Perturbations, Deformations, and Variations (and "Near-Misses") in Geometry, Physics, and Number Theory

Mathematic

Growth of Selmer Rank in Nonabelian Extensions of Number Fields

Let $p$ be an odd prime number, let E be an elliptic curve over a number field $k$, and let $F/k$ be a Galois extension of degree twice a power of p. We study the $Z_p$-corank $rk_p(E/F)$ of the $p$-power Selmer group of $E$ over $F$. We obtain lower bounds for $rk_p(E/F)$, generalizing the results in [MR], which applied to dihedral extensions. If $K$ is the (unique) quadratic extension of $k$ in $F$, if $G = Gal(F/K)$, if $G+$ is the subgroup of elements of $G$ commuting with a choice of involution of $F$ over $k$, and if $rk_p(E/K)$ is odd, then we show that (under mild hypotheses) $rkp(E/F)\ge[G:G+]$. As a very specific example of this, suppose that $A$ is an elliptic curve over $Q$ with a rational torsion point of order $p$ and without complex multiplication. If $E$ is an elliptic curve over $Q$ with good ordinary reduction at $p$ such that every prime where both $E$ and $A$ have bad reduction has odd order in $F\frac{x}{p}$ and such that the negative of the conductor of $E$ is not a square modulo $p$, then there is a positive constant $B$ depending on $A$ but not on $E$ or $n$ such that $rk_p(E/Q(A[p^n]))/geBp^{2n}$ for every $n$.Mathematic

Nearly Ordinary Galois Deformations over Arbitrary Number Fields

Let $K$ be an arbitrary number field, and let $\rho: Gal(K \bar/K) \rightarrow GL_2(E)$ be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of $\rho$. When $K$ is totally real and rho is modular, results of Hida imply that the nearly ordinary deformation space associated to rho 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 $\rho$ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of Leopoldt's 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 functorality", as well as some remarks on how our methods should apply to n-dimensional representations of Gal$(Q \bar/Q)$ when $n \lt 2$.Mathematic

On Mathematics, Imagination and the Beauty of Numbers

Mathematic

The mixing of interplanetary magnetic field lines: A significant transport effect in studies of the energy spectra of impulsive flares

Using instrumentation on board the ACE spacecraft we describe short-time scale (~3 hour) variations observed in the arrival profiles of ~20 keV nucleon^(–1) to ~2 MeV nucleon^(–1) ions from impulsive solar flares. These variations occurred simultaneously across all energies and were generally not in coincidence with any local magnetic field or plasma signature. These features appear to be caused by the convection of magnetic flux tubes past the observer that are alternately filled and devoid of flare ions even though they had a common flare source at the Sun. In these particle events we therefore have a means to observe and measure the mixing of the interplanetary magnetic field due to random walk. In a survey of 25 impulsive flares observed at ACE between 1997 November and 1999 July these features had an average time scale of 3.2 hours, corresponding to a length of ~0.03 AU. The changing magnetic connection to the flare site sometimes lead to an incomplete observation of a flare at 1 AU; thus the field-line mixing is an important effect in studies of impulsive flare energy spectra

Real Time Relativity: exploration learning of special relativity

Real Time Relativity is a computer program that lets students fly at relativistic speeds though a simulated world populated with planets, clocks, and buildings. The counterintuitive and spectacular optical effects of relativity are prominent, while systematic exploration of the simulation allows the user to discover relativistic effects such as length contraction and the relativity of simultaneity. We report on the physics and technology underpinning the simulation, and our experience using it for teaching special relativity to first year university students

Twisting Commutative Algebraic Groups

If $V$ is a commutative algebraic group over a field $k$, [View the MathML] source is a commutative ring that acts on $V$, and View the MathML source is a finitely generated free View the MathML source-module with a right action of the absolute Galois group of $k$, then there is a commutative algebraic group [View the MathML source] over $k$, which is a twist of a power of $V$. These group varieties have applications to cryptography (in the cases of abelian varieties and algebraic tori over finite fields) and to the arithmetic of abelian varieties over number fields. For purposes of such applications we devote this article to making explicit this tensor product construction and its basic properties.Mathematic

Visualizing elements of Sha[3] in genus 2 jacobians

Mazur proved that any element xi of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.Comment: 12 page