Modularity of CM elliptic curves over division fields

Efficient CM-constructions of elliptic curves over finite fields

We present an algorithm that, on input of an integer N ≥ 1 together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) has order N . Although it is unproved that this can be done for all N , a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2ω(N) logN , where ω(N) is the number of distinct prime fact...

Elliptic Curves over Finite Fields

In this chapter, we study elliptic curves defined over finite fields. Our discussion will include the Weil conjectures for elliptic curves, criteria for supersingularity and a description of the possible groups arising as E(Fq). We shall use basic algebraic geometry of elliptic curves. Specifically, we shall need the notion and properties of isogenies of elliptic curves and of the Weil pairing....

Elliptic Curves over Finite Fields

Elliptic curves and modularity

The discriminant Define b2 := a1 + 4a2, b4 := 2a4 + a1a3, b6 := a3 + 4a6, b8 := a1a6 + 4a2a6 − a1a3a4 + a2a3 − a4. Note that if char(K) 6= 2, then we can perform the coordinate transformation y 7→ (y− a1x− a3)/2 to arrive at an equation y = 4x + b2x +2b4x+ b6. Now for any Weierstrass equation we define its discriminant ∆ := −b2b8 − 8b4 − 27b6 + 9b2b4b6. Proposition 1. A curve C/K given by a Wei...

Quotients of Elliptic Curves over Finite Fields

Fix a prime `, and let Fq be a finite field with q ≡ 1 (mod `) elements. If ` > 2 and q ` 1, we show that asymptotically (`− 1)/2` of the elliptic curves E/Fq with complete rational `-torsion are such that E/〈P 〉 does not have complete rational `-torsion for any point P ∈ E(Fq) of order `. For ` = 2 the asymptotic density is 0 or 1/4, depending whether q ≡ 1 (mod 4) or 3 (mod 4). We also show t...