Nearly ordinary Galois deformations over arbitrary number fields

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${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$.

Lifting Galois representations over arbitrary number fields

The goal of this article is to give an explicit classification of the\npossible $p$-adic Galois representations that are attached to elliptic curves\n$E$ with CM defined over $\\mathbb{Q}(j(E))$. More precisely, let $K$ be an\nimaginary quadratic field, and let $\\mathcal{O}_{K,f}$ be an order in $K$ of\nconductor $f\\geq 1$. Let $E$ be an elliptic curve with CM by\n$\\mathcal{O}_{K,f}$, such that $E$ is defined by a model over\n$\\mathbb{Q}(j(E))$. Let $p\\geq 2$ be a prime, let $G_{\\mathbb{Q}(j(E))}$ be the\nabsolute Galois group of $\\mathbb{Q}(j(E))$, and let $\\rho_{E,p^\\infty}\\colon\nG_{\\mathbb{Q}(j(E))}\\to \\operatorname{GL}(2,\\mathbb{Z}_p)$ be the Galois\nrepresentation associated to the Galois action on the Tate module $T_p(E)$. The\ngoal is then to describe, explicitly, the groups of\n$\\operatorname{GL}(2,\\mathbb{Z}_p)$ that can occur as images of\n$\\rho_{E,p^\\infty}$, up to conjugation, for an arbitrary order\n$\\mathcal{O}_{K,f}$.\n

Let $${f(x)=(x-a_1)\cdots (x-a_m)}$$ , where a 1, . . . , a m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether $${(f(x))^{2^k}+1}$$ is irreducible for every k ≥ 1. In 1919 Polya proved that if $${P(x)\in\mathbb{Z}[x]}$$ is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2−N N! where $${N=\lceil m/2\rceil}$$ , then P(x) is irreducible. A great number of authors have published results of Schur-type or Polya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Polya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.

The purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

We determine the universal deformation ring, in the sense of Mazur, of a residual representation $\bar \rho :G_K\to {\rm GL}_2(k)$ , where k is a finite field of characteristic p and K is a local field of residue characteristic p . As one might hope for, but is not proven in the global case, the deformation ring is a complete intersection, flat over W(k) , with the exact number of equations given by the dimension of $H^2(G_K,{\rm ad}_{\bar \rho})$ . We then go on to determine the ordinary locus inside the deformation space and, using ideas of Mazur, apply this to compare the universal and the universal ordinary deformation spaces. Provided that the universal ring for ordinary deformations with fixed determinant is finite flat over W(k) , as was shown in many cases by Diamond, Fujiwara, Taylor–Wiles and Wiles, we show that the corresponding universal deformation ring – with no restriction of ordinariness or fixed determinant – is a complete intersection, finite flat over W(k) of the dimension conjectured by Mazur, provided that the restriction of $\det (\bar \rho)$ to the inertia subgroup is different from the inverse cyclotomic character.

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.

On the Waldspurger formula and the metaplectic Ramanujan conjecture over 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.

Kneser's method of constructing adjacent lattices will be used to determine class numbers of unimodular positive definite hermitian lattices of rank 2 and 3 over rings of integers in some imaginary quadratic fields. The same method will also be applied in order to construct indecomposable unimodular positive definite hermitian lattices of rank 2 and 3 over almost all orders in imaginary quadratic fields, even non-maximal ones. All exceptional cases will be determined explicitly.

The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known, see G. Lettl, A. Pethő and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1999) 1871–1894, On the arithmetic of simplest sextic fields and related Thue equations, in Number Theory: Diophantine, Computational and Algebraic Aspects, eds. K. Győry et al. (de Gruyter, Berlin, 1998), pp. 331–348. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper, we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic 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

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.