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Title:
On Eisenstein’s Jugendtraum for Complex Cubic Fields
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Abstract:
In the early 2000’s Ruijsenaars and Felder-Varchenko have introduced the elliptic gamma function, a remarkable multivariable meromorphic q-series that comes from mathematical physics. It satisfies modular functional equations under the group SL3(Z) which make it a higher dimensional analogue of the Jacobi theta function. In this work, we unveil the place that this function and its avatars play in number theory. Our main thesis is that these functions play the role of modular units in extending the theory of complex multiplication to complex cubic fields. In other words we propose a conjectural solution to Hilbert’s 12th problem for complex cubic fields, following a line of research actually initiated G. Eisenstein. We give a lot of numerical evidences that support this conjecture, and relate it to the Stark conjecture by proving an analogue of the Kronecker limit formula in this cubic setting. This is joint work with Nicolas Bergeron and Luis Garcia. This talk is dedicated to the memory of Nicolas Bergeron.
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