2017 Fiscal Year Annual Research Report
Algebraic aspects of elliptic multiple zeta values
| Project/Area Number |
17F17020
|
|---|
| Research Institution | Kyushu University |
|---|
Principal Investigator |
金子 昌信 九州大学, 数理学研究院, 教授 (70202017)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
MATTHES NILS 九州大学, 数理(科)学研究科(研究院), 外国人特別研究員
|
|---|
| Project Period (FY) |
2017-10-13 – 2020-03-31
|
| Keywords | Multiple zeta values / Associators / Double shuffle equations / Elliptic functions |
|---|
| Outline of Annual Research Achievements |
Multiple zeta values are generalizations of special values of the Riemann zeta function with a very interesting algebraic structure. I made progress on studying the double shuffle equations, which are a set of algebraic relations satisfied by the multiple zeta values. In joint work with Koji Tasaka (an assistant professor at Aichi Prefactural University, Japan), we have compared explicitly two sets of solutions to the double shuffle equations which have been constructed by Francis Brown and Jean Ecalle respectively. Our main result gives an explicit formula for the difference of the two solutions in terms of an explicit solution to the linearized double shuffle equation with an exotic pole structure. The tools used in our work should have further applications to the study of multiple zeta values. Related to this, in a second work I revisited the notion of flexion units, first introduced by Ecalle. In particular, I incorporated flexion units into the framework of representation theory of SL2(Z) and achieved a classification of flexion units in one variable. This helps clarifying some constructions around multiple zeta value theory, such as the explicit construction of rational associators.
|
| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
I am currently in the process of writing up the results of my research mentioned above which is roughly going according to plan. More specifically, the joint paper with Tasaka is (at the time of writing) almost completely finished while for the second one very advanced notes have been written.
|
| Strategy for Future Research Activity |
Besides finishing the two papers mentioned above, I am planning to expand my work on double shuffle equations to the case of higher depths. Furthermore, I am currently attempting to extend my classification results for flexion units to the two variable case. This could potentially help clarifying the functional equations satisfied by some classical elliptic functions such as Kronecker--Eisenstein series. I am also planning to study a suitable version of multiple zeta values in genus two (in particular in the hyperelliptic case) with an eye towards applications in perturbative string theory.
|
