ARITHMETIC PROGRESSIONS ON CONIC SECTIONS

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.

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 main result of this note is that there are at most seven rational points (including the one at infinity) on the curve C_A with the affine equation y^2 = x^5 + A (where A is a tenth power free integer) when the Mordell-Weil rank of the Jacobian of C_A is one. This bound is attained for A = 18^2.

Using an elementary argument, we show that for all rational numbers α such that neither α nor 3α is a rational square, the equationx 4 -4αx 2 -4αy 2 +y 4 =-6α 2 has no rational solutions. This answers Hindes’ two questions and generalizes his theorem (Theorem 1.1) in “Rational points on certain families of symmetric equations”, Int. J. Number Theory 11 (2015), no. 6, pp. 1821–1838.

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

A quiver representation assigns a vector space to each vertex, and a linear map to each arrow of a quiver. When one considers the category $\mathrm {Vect}(\mathbb {F}_1)$ of vector spaces “over $\mathbb {F}_1$ ” (the field with one element), one obtains $\mathbb {F}_1$ -representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficient quivers. To be precise, we prove that the category $\mathrm {Rep}(Q,\mathbb {F}_1)$ is equivalent to the (suitably defined) category of coefficient quivers over Q. This provides a conceptual way to see Euler characteristics of a class of quiver Grassmannians as the number of “ $\mathbb {F}_1$ -rational points” of quiver Grassmannians. We generalize techniques originally developed for string and band modules to compute the Euler characteristics of quiver Grassmannians associated with $\mathbb {F}_1$ -representations. These techniques apply to a large class of $\mathbb {F}_1$ -representations, which we call the $\mathbb {F}_1$ -representations with finite nice length: we prove sufficient conditions for an $\mathbb {F}_1$ -representation to have finite nice length, and classify such representations for certain families of quivers. Finally, we explore the Hall algebras associated with $\mathbb {F}_1$ -representations of quivers. We answer the question of how a change in orientation affects the Hall algebra of nilpotent $\mathbb {F}_1$ -representations of a quiver with bounded representation type. We also discuss Hall algebras associated with representations with finite nice length, and compute them for certain families of quivers.

Quantization of linear maps on a torus-fresnel diffraction by a periodic grating

We study F-theory compactifications with U(1)xU(1) gauge symmetry on elliptically fibered Calabi-Yau manifolds with a rank two Mordell-Weil group. We find that the natural presentation of an elliptic curve E with two rational points and a zero point is the generic Calabi-Yau onefold in dP_2. We determine the birational map to its Tate and Weierstrass form and the coordinates of the two rational points in Weierstrass form. We discuss its resolved elliptic fibrations over a general base B and classify them in the case of B=P^2. A thorough analysis of the generic codimension two singularities of these elliptic Calabi-Yau manifolds is presented. This determines the general U(1)xU(1)-charges of matter in corresponding F-theory compactifications. The matter multiplicities for the fibration over P^2 are determined explicitly and shown to be consistent with anomaly cancellation. Explicit toric examples are constructed, both with U(1)xU(1) and SU(5)xU(1)xU(1) gauge symmetry. As a by-product, we prove the birational equivalence of the two elliptic fibrations with elliptic fibers in the two blow-ups Bl_(1,0,0)P^2(1,2,3) and Bl_(0,1,0)P^2(1,1,2) employing birational maps and extremal transitions.

Consider strong approximation for algebraic varieties defined over a number field k k . Let S S be a finite set of places of k k containing all archimedean places. Let E E be an elliptic curve of positive Mordell–Weil rank and let A A be an abelian variety of positive dimension and of finite Mordell–Weil group. For an arbitrary finite set T \mathfrak {T} of torsion points of E × A E\times A , denote by X X its complement. Supposing the finiteness of ⨿ ⨿ ( E × A ) {\amalg \kern -.25pc\amalg } (E\times A) , we prove that X X satisfies strong approximation with Brauer–Manin obstruction off S S if and only if the projection of T \mathfrak {T} to A A contains no k 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.

Elementary Methods in Number Theory begins with first course in number for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary number theory. In the second and third parts of the book, deep results in number theory are proved using only elementary methods. Part II is about multiplicative number theory, and includes two of the most famous results in mathematics: the Erdos-Selberg elementary proof of the prime number theorem, and Dirichlets theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive number theory: Warings problems for polynomials, Liouvilles method to determine the number of representations of an integer as the sum of an even number of squares, and the asymptotics of partition functions. Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets.

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.