Elliptic curves with large intersection of projective torsion points
نویسندگان
چکیده
منابع مشابه
Torsion Points of Elliptic Curves
Elliptic curves as an area of mathematical study are initially simple to understand, but reveal startling complexity when considered over different fields. This paper discusses the general properties and characteristics of projective space, elliptic curves, and the group structure that arises with certain binary operations on the curve. We discuss elliptic curves over Q, including the topic of ...
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Let k be an algebraically closed field, char k 6= 2, 3, and let X ⊂ P be an elliptic curve with defining polynomial f . We show that any non-trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Φr of size 3r×3r with linear polynomial entries such that det Φr = f . We also show that the identity, thought of as the trivial torsion point of order r, determines up...
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i.e., the supremum of all orders of torsion points on elliptic curves defined over some degree d number field. Write T (d)′ for the set of prime divisors of elements of Td, and P (d) for the largest element of T (d)′. Let TCM(d) (resp. TIM(d)) be the subset of T (d) corresponding to elliptic curves with complex multiplication (resp. with algebraic integral modulus j(E)), and similarly adding th...
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Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρE : Gal(k/k) → GL2(b Z). For a fixed number field k, we describe the image of ρE for a “random” elliptic curve E over k. In particular, if k 6= Q is linearly disjoint from the cyclotomic extension of Q, then ρE will be surjective for “most” elliptic curves over k.
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Let E/Q be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Ω of the order. If the prime p splits completely in Ω, then we can reduce E modulo one the factors of p and get a curve E defined over Fp. The trace of the Frobenius of E is known up to sign and we need a fast way to find t...
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ژورنال
عنوان ژورنال: European Journal of Mathematics
سال: 2017
ISSN: 2199-675X,2199-6768
DOI: 10.1007/s40879-017-0207-8