Quantum effects on Lagrangian points and displaced periodic orbits in the Earth-Moon system

Recent work in the literature has shown that the one-loop long distance quantum corrections to the Newtonian potential imply tiny but observable effects in the restricted three-body problem of celestial mechanics, i.e., at the Lagrangian libration points of stable equilibrium the planetoid is not exactly at equal distance from the two bodies of large mass, but the Newtonian values of its coordinates are changed by a few millimeters in the Earth-Moon system. First, we assess such a theoretical calculation by exploiting the full theory of the quintic equation, i.e., its reduction to Bring-Jerrard form and the resulting expression of roots in terms of generalized hypergeometric functions. By performing the numerical analysis of the exact formulas for the roots, we confirm and slightly improve the theoretical evaluation of quantum corrected coordinates of Lagrangian libration points of stable equilibrium. Second, we prove in detail that also for collinear Lagrangian points the quantum corrections are of the same order of magnitude in the Earth-Moon system. Third, we discuss the prospects to measure, with the help of laser ranging, the above departure from the equilateral triangle picture, which is a challenging task. On the other hand, a modern version of the planetoid is the solar sail, and much progress has been made, in recent years, on the displaced periodic orbits of solar sails at all libration points, both stable and unstable. The present paper investigates therefore, eventually, a restricted three-body problem involving Earth, Moon and a solar sail. By taking into account the one-loop quantum corrections to the Newtonian potential, displaced periodic orbits of the solar sail at libration points are again found to exist

Torsion Subgroups of Rational Elliptic Curves Over Odd Degree Galois Fields

The Mordell-Weil Theorem states that if K is a number field and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of K = Q, it is not known which ranks are possible or if the ranks are bounded. However, there have been great strides in determining the sets E(K)_tors. Progress began in 1977 with Mazur\u27s classification of the possible torsion subgroups E(Q)_tors for rational elliptic curves, and there has since been an explosion of classifications. Inspired by work of Chou, González Jiménez, Lozano-Robledo, and Najman, the purpose of this work is to classify the set Φ_Q^{Gal}(9), i.e. the set of possible torsion subgroups for rational elliptic curves over nonic Galois fields. We not only completely determine the set Φ_Q^{Gal}(9), but we also determine the possible torsion subgroups based on the isomorphism type of Gal(K/Q). We then determine the possibilities for the growth of torsion from E(Q)_tors to E(K)_tors, i.e. what the possibilities are for E(K)_tors ⊇ E(Q)_tors given a fixed torsion subgroup E(Q)_tors. Extending the techniques used in the classification of Φ_Q^{Gal}(9), we then determine the possible structures over all odd degree Galois fields. Finally, we explicitly determine the sets Φ_Q^{Gal}(d) for all odd d based on the prime factorization for d while proving a number of other related results

Hurwitz Number Fields

The canonical covering maps from Hurwitz varieties to configuration varieties are important in algebraic geometry. The scheme-theoretic fiber above a rational point is commonly connected, in which case it is the spectrum of a Hurwitz number field. We study many examples of such maps and their fibers, finding number fields whose existence contradicts standard mass heuristics

Lines on the Dwork Pencil of Quintic Threefolds

We present an explicit parametrization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves which parametrize the lines. These curves are 125:1 covers of certain genus six curves. These genus six curves are first presented as curves in P^1*P^1 that have three nodes. It is natural to blow up P^1*P^1 in the three points corresponding to the nodes in order to produce smooth curves. The result of blowing up P^1*P^1 in three points is the quintic del Pezzo surface dP_5, whose automorphism group is the permutation group S_5, which is also a symmetry of the pair of genus six curves. The subgroup A_5, of even permutations, is an automorphism of each curve, while the odd permutations interchange the two curves. The ten exceptional curves of dP_5 each intersect each of the genus six curves in two points corresponding to van Geemen lines. We find, in this way, what should have anticipated from the outset, that the genus six curves are the curves of the Wiman pencil. We consider the family of lines also for the cases that the manifolds of the Dwork pencil become singular. For the conifold the genus six curves develop six nodes and may be resolved to a P^1. The group A_5 acts on this P^1 and we describe this action.Comment: 48 pages, 2 figure

Families of L-functions and their Symmetry

In [90] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [33], [107], [67] [47], [66] and especially [98] by the second and third-named authors which make it possible to give a conjectural answer for the symmetry type of a family and in particular the universality class predicted in [64] for the distribution of the zeros near s=1/2. In this note we carry this out after introducing some basic invariants associated to a family

Swampland conjectures as generic predictions of quantum gravity

The swampland program aims at answering the question of whether and how effective quantum field theories coupled to gravity can be UV-completed. Our limited understanding of the issues that can arise in this process is reflected in a steadily growing zoo of swampland conjectures. These are supposed to be necessary criteria for such a UV-completion and are motivated from our understanding of black hole thermodynamics, holography and string theory. The swampland conjectures can potentially have dramatic implications for physical models of real-world phenomena if they are proven in string theory. For example, they could rule out large field inflation, a cosmological constant, or non-vanishing masses of the standard model photon and graviton. The purpose of this work is to contribute to a better understanding of these conjectures by testing them in various corners of string theory. Furthermore, we reveal a complicated network of relations between the swampland conjectures. This hints at the existence of a deeper underlying structure, which is yet to be fully uncovered. One of the suggested swampland conjectures is the distance conjecture. It states that effective field theories have a finite range of validity in scalar field space, after which they necessarily break down due to an infinite tower of states becoming light. We quantify this range and identify the tower of states in the context of moduli spaces of Calabi-Yau compactifications with N = 2 supersymmetry. We claim that an analogous tower of states appears also in the limit where we send the mass of a spin-2 field to zero. We concretize this expectation in form of a spin-2 swampland conjecture. Finally, we investigate the question of whether the KKLT construction of de Sitter vacua in string theory is consistent. In this way, we challenge a recently proposed de Sitter swampland conjecture, which claims that de Sitter space cannot be a vacuum of string theory

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes asymptotics for field extensions and class numbers, random matrices and L-functions, rational points on curves and higher-dimensional varieties, and aspects of lattice basis reduction