On Elliptic K-curves

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.

The j-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over Q. This paper shows that the coordinate maps given by N. Elkies for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points with arbitrarily large negative discriminant.

Let K be the field of algebraic functions on an elliptic curve that can be described by an equation of the form y 2 = f ( x ) {y^2} = f(x) where f ( x ) f(x) is a quartic polynomial over a field k. Moreover, assume that the Riemann surface for K contains no points rational over k. When k is the field of real numbers it is well known that K may also be expressed as a quadratic extension of a function field L = k ( u , v ) L = k(u,v) of algebraic functions on a conic whose Riemann surface also contains no points rational over k. We extend this result to p-adic ground fields k. Moreover, we describe the various subfields of index two and genus zero (conic subfields) in terms of the k-rational points on the Jacobian of K. This is done for arbitrary ground fields. In particular, the embedding of the projective class group of K (over k) is seen to describe exactly those conic subfields that possess k-rational points.

For every number field k, we construct an affine algebraic surface X over k with a Zariski dense set of k-rational points, and a regular function f on X inducing an injective map $$X(k)\rightarrow k$$ on k-rational points. In fact, given any elliptic curve E of positive rank over k, we can take $$X=V\times V$$ with V a suitable affine open set of E. The method of proof combines value distribution theory for complex holomorphic maps with results of Faltings on rational points in sub-varieties of abelian varieties.

Let H be the Hilbert class field of an imaginary quadratic field K. An elliptic curve E over H with complex multiplication by K is called a Q-curve if E is isogenous over H to all its Galois conjugates. We classify Q-curves over H, relating them with the cohomology group H2(H/Q,±1). The structures of the abelian varieties over Q obtained from Q-curves by restriction of scalars are investigated.

Modular curves likeX0(N) andX1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here byXnsp(p) andXnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures ofp-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc.(3)77(1) (1998), 1–38;J. Algebra231(1) (2000), 414–448).

Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman and Siksek [Math. Comp. 88 (2019), pp. 2461–2484], and Box [Math. Comp. 90 (2021), pp. 321–343] described all the quadratic points on the modular curves of genus 2 ≤ g ( X 0 ( n ) ) ≤ 5 2\leq g(X_0(n)) \leq 5 . Since all the hyperelliptic curves X 0 ( n ) X_0(n) are of genus ≤ 5 \leq 5 and as a curve can have infinitely many quadratic points only if it is either of genus ≤ 1 \leq 1 , hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves X 0 ( n ) X_0(n) naturally arises; this question has recently also been posed by Mazur. We answer Mazur’s question completely and describe the quadratic points on all the bielliptic modular curves X 0 ( n ) X_0(n) for which this has not been done already. The values of n n that we deal with are n = 60 n=60 , 62 62 , 69 69 , 79 79 , 83 83 , 89 89 , 92 92 , 94 94 , 95 95 , 101 101 , 119 119 and 131 131 ; the curves X 0 ( n ) X_0(n) are of genus up to 11 11 . We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box’s relative symmetric Chabauty method and an application of a moduli description of Q \mathbb {Q} -curves of degree d d with an independent isogeny of degree m m , which reduces the problem to finding the rational points on several quotients of modular curves.

We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.

Rational points on elliptic curves with complex multiplication by the ring of integers in [formula omitted](√−7)

The growth of the Mordell-Weil rank of an elliptic curve in a tower of number fields can be discussed at many levels.On the one hand, the issues raised can be embedded in a broader Iwasawa theory of elliptic curves (Mazur [10]); on the other hand, they can be crystallized in a single easily stated question, namely whether the rank of the elliptic curve over subextensions of finite degree is bounded in the tower.But whatever one's point of view, the case of abelian towers has seen important advances in recent years, including notably the results of Kato on cyclotomic towers, of Cornut [2] and Vatsal [18] on anticyclotomic towers, and of Skinner and Urban on Mazur's conjecture.In the case of nonabelian towers, by contrast, virtually nothing is known, and even a conjectural framework has begun to emerge only recently [1].However Greenberg [7] observed more than twenty years ago that some insight could already be gained from the classical conjectures about L-functions: They imply that for certain pairs (E 1 , E 2 ) of elliptic curves of relatively prime conductor there exist primes p such that the rank of E 1 is unbounded in the p-division tower of E 2 (which can be chosen not to have complex multiplication, so that the division tower is nonabelian).Greenberg's idea was elaborated further by L. Howe [8], who gave a conjectural lower bound for the growth of the rank of E 1 in the division tower of E 2 .The aim of the present note is to show that the standard conjectures also have some bearing on the growth of the rank of an elliptic curve in its own division tower.For example, we shall see that if E is a semistable elliptic curve over Q and p is a sufficiently large prime congruent to 3 mod 4 then the rank of E should be unbounded in its p-division tower.As in the papers of Greenberg and Howe, the mechanism for arriving at such conclusions is a root number calculation.Fix a number field F , an elliptic curve E over F , and a prime p, and put, where E[p ∞ ] denotes the group of points on E of p-power order.The root numbers at issue here are associated to the L-functions L(s, E, τ ), where τ runs over irreducible self-dual representations of Gal(F ∞ /F ).For a precise definition of L(s, E, τ ) we refer the reader to [12], pp.151 and 156, or to [1], §5, but we should at least spell out our group-theoretic conventions: First of all, a representation of a topological group is understood to be continuous, finite-dimensional, and defined over the complex numbers.In particular, in the case of a profinite group like Gal(F ∞ /F ) a representation is trivial on an open subgroup and hence factors through a finite quotient.It also follows that when the term character is used in the sense of "one-dimensional representation" we are

This is a survey of some recent developments in the theory of elliptic curves. After an informal discussion of the main theorems of the arithmetic side of the theory and the open problems confronting the subject, we describe the recent work of K. Rubin, V. Koly vagin, K. Murty and the author which establishes the finiteness of the Shafarevic-Tate group for modular elliptic curves of rank zero and one.

On the modularity of supersingular elliptic curves over certain totally real number fields

The generalized Rédei-matrix for function fields

Given an elliptic curve E/k and a Galois extension k ' /k, we construct an exact functor from torsion-free modules over the endomorphism ring EndE k ' with a semilinear Gal(k ' /k) action to abelian varieties over k that are k ' -isogenous to a power of E. As an application, we give a simple proof that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.

For a fixed j–invariant j0 of an elliptic curve without complex multiplication we bound the number of j–invariants j that are algebraic units and such that elliptic curves corresponding to j and j0 are isogenous. Our bounds are explicit. We also modify the problem slightly by fixing a singular modulus α and looking at all j such that $j-\alpha$ is an algebraic unit and such that elliptic curves corresponding to j and j0 are isogenous. The number of such j can again be bounded explicitly.