On the class number of fields with complex multiplication

1966 ◽  
pp. 143-135
Author(s):  
A. P. Novikov
Keyword(s):  
Class Number ◽  
Complex Multiplication

scholarly journals Trace of Frobenius endomorphism of an elliptic curve with complex multiplication

2004 ◽  
Vol 70 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Noburo Ishii
Keyword(s):  
Prime Ideal ◽  
Elliptic Curve ◽  
Class Number ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Good Reduction ◽  
Absolute Value ◽  
Frobenius Endomorphism ◽  
Trace Of Frobenius ◽  
Positive Integers

Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D < −4 of an imaginary quadratic field K. If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal p of K dividing p and there exist positive integers u and υ such that 4p = u2 – Du;2. It is well known that the absolute value of the trace ap of the Frobenius endomorphism of the reduction of E modulo p is equal to u. We determine whether ap = u or ap = −u in the case where the class number of R is 2 or 3 and D is divisible by 3, 4 or 5.

scholarly journals An inverse Jacobian algorithm for Picard curves

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Joan-C. Lario ◽  
Anna Somoza ◽  
Christelle Vincent
Keyword(s):  
Class Number ◽  
Complex Multiplication ◽  
Hyperelliptic Curves ◽  
Period Matrix ◽  
Isomorphism Classes ◽  
Jacobian Problem ◽  
Picard Curves ◽  
The Given ◽  
Small Period ◽  
Jacobian Algorithm

AbstractWe study the inverse Jacobian problem for the case of Picard curves over $${\mathbb {C}}$$ C . More precisely, we elaborate on an algorithm that, given a small period matrix $$\varOmega \in {\mathbb {C}}^{3\times 3}$$ Ω ∈ C 3 × 3 corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most $$4$$ 4 . In particular, we obtain the complete list of maximal CM Picard curves defined over $${\mathbb {Q}}$$ Q . In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].

Complex Multiplication

2009 ◽  
Author(s):  
Reinhard Schertz
Keyword(s):  
Complex Multiplication

Note on class number parity of an abelian field of prime conductor, II

2019 ◽  
Vol 42 (1) ◽  
pp. 99-110 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Class Number ◽  
Abelian Field

scholarly journals Vertical Distribution Relations for Special Cycles on Unitary Shimura Varieties

2018 ◽  
Vol 2020 (13) ◽  
pp. 3902-3926
Author(s):  
Réda Boumasmoud ◽  
Ernest Hunter Brooks ◽  
Dimitar P Jetchev
Keyword(s):  
Vertical Distribution ◽  
Three Dimensional ◽  
Complex Multiplication ◽  
Horizontal Distribution ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Heegner Points ◽  
Universal Norm

Abstract We consider cycles on three-dimensional Shimura varieties attached to unitary groups, defined over extensions of a complex multiplication (CM) field $E$, which appear in the context of the conjectures of Gan et al. [6]. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of [8], and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda $-module constructed from Heegner points.

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