Number Theory Seminar

Multiple zeta values (MZV) were first defined by Euler for depth 2 in late 1700's, and were popularized by Zagier in 1990's. Gangl, Kaneko and Zagier found an infinite series of relation between MZVs of depth 2 which are related to cusp forms of weight k and level 1. Their result gives a nice explanation that why cusp form would appear in the Broadhurst Krimer conjecture, which is the conjecture about the generating series of the dimension of MZVs for each depth and weight. Recently, people generalize the study of MZVs to multiple zeta star values (MZSV) and multiple zeta half values (MZHV). In this talk, I will talk about the appearance of cusp form in the case of multiple zeta half values. I will compare these results with the corresponding results in the case of MZVs. From the comparison, we can see the depth filtration for MZVs and MZHVs are different. The proof uses Francis Brown's solution to the double shuffle equation, and I will also discuss his result in the talk.