Seminar
An algebraic characterization of the Kronecker function

Hold Date 
20180615 16:00～20180615 17:00 


Place 
Lecture Room M W1C512, West Zone 1, Ito campus, Kyushu University 


Object person 



Speaker 
Nils Matthes (Kyushu University) 

Abstract:
In this work on elliptic functions, Kronecker introduced a meromorphic function $F_{¥tau}(u,v)$ of two complex variables $u$ and $v$ (with $¥tau$ in the upper halfplane) that can be expressed a quotient of Jacobi theta functions. About a century later, the function $F_{¥tau}(u,v)$ was rediscovered by Zagier who demonstrated its central role in the context of periods of modular forms for $¥Gamma:=PSL_2(¥mathbb{Z})$. The Kronecker function satisfies a threeterm functional equation, the "Fay identity", which can be interpreted in terms of the cohomology of $¥Gamma$. We show that, conversely, any solution to the Fay identity which is meromorphic in a small neighborhood of $(0,0) ¥in ¥mathbb{C}^2$ belongs to a fiveparameter family of deformations of $F_{¥tau}(u,v)$.