Rational Points Research Articles

Torsion points on elliptic curves over number fields of small degree

We determine the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$, for $d = 4$, $5$, $6$, and $7$.

Rational points on Atkin–Lehner quotients of geometrically hyperelliptic Shimura curves

Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves X0(D,N). In this paper we compute the Q-rational points on the Atkin–Lehner quotients of these curves using a variety of techniques. We also determine which rational points are CM for many of these curves.

Counting rational points on projective varieties

AbstractWe develop a global version of Heath‐Brown's p‐adic determinant method to study the asymptotic behaviour of the number N(W; B) of rational points of height at most B on certain subvarieties W of Pn defined over Q. The most important application is a proof of the dimension growth conjecture of Heath‐Brown and Serre for all integral projective varieties of degree d ≥ 2 over Q. For projective varieties of degree d ≥ 4, we prove a uniform version N(W; B) = Od,n,ε(Bdim W+ε) of this conjecture. We also use our global determinant method to improve upon previous estimates for quasi‐projective surfaces. If, for example, is the complement of the lines on a non‐singular surface X ⊂ P3 of degree d, then we show that . For surfaces defined by forms with non‐zero coefficients, then we use a new geometric result for Fermat surfaces to show that for B ≥ e.

Rational points on [formula omitted

We compute the rational points on the Atkin–Lehner quotient X0+(125) using the quadratic Chabauty method. Our work completes the study of exceptional rational points on the curves X0+(N) of genus between 2 and 6. Together with the work of several authors, this completes the proof of a conjecture of Galbraith.

Quadratic Chabauty and 𝑝-adic Gross–Zagier

Let X X be a quotient of the modular curve X 0 ( N ) X_0(N) whose Jacobian J X J_X is a simple factor of J 0 ( N ) n e w J_0(N)^{new} over Q \mathbf {Q} . Let f f be the newform of level N N and weight 2 2 associated with J X J_X ; assume f f has analytic rank 1 1 . We give analytic methods for determining the rational points of X X using quadratic Chabauty by computing two p p -adic Gross–Zagier formulas for f f . Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotomic p p -adic L L -function of f f constructed by Bertolini, Darmon, and Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148], which lies outside of the range of interpolation.

A unifying approach to tropicalization

In this paper, we introduce ordered blueprints and ordered blue schemes, which serve as a common language for the different approaches to tropicalizations and which enhance tropical varieties with a schematic structure. As an abstract concept, we consider a tropicalization as a moduli problem about extensions of a given valuation v : k → T v:k\to T between ordered blueprints k k and T T . If T T is idempotent, then we show that a generalization of the Giansiracusa bend relation leads to a representing object for the tropicalization, and that it has yet another interpretation in terms of a base change along v v . We call such a representing object a scheme theoretic tropicalization. This theory recovers and improves other approaches to tropicalizations as we explain with care in the second part of this text. The Berkovich analytification and the Kajiwara-Payne tropicalization appear as rational point sets of a scheme theoretic tropicalization. The same holds true for its generalization by Foster and Ranganathan to higher rank valuations. The scheme theoretic Giansiracusa tropicalization can be recovered from the scheme theoretic tropicalizations in our sense. We obtain an improvement due to the resulting blueprint structure, which is sufficient to remember the Maclagan-Rincón weights. The Macpherson analytification has an interpretation in terms of a scheme theoretic tropicalization, and we give an alternative approach to Macpherson’s construction of tropicalizations. The Thuillier analytification and the Ulirsch tropicalization are rational point sets of a scheme theoretic tropicalization. Our approach yields a generalization to any, possibly non-trivial, valuation v : k → T v:k\to T with idempotent T T and enhances the tropicalization with a schematic structure.

Effect of clopidogrel on post extraction clotting in patients on dual antiplatelet therapy

Purpose: To evaluate the association between the platelet inhibition rate of clopidogrel (CLO-PIR) and post-extraction clotting status in DAPT patients.
 Methods: Ninety (90) eligible patients scheduled for a single tooth extraction were enrolled in this study. The CLO-PIR and platelet inhibition rate of aspirin (ASA-PIR) were determined by thromboelastography platelet mapping assay. Post-extraction clotting assessments were performed, and a complete intra-alveolar clot formation within 30 min post-operation was defined as normal clotting. For clot formation exceeding 30 min, it was defined as prolonged bleeding.
 Results: At a similar level of ASA-PIR, a higher proportion of patients with normal CLO-PIR (≤ 75 %) exhibited normal clotting, compared with those featuring high CLO-PIR (>75 %, p < 0.001). However, in patients with similar CLO-PIRs, the clotting results varied insignificantly, with increase in ASA-PIR. The effect of CLO-PIR was further validated using logistic regression analysis (odds ratio = 1.071, 95 % confidence interval: 1.024 - 1.120, p = 0.003), and receiver operating characteristic curve analysis revealed that a 78.6 % CLO-PIR was the rational cut-off point.
 Conclusion: This study preliminarily demonstrates the prominence of high clopidogrel responsiveness in slowing the post-extraction clotting process in DAPT patients.

Rational points on certain homogeneous varieties

Let L be a simply-connected simple connected algebraic group over a number field F, and H be a semisimple absolutely maximal connected F-subgroup of L. Let $$\Delta (H)$$ be the image of H diagonally embedded in $$L^n$$ . Under a cohomological condition, we prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of $$\Delta (H)\backslash L^n$$ with respect to a balanced line bundle.

Diophantine approximation on conics

Given a conic C \mathcal {C} over Q \mathbb {Q} , it is natural to ask what real points on C \mathcal {C} are most difficult to approximate by rational points of low height. For the analogous problem on the real line (for which the least approximable number is the golden ratio, by Hurwitz’s theorem), the approximabilities comprise the classically studied Lagrange and Markoff spectra, but work by Cha–Kim [Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum, arXiv: 1903.02882, 2021] and Cha–Chapman–Gelb–Weiss [Monatsh. Math. 197 (2022), pp. 1–55] shows that the spectra of conics can vary. We provide notions of approximability, Lagrange spectrum, and Markoff spectrum valid for a general C \mathcal {C} and prove that their behavior is exhausted by the special family of conics C n : X Z = n Y 2 \mathcal {C}_n : XZ = nY^2 , which has symmetry by the modular group Γ 0 ( n ) \Gamma _0(n) and whose Markoff spectrum was studied in a different guise by A. Schmidt and Vulakh [Trans. Amer. Math. Soc. 352 (2000), pp. 4067–4094]. The proof proceeds by using the Gross-Lucianovic bijection to relate a conic to a quaternionic subring of Mat 2 × 2 ⁡ ( Z ) \operatorname {Mat}^{2\times 2}(\mathbb {Z}) and classifying invariant lattices in its 2 2 -dimensional representation.

Quartic surfaces, their bitangents and rational points

Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface, having vanishing irregularity, contains only finitely many rational points. In our proof, we use the geometry of lines of the quartic double solid associated to X. In a somewhat opposite direction, we show that on any quartic surface X over a number field K, the set of algebraic points in X(\overeline K) which are quadratic over a suitable finite extension K' of K is Zariski-dense.

Four-loop HQET propagators from the DRA method

We use dimensional recurrence relations and analyticity to calculate four-loop propagator-type master integrals in the heavy-quark effective theory. Compared to previous applications of the DRA method, we apply a new technique of fixing homogeneous solutions from pole parts of integrals evaluated in different rational space-time dimension points. The latter were calculated from the integration-by-parts reduction of finite integrals in shifted space-time dimension and/or with increased propagators powers. We provide results for epsilon expansions of master integrals near d = 4 and d = 3 using constructed alternative sets of integrals with expansion coefficients having conjectural uniform transcendental weight.

Quadratic points on bielliptic modular curves

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.

Height pairings of 1-motives

The purpose of this work is to generalize, in the context of 1-motives, the p -adic height pairings constructed by B. Mazur and J. Tate on abelian varieties. Following their approach, we define a global pairing between the rational points of a 1-motive and its dual. We also provide a local pairing between disjoint zero-cycles of degree zero on a curve, which is done by considering the Picard and Albanese 1-motives associated to the curve.

Geometric quadratic Chabauty over number fields

This article generalizes the geometric quadratic Chabauty method, initiated over Q \mathbb {Q} by Edixhoven and Lido, to curves defined over arbitrary number fields. The main result is a conditional bound on the number of rational points on curves that satisfy an additional Chabauty type condition on the Mordell–Weil rank of the Jacobian. The method gives a more direct approach to the generalization by Dogra of the quadratic Chabauty method to arbitrary number fields.

Quadrature formulas for Bessel polynomials

A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring’s problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz–Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves.

Bounds on the number of rational points of curves in families

AbstractIn this note, we give an alternative proof of uniform boundedness of the number of integral points of smooth projective curves over a fixed number field with good reduction outside of a fixed set of primes. We use that due to Bertin–Romagny, the Kodaira–Parshin families constructed by Lawrence–Venkatesh can themselves be assembled into a family. We then repeat Lawrence–Venkatesh's study of the ‐adic period map, together with the comparison of nearby fibres.

Heights on stacks and a generalized Batyrev–Manin–Malle conjecture

Abstract We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $\mathcal {X}$ , which specializes to the Batyrev–Manin conjecture when $\mathcal {X}$ is a scheme and to Malle’s conjecture when $\mathcal {X}$ is the classifying stack of a finite group.

Rational points of algebraic varieties: a homotopical approach

This article, dedicated to the 100-th anniversary of I. R. Shafarevich, is a survey of techniques of homotopical algebra, applied to the problem of distribution of rational points on algebraic varieties. We due to I. R. Shafarevich, jointly with J. Tate, one of the breakthrough discoveries in this domain: construction of the so-called Shafarevich-Tate groups and the related obstructions to the existence of rational points. Later it evolved into the theory of Brauer-Manin obstructions. Here we focus on some facets of the later developments in Diophantine geometry: the study of the distribution of rational points on them. More precisely, we show how the definition of accumulating subvarieties, based upon counting the number of points whose height is bounded by varying $H$, can be encoded by a special class of categories in such a way that the arithmetical invariants of varieties are translated into homotopical invariants of objects and morphisms of these categories. The central role in this study is played by the structure of an assembler (I. Zakharevich) in general, and a very particular case of it, an assembler on the family of unions of half-open intervals $(a,b]$ with rational ends.

Rational points of algebraic varieties: a homotopical approach

This article, dedicated to the 100 th anniversary of I. R. Shafarevich, is a survey of techniques of homotopical algebra, applied to the problem of distribution of rational points on algebraic varieties. We due to I. R. Shafarevich, jointly with J. Tate, one of the breakthrough discoveries in this domain: construction of the so-called Shafarevich-Tate groups and the related obstructions to the existence of rational points. Later it evolved into the theory of Brauer-Manin obstructions. Here we focus on some facets of the later developments in Diophantine geometry: the study of the distribution of rational points on them. More precisely, we show how the definition of accumulating subvarieties, based upon counting the number of points whose height is bounded by varying $H$, can be encoded by a special class of categories in such a way that the arithmetical invariants of varieties are translated into homotopical invariants of objects and morphisms of these categories. The central role in this study is played by the structure of an assembler (I. Zakharevich) in general, and a very particular case of it, an assembler on the family of unions of half-open intervals $(a,b]$ with rational ends.

Equivariant derived equivalence and rational points on K3 surfaces

Equivariant derived equivalence and rational points on K3 surfaces