On a Class of Elliptic Functions Associated with Imaginary Quadratic Fields
نویسنده
چکیده
Let −D be the field discriminant of an imaginary quadratic field. We construct a class of elliptic functions associated naturally with the quadratic field Q( √ −D) which, combined with the general theory of elliptic functions, allows us to provide a unified theory for two fundamental results (one classical and one due to Ramanujan) about the elliptic functions. §
منابع مشابه
Class number formulas via 2-isogenies of elliptic curves
A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a 2-isogeny of elliptic curves.
متن کاملSingular values of multiple eta-quotients for ramified primes
We determine the conditions under which singular values of multiple η-quotients of square-free level, not necessarily prime to 6, yield class invariants, that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. We show that the singular values lie in subfields of the ring class fields of index 2 ′ −1 when k > 2 primes dividing the level are ramified in the imaginary...
متن کاملIndivisibility of class numbers of imaginary quadratic fields
We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to −X whose class numbers are indivisible by a given prime and whose imaginary quadratic fields satisfy any given set of local conditions. This estimate matches the best results in the direction of the Cohen–Lenstra heuristics for ...
متن کاملComputations of elliptic units for real quadratic fields
Elliptic units, which are obtained by evaluating modular units at quadratic imaginary arguments of the Poincaré upper half-plane, allow the analytic construction of abelian extensions of imaginary quadratic fields. The Kronecker limit formula relates the complex absolute values of these units to values of zeta functions, and allowed Stark to prove his rank one archimedean conjecture for abelian...
متن کاملClass numbers of ray class fields of imaginary quadratic fields
Let K be an imaginary quadratic field with class number one and let p ⊂ OK be a degree one prime ideal of norm p not dividing 6dK . In this paper we generalize an algorithm of Schoof to compute the class numbers of ray class fields Kp heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura’s reciprocity law. We have...
متن کامل