Some new results concerning the equation $sigma(N)=aM, sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.
These are notes of my lectures at the summer school Higher-dimensional geometry over finite fields in Goettingen, June--July 2007. We present a proof of Tates theorem on homomorphisms of abelian varieties over finite fields (including the $ell=p$ c
ase) that is based on a quaternion trick. In fact, a a slightly stronger version of those theorems with finite coefficients is proven.
We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Sh
imura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic $(0,p)$ of integral canonical models of projective Shimura varieties of Hodge type with respect to h--hyperspecial subgroups as pro-etale covers of Neron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.
Sums of the form add((-1)^n q^(n(n-1)/2) x^n, n>=0) are called partial theta functions. In his lost notebook, Ramanujan recorded many identities for those functions. In 2003, Warnaar found an elegant formula for a sum of two partial theta functions.
Subsequently, Andrews and Warnaar established a similar result for the product of two partial theta functions. In this note, I discuss the relation between the Andrews-Warnaar identity and the (1986) product formula due to Gasper and Rahman. I employ nonterminating extension of Sears-Carlitz transformation for 3phi_2 to provide a new elegant proof for a companion identity for the difference of two partial theta series. This difference formula first appeared in the work of Schilling-Warnaar (2002). Finally, I show that Schilling-Warnnar (2002) and Warnaar (2003) formulas are, in fact, equivalent.
Yu. I. Manin conjectured that the maximal abelian extensions of the real quadratic number fields are generated by the pseudo-lattices with real multiplication. We prove this conjecture using theory of measured foliations on the modular curves.
We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the Neron-Tate height of generators of its Mordell-Weil group. The bounds are implie
d by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators of the free part of the Mordell-Weil group. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals.
Let $M$ be a T-motive. We introduce the notion of duality for $M$. Main results of the paper (we consider uniformizable $M$ over $F_q[T]$ of rank $r$, dimension $n$, whose nilpotent operator $N$ is 0): 1. Algebraic duality implies analytic duality
(Theorem 5). Explicitly, this means that the lattice of the dual of $M$ is the dual of the lattice of $M$, i.e. the transposed of a Siegel matrix of $M$ is a Siegel matrix of the dual of $M$. 2. Let $n=r-1$. There is a 1 -- 1 correspondence between pure T-motives (all they are uniformizable), and lattices of rank $r$ in $C^n$ having dual (Corollary 8.4).
The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculationa
l strategy is adopted from Cohens work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.
Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types of polynom
ials modulo primes, and cycle types of the Galois groups of polynomials. One remarkable example is the removal of all artificial constraints from the Kummer-Dedekind Theorem that relates splitting and factorization patterns. Finally, we present an elementary proof that the discriminant of the splitting field of a monic irreducible polynomial with integer coefficients has a computable upper bound in terms of the coefficients. This result, combined with one of Lagarias et al., shows that tests of polynomials for the cycle types of the Galois group are conclusive. In particular, the Galois groups of monic irreducible cubics, quartics, and quintics with integer coefficients can be completely determined in finitely many steps (though not necessarily in ones lifetime).
The goal of this paper is to calculate the trace of the composition of a Hecke correspondence and a (high enough) power of the Frobenius at a good place on the intersection cohomology of the Satake-Baily-Borel compactification of certain Shimura vari
eties, to stabilize the result for Shimura varieties associated to unitary groups over $mathbb{Q}$ and to give applications of this calculations using base change from these unitary groups to $GL_n$. ----- Le but de ce texte est de calculer la trace dune correspondance de Hecke composee avec une puissance (assez grande) du Frobenius en une bonne place sur la cohomologie dintersection de la compactification de Satake-Baily-Borel de certaines varietes de Shimura, de stabiliser le resultat obtenu pour les varietes de Shimura associees aux groupes unitaires sur $mathbb{Q}$, et de donner des applications de ces calculs en utilisant le changement de base de ces groupes unitaires a $GL_n$.
