Annales Mathematiques du Quebec最新文献

Let (f(z)=q+sum _{nge 2}a(n)q^n) be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for (k=2) by ruling out or locating all odd prime values (|ell |<100) of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights (kge 1) newforms where the nebentypus is given by a quadratic Dirichlet character.

On the canonical 2-sphere and for Schrödinger eigenfunctions, we obtain a simple geometric criterion on the potential under which we can improve, near a given point and for every (pne 6), Sogge’s estimates by a power of the eigenvalue. This criterion can be formulated in terms of the critical points of the Radon transform of the potential and it is independent of the choice of eigenfunctions.

In a previous article [7], the author proves that the value of the root number varies in a non-isotrivial family of elliptic curves indexed by one parameter t running through ({mathbb {Q}}). However, a well-known example of Washington has root number (-1) for every fiber when t runs through ({mathbb {Z}}). Such examples are rare since, as proven in this paper, the root number of the integer fibers varies for a large class of families of elliptic curves. This result depends on the squarefree conjecture and Chowla’s conjecture, and is unconditional in many cases.

In this paper we consider the Diophantine equation (U_n=p^x) where (U_n) is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on (U_n), we show that, for any p outside of an effectively computable finite set of prime numbers, there exists at most one solution (n, x) to that Diophantine equation. We compute this exceptional set for the Tribonacci sequence and for the Lucas sequence plus one.

In this paper, we study an almost coKähler manifold admitting certain metrics such as (*)-Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKähler 3-manifold (M, g) admitting a (*)-Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKähler ((kappa ,mu ))-almost coKähler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a ((kappa , mu ))-almost coKähler manifold (M, g) is coKähler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKähler manifolds which are non-coKähler.

Based on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp (tau = 0). As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

We give a criterion for the transversality of a curve embedded in a product of elliptic curves. We then apply our criterion to some explicit classes of curves. The transversality allows us to apply theorems that produce explicit and implementable bounds for the height of the rational points on the curves.

We give an example of a bounded divergence free autonomous vector field in ({mathbb {R}}^3) (and of a nonautonomous bounded divergence free vector field in ({mathbb {R}}^2)) and of a smooth initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.

Given a prime p, a number field ({K}) and a finite set of places S of ({K}), let ({K}_S) be the maximal pro-p extension of ({K}) unramified outside S. Using the Golod–Shafarevich criterion one can often show that ({K}_S/{K}) is infinite. In both the tame and wild cases we construct infinite subextensions with bounded ramification using the refined Golod–Shafarevich criterion. In the tame setting we are able to produce infinite asymptotically good extensions in which infinitely many primes split completely, and in which every prime has Frobenius of finite order, a phenomenon that had been expected by Ihara. We also achieve new records on Martinet constants (root discriminant bounds) in the totally real and totally complex cases.