Bivariate polynomial injections and elliptic curves

We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the étale Brauer–Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then further consider properties of the distribution of the set of rational points with respect to its image in the rational points of the elliptic curve. In the process, we prove results on a local-to-global principle for torsion points on elliptic curves over \({{{\mathbb {Q}}}}\).

The General Case of S. Lang's Conjecture

We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic varietyXX. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms ofXX(existence of rational points, existence of a dense set of rational points, etc.). We give applications of this equivalence in both directions to study versality of group actions and rational points on algebraic varieties. We obtain similar results onpp-versality for a prime integerpp. An appendix, containing a letter from J.-P. Serre, puts the notion of versality in a historical perspective.

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

An elliptic curve can be defined as a smooth projective curve of degree 3 in the projective plane with a distinguished point chosen on it. The set of points on the curve can thus be endowed with a natural additive group structure. The most concrete description of an elliptic curve comes from its affine equation, written as $$y^2=x^3+ax+b,\quad \text{with}\ 4a^3+27b^2\not=0.$$ The theory of elliptic curves is a marvelous mixture of elementary mathematics and profound, advanced mathematics, a mixture which moreover lies on the crossroads of multiple theories: arithmetic, algebraic geometry, group representations, complex analysis, etc. Here, we will provide an introduction to the subject and prove the main Diophantine theorems: the group of rational points is finitely generated (the Mordell-Weil theorem) and the set of integral points is finite (Siegel’s theorem). Finally, we will evoke the famous theorem of Wiles—whose proof resulted in the proof of Fermat’s last theorem—and the Birch & Swinnerton-Dyer conjecture.

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.

In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion point. This family is a small family, as opposed to the large families considered by the aforementioned authors. We prove the analogous counting theorem for elliptic curves with so-called square-free index as well as for curves with suitably bounded Szpiro ratios. We note that the assumptions we impose on the size of the Szpiro ratios are less stringent than would be expected by the naive generalization of their approach; this requires an application of a theorem of Browning and Heath-Brown. Our proof also relies on linear programming techniques.

Let C be a smooth genus one curve described by a quartic polynomial equation over the rational field \({{\mathbb {Q}}}\) with \(P\in C({{\mathbb {Q}}})\). We give an explicit criterion for the divisibility-by-2 of a rational point on the elliptic curve (C, P). This provides an analogue to the classical criterion of the divisibility-by-2 on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational D(q)-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that although the rational \( D(16t+9) \)-quadruple \(\{t, 16t+8,2 25t+14, 36t+20 \}\) can not be extended to a polynomial \( D(16t+9) \)-quintuple using a linear polynomial, there are infinitely many rational values of t for which the aforementioned rational \( D(16t+9) \)-quadruple can be extended to a rational \( D(16t+9) \)-quintuple. Moreover, these infinitely many values of t are parametrized by the rational points on a certain elliptic curve of positive Mordell–Weil rank.

For any field K, a “K-curve” is an elliptic curve E defined over some finite separable extension of K such that every Galois conjugate of E is isogenous with E over a separable closure of K. A K-curve must either have complex multiplication (CM) or be isogenous to a collection of K-curves parametrized by a K-rational point on some modular curve X* (N). We give the first proof of this result, obtained in 1993 but not heretofore published. We also indicate some of the reasons for interest in K-curves, and in particular in Q-curves, and give some computational examples and open questions.

We construct rational Poncelet configurations, which means finite sets of pairwise distinct [Formula: see text]-rational points [Formula: see text] in the plane such that all [Formula: see text] are on a fixed conic section defined over [Formula: see text], and moreover the lines [Formula: see text] are all tangent to some other fixed conic section defined over [Formula: see text]. This is done for [Formula: see text] in which case only [Formula: see text] and [Formula: see text] are possible, and for certain real quadratic number fields [Formula: see text]; here moreover [Formula: see text] and [Formula: see text] and [Formula: see text] occur, but no further new values of [Formula: see text]. In fact, for every pair [Formula: see text] presented here, we show that infinitely many such tuples [Formula: see text] exist. The construction uses elliptic curves [Formula: see text] over [Formula: see text] such that the group [Formula: see text] is infinite and moreover contains a point of exact order [Formula: see text]. As an aside, a formulation of Mazur’s theorem/Ogg’s conjecture in terms of arbitrary genus one curves over the rational numbers (so not necessarily containing any rational point) is presented, since this occurs naturally in the context of Poncelet configurations.

Given a quintic number field K/ℚ, we study the set of irreducible trinomials, polynomials of the form x5 + ax + b, that have a root in K. We show that there is a genus 4 curve CK whose rational points are in bijection with such trinomials. This curve CK maps to an elliptic curve defined over a number field, and using this map, we are able (in some cases) to determine all the rational points on CK using elliptic curve Chabauty.

On the elliptic curve analogue of the sum-product problem

Injectivity of the specialization homomorphism of elliptic curves

We consider smooth projective curves C/𝔽 over a finite field and their symmetric squares C (2) . For a global function field K/𝔽, we study the K-rational points of C (2) . We describe the adelic points of C (2) surviving Frobenius descent and how the K-rational points fit there. Our methods also lead to an explicit bound on the number of K-rational points of C (2) satisfying an additional condition. Some of our results apply to arbitrary constant subvarieties of abelian varieties, however we produce examples which show that not all of our stronger conclusions extend.

Richard Guy asked for the largest set of points which can be placed in the plane so that their pairwise distances are rational numbers. In this article, we consider such a set of rational points restricted to a given hyperbola. To be precise, for rational numbers a, b, c, and d such that the quantity D = (ad - bc)/(2a2) is defined and non-zero, we consider rational distance sets on the conic section axy + bx + cy + d = 0. We show that, if the elliptic curve Y2 = X3 - D2X has infinitely many rational points, then there are infinitely many sets consisting of four rational points on the hyperbola such that their pairwise distances are rational numbers. We also show that any rational distance set of three such points can always be extended to a rational distance set of four such points.