On some notions of good reduction for endomorphisms of the projective line

On the number of isomorphism classes of CM elliptic curves defined over a number field

We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the Generalized Riemann Hypothesis, we show that the average analytic rank of isomorphism classes of elliptic curves over K is bounded above by $(9\deg (K)+1)/2$ , when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.

This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has received relatively little attention, but now it appears to be developing into one of the central topics in the emerging arithmetic theory of (linear) algebraic groups over higher-dimensional fields. The focus of this article is on the Main Conjecture (Conjecture 5.7) asserting the finiteness of the number of isomorphism classes of forms of a given reductive group over a finitely generated field that have good reduction at a divisorial set of places of the field. Various connections between this conjecture and other problems in the theory of algebraic groups (such as the analysis of the global-to-local map in Galois cohomology and the genus problem) are discussed in detail. The article also includes a brief review of the required facts about discrete valuations, forms of algebraic groups, and Galois cohomology.

Let p be an odd prime and let E be an elliptic curve defined over a number field F with good reduction at the primes above p. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over ℤ p 2 -extensions of an imaginary quadratic field where p splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the μ-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups, when E has good supersingular reduction, which are completely analogous to the classical Selmer group, when E has good ordinary reduction. In this survey paper we do not present any proofs, however, we have tried to give references of the discussed results for the interested reader.

We prove the following.Theorem. Let be a number field, and the Jacobian of the curve parametrizing the elliptic curves with distinguished cyclic subgroups of order . If the number is written as , where contains a -simple abelian subvariety such that {\operatorname{rk}} A_k,$ SRC=http://ej.iop.org/images/0025-5734/11/2/A12/tex_sm_2058_img8.gif/> then the set of -isomorphism classes of elliptic curves over the field possessing -points of order is finite.Bibliography: 4 items.

Edwards recently introduced a new normal form for elliptic curves. Every elliptic curve over a non-binary field is birationally equivalent to a curve in Edwards form over an extension of the field, and in many cases over the original field.This paper presents fast explicit formulas (and register allocations) for group operations on an Edwards curve. The algorithm for doubling uses only 3M + 4S, i.e., 3 field multiplications and 4 field squarings. If curve parameters are chosen to be small then the algorithm for mixed addition uses only 9M + 1S and the algorithm for non-mixed addition uses only 10M + 1S. Arbitrary Edwards curves can be handled at the cost of just one extra multiplication by a curve parameter.For comparison, the fastest algorithms known for the popular “a 4 = −3 Jacobian” form use 3M + 5S for doubling; use 7M + 4S for mixed addition; use 11M + 5S for non-mixed addition; and use 10M + 4S for non-mixed addition when one input has been added before.The explicit formulas for non-mixed addition on an Edwards curve can be used for doublings at no extra cost, simplifying protection against side-channel attacks. Even better, many elliptic curves (approximately 1/4 of all isomorphism classes of elliptic curves over a non-binary finite field) are birationally equivalent — over the original field — to Edwards curves where this addition algorithm works for all pairs of curve points, including inverses, the neutral element, etc.This paper contains an extensive comparison of different forms of elliptic curves and different coordinate systems for the basic group operations (doubling, mixed addition, non-mixed addition, and unified addition) as well as higher-level operations such as multi-scalar multiplication.

Let Ki be a number field for all i ∈ ℤ>0 and let ℰ be a family of elliptic curves containing infinitely many members defined over Ki for all i. Fix a rational prime p. We give sufficient conditions for the existence of an integer i0 such that, for all i > i0 and all elliptic curve E ∈ ℰ having good reduction at all 𝔭 | p in Ki, we have that E has good ordinary reduction at all primes 𝔭 | p. We illustrate our criteria by applying it to certain Frey curves in [Recipes to Fermat-type equations of the form xr + yr = Czp, to appear in Math. Z.; http://arXiv.org/abs/1203.3371 ] attached to Fermat-type equations of signature (r, r, p).

Let E / Q E/\mathbb {Q} be an elliptic curve and let Q ( 3 ∞ ) \mathbb {Q}(3^\infty ) be the compositum of all cubic extensions of Q \mathbb {Q} . In this article we show that the torsion subgroup of E ( Q ( 3 ∞ ) ) E(\mathbb {Q}(3^\infty )) is finite and we determine 20 possibilities for its structure, along with a complete description of the Q ¯ \overline {\mathbb {Q}} -isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many Q ¯ \overline {\mathbb {Q}} -isomorphism classes of elliptic curves, and a complete list of j j -invariants for each of the 4 that do not.

In this chapter, we count the isomorphism classes of elliptic curves over finite fields K. For the case K = F 2 m, we list a representative, in Weierstrass form, of each isomorphism class. We determine #E(F 2 m) for each supersingular curve E defined over F 2 m.

This paper presents explicit formulas for the number of isomorphism classes of elliptic curves with 2-torsion points over finite fields. These results also can be used in the elliptic curve cryptosystems and classification problems.Keywordselliptic curvecryptographyisomorphism classesfinite field

We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quadratic fields Q(m) for m≤200. These results of computations give best-possible data including structures of Mordell-Weil groups over some real quadratic fields via two-descent. We also prove similar results for the case of certain cubic fields. Especially, we give the first example of elliptic curve having everywhere good reduction over a pure cubic field using our method.

Let $C$ be a hyperelliptic curve of genus $g\geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\geq 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$.

Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:\mathbb{P}^1_K\to\mathbb{P}^1_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism $\mathbb{P}^1_{R_S}\to\mathbb{P}^1_{R_S}$. A finite Galois invariant subset $X\subset\mathbb{P}^1_K(\bar{K})$ has good reduction outside $S$ if its closure in $\mathbb{P}^1_{R_S}$ is \'etale over $R_S$. We study triples $(f,Y,X)$ with $X=Y\cup f(Y)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many $\text{PGL}_2(R_S)$-equivalence classes of triples with $\text{deg}(f)=d$ and $\sum_{P\in Y}e_f(P)\ge2d+1$ and $X$ having good reduction outside $S$. We consider refined questions in which the weighted directed graph structure on $f:Y\to X$ is specified, and we give an exhaustive analysis for degree $2$ maps on $\mathbb{P}^1$ when $Y=X$.

We show the existence of good hyperplane sections for schemes over discrete valuation rings with good or (quasi-)semi-stable reduction, and the existence of good Lefschetz pencils for schemes with good reduction or ordinary quadratic reduction. As an application, we prove that the reciprocity map introduced for smooth projective varieties over local fields K K by Bloch, Kato and Saito is an isomorphism after ℓ \ell -adic completion, if the variety has good or ordinary quadratic reduction and ℓ ≠ c h a r ( K ) \ell \neq \mathrm {char}(K) .

Fix a number field [Formula: see text] and a rational prime [Formula: see text]. We consider abelian varieties whose [Formula: see text]-power torsion generates a pro-[Formula: see text] extension of [Formula: see text] which is unramified away from [Formula: see text]. It is a necessary, but not generally sufficient, condition that such varieties have good reduction away from [Formula: see text]. In the special case of [Formula: see text], we demonstrate that for abelian surfaces [Formula: see text], good reduction away from [Formula: see text] does suffice. The result is extended to elliptic curves and abelian surfaces over certain number fields unramified away from [Formula: see text]. An explicit example is constructed to demonstrate that good reduction away from [Formula: see text] is not sufficient, at [Formula: see text], for abelian varieties of sufficiently high dimension.