-adic images of Galois for elliptic curves over (and an appendix with John Voight)

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

Bruin and Najman [LMS J. Comput. Math. 18 (2015), no. 1, 578–602] and Ozman and Siksek [Math. Comp. 88 (2019), no. 319, 2461–2484] have recently determined the quadratic points on each modular curve X 0 ( N ) X_0(N) of genus 2, 3, 4, or 5 whose Mordell–Weil group has rank 0. In this paper we do the same for the X 0 ( N ) X_0(N) of genus 2, 3, 4, and 5 and positive Mordell–Weil rank. The values of N N are 37, 43, 53, 61, 57, 65, 67, and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell–Weil sieve. Often the quadratic points are not finite, as the degree 2 map X 0 ( N ) → X 0 ( N ) + X_0(N)\to X_0(N)^+ can be a source of infinitely many such points. In such cases, we describe this map and the rational points on X 0 ( N ) + X_0(N)^+ , and we specify the exceptional quadratic points on X 0 ( N ) X_0(N) not coming from X 0 ( N ) + X_0(N)^+ . In particular, we determine the j j -invariants of the corresponding elliptic curves and whether they are Q {\mathbb {Q}} -curves or have complex multiplication.

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.

On the one hand, it is well known that Jacobians of (hyper)elliptic curves defined over ℚ having a rational point of orderI can be used in many applications, for instance in the constructionof class groups of quadratic fields with a nontrivial l-rank.On the other hand, it is also well known that 11 is the leastprime number that is not the order of a rational point of an ellipticcurve defined over ℚ. It is therefore interesting to look forcurves of higher genus whose Jacobians have a rational point oforder 11. This problem has already been addressed, and Flynnfound such a family 𝔉 t of genus-2 curves. Now it turns out thatthe Jacobian J 0(23) of the modular genus-2 curve X 0(23) hasthe required property, but does not belong to 𝔉 t . The study ofX 0(23) leads to a method giving a partial solution of the consideredproblem. Our approach allows us to recover X 0(23) and toconstruct another 18 distinct explicit curves of genus 2 definedover ℚ whose Jacobians have a rational point of order 11. Ofthese 19 curves, 10 do not have any rational Weierstrass point,and 9 have a rational Weierstrass point. None of these curvesare ℚ̄-isomorphic to each other, nor ℚ̄-isomorphic to an elementof Flynn's family 𝔉 t . Finally, the Jacobians of these new curvesare absolutely simple.

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

Let $\mathcal{O}$ be an order in the imaginary quadratic field $K$. For positive integers $M \mid N$, we determine the least degree of an $\mathcal{O}$-CM point on the modular curve $X(M,N)_{/K(\zeta_M)}$ and also on the modular curve $X(M,N)_{/\mathbb{Q}(\zeta_M)}$: that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally defined. To prove these results we establish several new theorems on rational cyclic isogenies of CM elliptic curves. In particular, we extend a result of Kwon that determines the set of positive integers $N$ for which there is an $\mathcal{O}$-CM elliptic curve $E$ admitting a cyclic, $\mathbb{Q}(j(E))$-rational $N$-isogeny.

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.

2-Descent on elliptic curves and rational points on certain Kummer surfaces

Let C be a curve of genus 2 that admits a nonhyperelliptic involution. We show that there are at most 2 isomorphism classes of elliptic curves that are quotients of degree 2 of the Jacobian of C. Our proof is constructive, and we present explicit formulae, classified according to the involutions of C, that give the minimal polynomial of the j-invariant of these curves in terms of the moduli of C. The coefficients of these minimal polynomials are given as rational functions of the moduli.

For each open subgroup $G$ of ${\rm GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in $G$. Up to conjugacy, we determine a complete list of the $248$ such groups $G$ of prime power level for which $X_G(\mathbb{Q})$ is infinite. For each $G$, we also construct explicit maps from each $X_G$ to the $j$-line. This list consists of $220$ modular curves of genus $0$ and $28$ modular curves of genus $1$. For each prime $\ell$, these results provide an explicit classification of the possible images of the $\ell$-adic Galois representations arising from elliptic curves over $\mathbb{Q}$ that is complete except for a finite set of exceptional $j$-invariants.

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.

On automorphisms of geometric Goppa codes

Faltings’ long awaited proof of the Mordell conjecture completes, roughly speaking, the question of whether a given curve has only finitely many integral or rational points. Indeed, if a complete curve has genus g ≥ 2, then it has finitely many rational points; any affine curve whose projective closure is a curve of genus at least two will, a fortiori, have only finitely many integral points. A curve of genus 1 is an elliptic curve; it will have infinitely many rational points over a sufficiently large ground field, but no affine subvariety has an infinite number of integral points. Finally, a curve of genus zero is, after a base change, the projective line, which has an infinite number of rational points; affine sub-varieties omitting at most two points will have infinitely many integral points over a sufficiently large ring; but affine sub-varieties omitting at least three points will have only finitely many integral points. Thus the answer to the finiteness question is given entirely by the structure of the curve over the complex numbers.

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.

We give bounds on the primes of geometric bad reduction for curves of genus 3 3 of primitive complex multiplication (CM) type in terms of the CM orders. In the case of elliptic curves, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least 2 2 , the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus 2 2 . In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are important for algorithmic construction of curves with given characteristic polynomials over finite fields.