adic Lie extensions which are not so complicated.
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Abstract: The Iwasawa main conjecture is
a very mysterious conjecture and one of the hottest topics in number theory,
which insists that the characteristic ideals of the Pontrjagin duals
of Selmer groups
(«algebraic invariants») and p-adic L-functions
(«analytic invariants») essentially coincide
in the p-adic world. Recently the study of the main conjecture
for elliptic modular forms has made outstanding developments after results
of Kazuya Kato, Christopher Skinner, Eric Urban and so on. On the other hand,
one can adopt a rather different approach to the main conjecture
for modular forms with complex multiplication (as was observed
by Karl Rubin and Kazuya Kato): that is, one can relate the main conjecture
for them with that for imaginary quadratic fields via theta lifts
of associated größencharacters.
In this talk, we will generalise the latter strategy
to Hilbert cuspforms with complex multiplication, and try to approach
the main conjecture for them by relating it to that for CM fields.
If time permits, we would like to mention technical difficulties arising
in the specialisation procedure of the main conjecture for CM fields.
This is a joint work with Tadashi Ochiai (Osaka University).
Takashi Hara: Introductional lecture on Kazuya Kato's paper (4th part),
Workshop on arithmetic geometry at Tambara 2011,
Tambara Institute of Mathematical Sciences (the University of Tokyo)
1st/June/2011.
Title of the paper:
Kazuya Kato, p-adic Hodge theory and values of
zeta functions of modular forms, in: Cohomologies p-adiques et
applications arithmétiques III, Astérisque 295 (2004) 117--290.
I gave a lecture on the proof of Zeta Value Formulae
on Beĭlinson-Kato elements.
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Abstract: We will sketch the proof of
non-commutative Iwasawa main conjecture for totally real number fields
and explain key ideas in the proof based on the works of Kazuya Kato,
David Burns, Ritter-Weiss, Mahesh Kakde and the speaker.
We especially focus on how induction works in the construction of
the p-adic zeta functions for non-commutative p-extensions.
Takashi
Hara:
Inductive construction of the p-adic zeta functions for non-commutative p-extensions of exponent p of totally real fields,
Thesis Presentation (Ph.D. Defense),
the University of Tokyo, 4th/February/2011.
Takashi Hara: On non-commutative Iwasawa main conjecture and
related topics,
Number Theory Seminal at Waseda University,
Waseda University, 14th/January/2011.
Takashi
Hara:
Iwasawa theory of totally real fields
for certain non-commutative p-extensions,
Algebraic Number Theory and Related Topics,
Research Institute
for Mathematical Sciences (RIMS), Kyoto University, 12th/December/2008.
Takashi Hara: On non-commutative Iwasawa main conjecture
of totally real fields for p-extensions,
Algebra Seminar,
Kyushu University, 24th/October/2008.
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Takashi Hara: Iwasawa theory of totally real fields
for certain non-commutative p-extensions,
The 7-th conference of number theory in Hiroshima,
Hiroshima University, 24th/July/2008.
Abstract: Recently, Kazuya Kato has proven
the non-commutative Iwasawa
main conjecture (in the sense of Coates, Fukaya, Kato, Sujatha and Venjakob)
for non-commutative Galois extensions of “Heisenberg type”
of totally real fields, using integral logarithmic homomorphisms introduced
by Oliver and Taylor. His method was based on Burns' outstanding idea,
“construct the p-adic zeta functions for non-commutative extensions by
patching p-adic zeta functions associated to commutative subextensions.&rquo;
In this talk, we apply Kato's method (and Burns' technique) to certain
non-commutative p-extensions which are more complicated than those of
Heisenberg type, and sketch the proof of the main conjecture for them.
Takashi
Hara:
Iwasawa theory of totally real fields for
non-commutative p-extensions of strictly upper triangular type,
Iwasawa 2008,
Kloster Irsee (Augusburg, Germany) 3rd/July/2008.
Abstract: Recently, Kazuya Kato has proven the Iwasawa
main conjecture (in the sense of Coates, Fukaya, Kato, Sujatha and Venjakob)
for non-commutative Galois extensions of “Heisenberg type”
of totally real algebraic fields. In this talk, we apply Kato&aopos;s method
to non-commutative p-extensions of “strictly upper triangular type.”
This is another generalization of Kato's result than M. Kakde.
Takashi
Hara:
Noncommutative Iwasawa theory of totally real fields,
Iwasawa Theory Seminar,
Keio University, 10th/May/2010.
Takashi Hara: Iwasawa theory of totally real fields for certain
non-commutative p-extensions,
Number Theory Seminal at Waseda University,
Waseda University, 9th/May/2008.
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Abstract: Recently, Kazuya Kato has proven the non-commutative
Iwasawa main conjecture (in the sense of Coates, Fukaya, Kato,
Sujatha and Venjakob) for non-commutative Galois extensions
of “Heisenberg type” of totally real fields, using integral logarithmic
homomorphisms. In this talk, we apply Kato's method to certain
non-commutative p-extensions which are more complicated than those
of Heisenberg type, and prove the main conjecture for them.
Takashi Hara: The development of non-commutative Iwasawa theory
for totally real number fields,
The 5-th Kinosaki Shinjin Seminar,
Kinosaki Health Welfare center (Toyooka City, Hyogo Prefecture),
21st/February/2008.