Sharp Szemerédi-Trotter Constructions from Arbitrary Number Fields

Estimating averages of Dirichlet convolutions $1\ast \unicode[STIX]{x1D712}$, for some real Dirichlet character $\unicode[STIX]{x1D712}$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin’s conjecture for Châtelet surfaces. We introduce a far-reaching generalisation of this problem, in particular replacing $\unicode[STIX]{x1D712}$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin’s conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number.

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.

In this paper we generalize, to an arbitrary number field, the theorem which gives the probability that two integers are relatively prime. The probability that two integers are relatively prime is 1/ζ(2), where ζ is the Riemann zeta function and 1/ζ(2)=6/π 2. The theorem for an arbitrary number field states that the probability that two ideals are relatively prime is the reciprocal of the zeta function of the number field evaluated at two. In particular, since the Gaussian integers are an unique factorization domain, we get the probability that two Gaussian integers are relatively prime is 1/ζ G (2) where ζ G is the zeta function associated with the Gaussian integers. In order to calculate the Gaussian probability, we use a theorem that enables us to factor the zeta function into a product of the Riemann zeta function and a Dirichlet series called an L-series. For the Gaussian integers we get: ζ G (2)=ζ(2)L(2,χ), where $$L\left( {2,\chi } \right) = 1 - \frac{1}{{3^2 }} + \frac{1}{{5^2 }} - \frac{1}{{7^2 }} + \cdots .$$ We use this factorization to approximate the Gaussian probability to 17 decimal places.

We present a new proof of a theorem of Schur's determining the least common multiple of the orders of all finite groups of complex $n \times n$-matrices whose elements have traces in the field of rational numbers. The basic method of proof goes back to Minkowski and proceeds by reduction to the case of finite fields. For the most part, we work over an arbitrary number field rather than the rationals. The first half of the article is expository and is intended to be accessible to graduate students and advanced undergraduates. It gives a self-contained treatment, following Schur, over the field of rational numbers.

In this paper, over an arbitrary number field, we prove subconvexity bounds for selfdual \operatorname{GL}_3 L -functions in the t -aspect and for self-dual \operatorname{GL}_3 \times \operatorname{GL}_2 L -functions in the \operatorname{GL}_2 Archimedean aspect.

Let K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

For an arbitrary (not totally real) number field K of degree ≥3, we ask how many perfect powers γ p of algebraic integers γ in K exist, such that μ(τ(γ p ))≤X for each embedding τ of K into the complex field. (X a large real parameter, p≥2 a fixed integer, and μ(z)=max(| Re (z)|,| Im (z)|) for any complex z.) This quantity is evaluated asymptotically in the form c p,K X n/p +R p,K (X), with sharp estimates for the remainder R p,K (X). The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation of algebraic numbers by rationals.

On the Waldspurger formula and the metaplectic Ramanujan conjecture over number fields

We give archimedean and non-archimedean constructions of Darmon points on modular abelian varieties attached to automorphic forms over arbitrary number fields and possibly non-trivial central character. An effort is made to present a unifying point of view, emphasizing the automorphic nature of the construction.

abstract: We construct $p$-adic $L$-functions for torsion classes for $\GL(n+1)\times\GL(n)$ and along the way prove new congruences between special values of Rankin-Selberg $L$-functions for $\GL(n+1)\times\GL(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\GL(n\!+\!1)\linebreak\times\GL(n)$. As an application, we establish generic non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.

Plectic Stark–Heegner points

Singular curves and the étale Brauer-Manin obstruction for surfaces

Plectic points were introduced by Fornea and Gehrmann [Adv. Math. 414 (2023)] as certain tensor products of local points on elliptic curves over arbitrary number fields F F . They are constructed in rank r ≤ [ F : Q ] r\leq [F:\mathbb {Q}] -situations, and they conjecturally come from p p -adic regulators of basis of the Mordell-Weil group defined over dihedral extensions of F F . In this article we define anticyclotomic p p -adic L-functions with variables of type weight and level attached to a family of overconvergent modular symbols defined over totally real fields F F and a quadratic extension K / F K/F . Their restriction to the weight space provides Hida-Rankin p p -adic L-functions. If such a family passes through an overconvergent modular symbol attached to a modular elliptic curve E E , we obtain a p p -adic Gross-Zagier formula that computes higher derivatives of such Hida-Rankin p p -adic L-functions in terms of plectic points. This result generalizes that of Bertolini and Darmon [Ann. of Math. (2) 170 (2009), pp. 343–370], which has been key to the recent approach towards the algebraicity of Darmon points.

A note on Roth's Theorem

A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields$K$for the quartic del Pezzo surface$S$of singularity type${\boldsymbol{A}}_{3}$with five lines given in${\mathbb{P}}_{K}^{4}$by the equations${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$.