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V. Vu has recently shown that when k≥2 and s is sufficiently large in terms of k, then there exists a set (k), whose number of elements up to t is smaller than a constant times (t log t)1/s, for which all large integers n are represented as the sum of skth powers of elements of (k) in order log n ways. We establish this conclusion with s∼k log k, improving on the constraint implicit in Vu's work which forces s to be as large as k48k. Indeed, the methods of this paper show, roughly speaking, that whenever existing methods permit one to show that all large integers are the sum of H(k) kth powers of natural numbers, then H(k)+2 variables suffice to obtain a corresponding conclusion for "thin sets," in the sense of Vu.
The problem of estimating the number of imaginary quadratic fields whose ideal class group has an element of order ℓ≥2$ is classical in number theory. Analogous questions for quadratic twists of elliptic curves have been the focus of recent interest. Whereas works of C. Stewart and J. Top [ST], and of F. Gouvêa and B. Mazur [GM] address the nontriviality of Mordell-Weil groups, less is known about the nontriviality of Shafarevich-Tate groups. Here we obtain a new type of result for the nontriviality of class groups of imaginary quadratic fields using the circle method, and then we combine it with works of G. Frey [F], V. Kolyvagin [K], and K. Ono [O2] to obtain results on the nontriviality of Shafarevich-Tate groups of certain elliptic curves. For E=X0 (11), these results imply that
Un cône ouvert proprement convexe C de ℝm est dit divisible si il existe un sousgroupe discret Γ de GL(ℝm) qui préserve C et tel que quotient Γ\C est compact. Nous décrivons l'adhérence de Zariski d'un tel groupe Γ.
Nous montrons que si C n'est ni un produit ni un cône symétrique alors Γ est Zariski dense dans GL(ℝm).
A properly convex open cone in ℝm is called divisible if there exists a discrete subgroup Γ of GLℝm preserving C such that the quotient Γ\C is compact. We describe the Zariski closure of such a group Γ.
We show that if C is divisible but is neither a product nor a symmetric cone, then Γ is Zariski dense in GLℝm.
Let G=ResE/FH, where H is a connected reductive group over a number field F and E/F is a quadratic extension. We define the regularized period of an automorphic form of G relative to H, and we express the regularized period of cuspidal Eisenstein series in terms of intertwining periods, which are relative analogues of the standard intertwining operators. This leads to an analogue of the Maass-Selberg relations. The regularized periods appear in the contribution of the continuous spectrum to the relative trace formula.