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Pré-Publication, Document De Travail Année : 2023

Elliptic Units For Complex Cubic Fields

Résumé

We propose a conjecture extending the classical construction of elliptic units to complex cubic number fields K. The conjecture concerns special values of the elliptic gamma function, a holomorphic function of three complex variables arising in mathematical physics whose transformation properties under SL3(Z) were studied by Felder and Varchenko in the early 2000s. Using this function we construct complex numbers that we conjecture to be units in narrow ray class fields of K. We also propose a reciprocity law for the action of the Galois group on these units in the style of Shimura. To support our conjecture we offer numerical evidence and also prove a new type of Kronecker limit formula relating the logarithm of the modulus of these complex numbers to the derivatives at s = 0 of partial zeta functions of K. Our constructions unveil the role played by the elliptic gamma function in Hilbert's twelfth problem for complex cubic fields.
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Dates et versions

hal-04273703 , version 1 (07-11-2023)

Identifiants

  • HAL Id : hal-04273703 , version 1

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Nicolas Bergeron, Pierre Charollois, Luis E Garcia. Elliptic Units For Complex Cubic Fields: (ON EISENSTEIN’S JUGENDTRAUM). 2023. ⟨hal-04273703⟩
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