The Chowla-Selberg Formula for Quartic Abelian CM Fields
نویسنده
چکیده
We provide explicit analogues of the Chowla-Selberg formula for quartic abelian CM fields. This consists of two main parts. First, we implement an algorithm to compute the CM points at which we will evaluate a certain Hilbert modular function. Second, we exhibit families of quartic fields for which we can determine the precise form of the analogue of the product of gamma values.
منابع مشابه
An arithmetic intersection formula on Hilbert modular surfaces
In this paper, we obtain an explicit arithmetic intersection formula on a Hilbert modular surface between the diagonal embedding of the modular curve and a CM cycle associated to a nonbiquadratic CM quartic field. This confirms a special case of the author’s conjecture with J. Bruinier, and is a generalization of the beautiful factorization formula of Gross and Zagier on singular moduli. As an ...
متن کاملThe Chowla–Selberg Formula and The Colmez Conjecture
In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.
متن کاملChowla-selberg Formula and Colmez’s Conjecture
In this paper, we reinterpret the Colmez conjecture on Faltings’ height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving Faltings’ height of a CM abelian surface and arithmetic intersection numbers, and prove that Colmez’s conjecture for CM abelian surfaces is equivalent to the cuspitality of this modular form.
متن کاملArithmetic Intersection on a Hilbert Modular Surface and the Faltings Height
In this paper, we prove an explicit arithmetic intersection formula between arithmetic Hirzebruch-Zagier divisors and arithmetic CM cycles in a Hilbert modular surface over Z. As applications, we obtain the first ‘non-abelian’ Chowla-Selberg formula, which is a special case of Colmez’s conjecture; an explicit arithmetic intersection formula between arithmetic Humbert surfaces and CM cycles in t...
متن کاملAn Analogue of the Chowla–selberg Formula for Several Automorphic L-functions
In this paper, we will give a certain formula for the Riemann zeta function that expresses the Riemann zeta function by an infinte series consisting of KBessel functions. Such an infinite series expression can be regarded as an analogue of the Chowla-Selberg formula. Roughly speaking, the Chowla-Selberg formula is the formula that expresses the Epstein zeta-function by an infinite series consis...
متن کامل