scholarly journals Integrality of Seshadri constants and irreducibility of principal polarizations on products of two isogenous elliptic curves

2021 ◽  
Author(s):  
Maximilian Schmidt
Keyword(s):  
Elliptic Curves ◽  
Quadratic Forms ◽  
Complex Multiplication ◽  
Minimal Degree ◽  
Seshadri Constants ◽  
Numerical Problem

AbstractIn this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves $$E_1\times E_2$$ E 1 × E 2 without complex multiplication are integers. By studying elliptic curves on $$E_1\times E_2$$ E 1 × E 2 we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on $$E_1\times E_2$$ E 1 × E 2 are integers if and only if the minimal degree of an isogeny $$E_1\rightarrow E_2$$ E 1 → E 2 equals 1 or 2. Furthermore, this method enables a characterization of irreducible principal polarizations on $$E_1\times E_2$$ E 1 × E 2 .

scholarly journals Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields

2021 ◽  
Vol 56 (1) ◽  
pp. 47-61
Author(s):  
Enrique González-Jiménez ◽  
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Complex Multiplication ◽  
Number Fields ◽  
Quadratic Fields ◽  
Fixed Degree ◽  
Explicit Characterization

In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.

Binary theta series and modular forms with complex multiplication

2014 ◽  
Vol 10 (04) ◽  
pp. 1025-1042 ◽  
Author(s):  
Ernst Kani
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Complex Multiplication ◽  
Theta Series ◽  
Character Formula ◽  
Binary Quadratic Forms

The main purpose of this paper is to give an intrinsic interpretation of the space Θ(D) generated by the binary theta series ϑf attached to the positive binary quadratic forms f whose discriminant has the form D(f) = D/t2, for some integer t. It turns out that [Formula: see text], the space of modular forms of weight 1 and of level |D| which have complex multiplication (CM) by their Nebentypus character [Formula: see text]. As an application, we obtain a structure theorem of the space [Formula: see text]. The proof of this theorem rests on the results of [The space of binary theta series, Ann. Sci. Math. Québec36 (2012) 501–534] together with a characterization of the newforms f which have CM by their Nebentypus character in terms of properties of the associated Deligne–Serre Galois representationρf.

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

scholarly journals On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

1971 ◽  
Vol 43 ◽  
pp. 199-208 ◽  
Author(s):  
Goro Shimura
Keyword(s):  
Elliptic Curves ◽  
Cusp Form ◽  
Mellin Transform ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Function Fields ◽  
Complex Multiplication ◽  
Modular Function ◽  
Imaginary Quadratic Field

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

scholarly journals On Hyperbolic Clifford Algebras with Involution

2013 ◽  
Vol 20 (02) ◽  
pp. 251-260
Author(s):  
M. G. Mahmoudi
Keyword(s):  
Quadratic Forms ◽  
Clifford Algebras ◽  
Low Dimension

The aim of this article is to provide a characterization of quadratic forms of low dimension such that the canonical involutions of their Clifford algebras are hyperbolic.

scholarly journals Linnik’s theorem for Sato-Tate laws on elliptic curves with complex multiplication

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
Evan Chen ◽  
Peter S. Park ◽  
Ashvin A. Swaminathan
Keyword(s):  
Elliptic Curves ◽  
Complex Multiplication
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