PGL\u3csub\u3e2\u3c/sub\u3e(F\u3csub\u3el\u3c/sub\u3e) Number Fields with Rational Companion Forms

We give a list of PGL2(Fl) number fields for ℓ ≥ 11 which have rational companion forms. Our list has fifty-three fields and seems likely to be complete. Some of the fields on our list are very lightly ramified for their Galois group

Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.Comment: 35 pages, 4 figures, LaTe

Ranks of twists of elliptic curves and Hilbert's Tenth Problem

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert's Tenth Problem has a negative answer over the ring of integers of every number field.Comment: Minor changes. To appear in Inventiones mathematica

Congruences for Fourier coefficients of half-integral weight modular forms and special values of L-functions

Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall ex-hibit congruences for Fourier coefficients of a slightly different type. Let f(z) =P∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coef-ficients. If ` is prime, then we shall be interested in congruences of the form a(`N) ≡ 0 mod ` where N is any quadratic residue (resp. non-residue) modulo `. For every prime `> 3 we exhibit a natural holomorphic weight ` 2 +1 modular form whose coefficients satisfy the congruence a(`N) ≡ 0 mod ` for every N satisfying `−

Counting points on cubic surfaces, I

Let V be a nonsingular cubic surface defined over Q, let U be the open subset of V obtained by deleting the 27 lines, and denote by N(U, H) the number of rational points in U of height less than H. Manin has conjectured that if V(Q) is not empty then (1) N(U, H) = C1H(log H)(r-1)(1 + o(1)) for some C-1 > 0, where r is the rank of NS(V/Q), the Neron-Severi group of V over Q. In this note we consider the special case when V contains two rational skew lines; and we prove that for some C-2 > 0 and all large enough H, N(U, H) > C2H(log H)(r-1) This is the one-sided estimate corresponding toll). It seems probable that the arguments in this paper could be modified to prove the corresponding result when V contains two skew lines conjugate over Q and each defined over a quadratic extension of Q but we have not attempted to write out the details

Analytic theory of Abelian varieties

This introduction presupposes little more than a basic course in complex variables

A brief guide to algebraic number theory

Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author