The yoga of the Cassels-Tate pairing

Cassels has described a pairing on the 2-Selmer group of an elliptic curve which shares some properties with the Cassels-Tate pairing. In this article, we prove that the two pairings are the same.Comment: 8 page

Sato-Tate distributions of twists of y^2=x^5-x and y^2=x^6+1

We determine the limiting distribution of the normalized Euler factors of an abelian surface A defined over a number field k when A is isogenous to the square of an elliptic curve defined over k with complex multiplication. As an application, we prove the Sato-Tate Conjecture for Jacobians of Q-twists of the curves y^2=x^5-x and y^2=x^6+1, which give rise to 18 of the 34 possibilities for the Sato-Tate group of an abelian surface defined over Q. With twists of these two curves one encounters, in fact, all of the 18 possibilities for the Sato-Tate group of an abelian surface that is isogenous to the square of an elliptic curve with complex multiplication. Key to these results is the twisting Sato-Tate group of a curve, which we introduce in order to study the effect of twisting on the Sato-Tate group of its Jacobian.Comment: minor edits, 42 page

Expansion of a compressible gas in vacuum

Tai-Ping Liu \cite{Liu\_JJ} introduced the notion of "physical solution' of the isentropic Euler system when the gas is surrounded by vacuum. This notion can be interpreted by saying that the front is driven by a force resulting from a H\"older singularity of the sound speed. We address the question of when this acceleration appears or when the front just move at constant velocity. We know from \cite{Gra,SerAIF} that smooth isentropic flows with a non-accelerated front exist globally in time, for suitable initial data. In even space dimension, these solutions may persist for all $t\in\R$ ; we say that they are {\em eternal}. We derive a sufficient condition in terms of the initial data, under which the boundary singularity must appear. As a consequence, we show that, in contrast to the even-dimensional case, eternal flows with a non-accelerated front don't exist in odd space dimension. In one space dimension, we give a refined definition of physical solutions. We show that for a shock-free flow, their asymptotics as both ends $t\rightarrow\pm\infty$ are intimately related to each other

Endomorphisms of abelian varieties, cyclotomic extensions and Lie algebras

We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.Comment: 9 page

Essential dimension of abelian varieties over number fields

We affirmatively answer a conjecture in the preprint ``Essential dimension and algebraic stacks,'' proving that the essential dimension of an abelian variety over a number field is infinite.Comment: 4 pages. To appear in C. R. Math. Acad. Sci. Paris. Preprint posted earlier to http://www.mathematik.uni-bielefeld.de/LAG

Modeling aeroacoustic excitations by subsonic wave packets in the Kirchhoff formalism

The present work aims at linking the shape of a wave packet to its acoustic e ciency through numerical integration of the homogeneous Helmholtz equation in the Kirchho formalism. We shall consider the case of a bidimensional, rectangular domain where one side is a ected by a spatially evolving wave packet as a boundary condition with subsonic convection velocity. In the rst place, the numerical tool is validated by comparison with analytical developments available in literature. Then, an extension to more advanced forms of wave packet is approached. The appearance of a criterion for acoustic e ciency is discussed, as well as the need for a proper de nition regarding the acoustic energy of a wave packet and its radiated energy output