On Tate’s refinement for a conjecture of Gross and its generalization

Aoki, Noboru

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 457-486.

Nous étudions un raffinement dù à Tate de la conjecture de Gross sur les valeurs de fonctions $L$ abéliennes en $s=0$ et formulons sa généralisation à une extension cyclique abitraire. Nous prouvons que notre conjecture généralisée est vraie dans le cas des corps de nombres. Cela entraine en particulier que le raffinement de Tate est vrai pour tout corps de nombres.

We study Tate’s refinement for a conjecture of Gross on the values of abelian $L$-function at $s=0$ and formulate its generalization to arbitrary cyclic extensions. We prove that our generalized conjecture is true in the case of number fields. This in particular implies that Tate’s refinement is true for any number field.

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