Calculating formula and divisibility for relative class numbers of abelian function fields

In this paper abelian function fields are restricted to the subfields of cyclotomic function fields. For any abelian function field K/k with conductor an irreducible polynomial over a finite field of odd characteristic, we give a calculating formula of the relative divisor class number of K. And using the given calculating formula we obtain a criterion for checking whether or not the relative divisor class number is divisible by the characteristic of k.     

Properties of the Beurling generalized primes

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In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that , where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit exists. We also show that this limit coincides with ; for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small |N(x)−[x]| can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that and conjecture that this is the best possible bound.

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For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Kw3iNo3fAbk/.     

The hyperring of adèle classes

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We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK=AK/K^× of a global field K. After promoting F₁ to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G⊂R^× is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G∪{0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field K.

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Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

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In a previous paper Najman (in press) [9], the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and . Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over Q(i) and , we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves over Q(i) and .

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For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=VPhCkJTGB_o.     

Quasi-co-local subgroups of abelian groups

Let {0}≠K be a subgroup of the abelian group G. In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Appl. Math., vol. 249, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 29-37], K was called a co-local (cl) subgroup of G if is naturally isomorphic to . We generalize this notion to the quasi-category of abelian groups and call the subgroup K≠{0} of G a quasi-co-local (qcl) subgroup of G if is naturally isomorphic to . We show that qcl subgroups behave quite differently from cl subgroups. For example, while cl subgroups K are pure in G, i.e. G/K is torsion-free if G is torsion-free, any reduced torsion group T can be the torsion subgroup t(G/K) of G/K where G is torsion-free and K is a qcl subgroup of G.     

On the independence of Heegner points in the function field case

Let ∞ be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at ∞ and fix a modular parametrization ΦE:X₀(N)→E. Let be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K₁,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K₁,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in . Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C.     

Algebraic points of small height missing a union of varieties

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Let K be a number field, , or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of KN, N?2. Let ZK be a union of varieties defined over K such that V?ZK. We prove the existence of a point of small height in V?ZK, providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of a hypersurface containing ZK, where dependence on both is optimal. This generalizes and improves upon the results of Fukshansky (2006) [6] and [7]. As a part of our argument, we provide a basic extension of the function field version of Siegel's lemma (Thunder, 1995) [21] to an inequality with inhomogeneous heights. As a corollary of the method, we derive an explicit lower bound for the number of algebraic integers of bounded height in a fixed number field.

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Normal integral bases and strict ray class groups modulo 4

Let F be a number field. We construct three tamely ramified quadratic extensions which are ramified at most at some given set of finite primes, such that K₃⊂K₁K₂, both K₁/F and K₂/F have normal integral bases, but K₃/F has no normal integral basis. Since Hilbert-Speiser's theorem yields that every finite and tamely ramified abelian extension over the field of rational numbers has a normal integral basis, it seems that this example is interesting (cf. [5] J. Number Theory 79 (1999) 164; Theorem 2). As we shall explain below, the previous papers (Acta Arith. 106 (2) (2003) 171-181; Abh. Math. Sem. Univ. Hamburg 72 (2002) 217-233) motivated the construction. We prove that if the class number of F is bigger than 1, or the strict ray class group of F modulo 4 has an element of order ?3, then there exist infinitely many triplets (K₁,K₂,K₃) of such fields.     

Five peculiar theorems on simultaneous representation of primes by quadratic forms

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It is a theorem of Kaplansky that a prime is representable by both or none of x²+32y² and x²+64y², whereas a prime is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved. As an example, one theorem states that a prime is representable by both or none of x²+20y² and x²+100y², whereas a prime is representable by exactly one of these forms. A heuristic argument is given why there are no other results of the same kind. This argument relies on the (plausible) conjecture that there are exactly 485 negative discriminants Δ such that the class group C(Δ) has exponent 4.

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For a video summary of this paper, please visit http://www.youtube.com/watch?v=l_yRq0oqKx4.     

Sequences with small subsum sets

A conjecture of Gao and Leader, recently proved by Sun, states that if is a sequence of length n in a finite abelian group of exponent n, then either some subsequence of X sums to zero or the set of all sums of subsequences of X has cardinality at least 2n−1. This conjecture turns out to be a simple consequence of a theorem of Olson and White; we investigate generalizations that are not implied by this theorem. In particular, we prove the following result: if is a sequence of length n, the terms of which generate a finite abelian group of rank at least 3, then either some subsequence of X sums to zero or the set of all sums of subsequences of X has cardinality at least 4n−5.     

Configurations in abelian categories. III. Stability conditions and identities

This is the third in a series on configurations in an abelian category A. Given a finite poset (I,?), an (I,?)-configuration(σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects.The first paper defined configurations and studied moduli spaces of configurations in A, using the theory of Artin stacks. It showed well-behaved moduli stacks ObjA,MA(I,?) of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod-KQ of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on ObjA.This paper introduces (weak) stability conditions(τ,T,?) on A. We show the moduli spaces , , of τ-semistable, indecomposable τ-semistable and τ-stable objects in class α are constructible sets in ObjA, and some associated configuration moduli spaces constructible in MA(I,?), so their characteristic functions and are constructible.We prove many identities relating these constructible functions, and their stack function analogues, under pushforwards. We introduce interesting algebras of constructible and stack functions, and study their structure. In the fourth paper we show are independent of (τ,T,?), and construct invariants of A,(τ,T,?).     

Slopes and Abelian Subvariety Theorem

In this article, we show how to modify the proof of the Abelian Subvariety Theorem by Bost (Périodes et isogénies des variétés abeliennes sur les corps de nombres, Séminaire Bourbaki, 1994-95, Theorem 5.1) in order to improve the bounds in a quantitative respect and to extend the theorem to subspaces instead of hyperplanes. Given an abelian variety A defined over a number field κ and a non-trivial period γ in a proper subspace W of t_AK with K a finite extension of κ, the Abelian Subvariety Theorem shows the existence of a proper abelian subvariety B of , whose degree is bounded in terms of the height of W, the norm of γ, the degree of κ and the degree and dimension of A. If A is principally polarized then the theorem is explicit.     

Cohomology and coquasi-bialgebra extensions associated to a matched pair of bialgebras

By using braid diagrams, we explicitly reconstruct the cohomology associated to a matched pair of cocommutative bialgebras, in order to give a method of constructing coquasi-bialgebras, which generalize bialgebras, and classifying them up to monoidal equivalence of their comodule categories. An alternative, homological proof is given for Schauenburg's generalized Kac Sequence involving the abelian group Opext(H,K) of bialgebra extensions. We define an abelian group, Opext″(H,K), of coquasi-bialgebra extensions associated to a Singer pair (H,K) of bialgebras, and prove a variant of Schauenburg's sequence which involves the group. It is also proved that there is a natural isomorphism that preserves monoidal equivalence classes, if (H₁,K) and (H₂,K) arise from such matched pairs that are shifts of each other.     

Waring's number mod m

Let p be an odd prime and γ(k,pn) be the smallest positive integer s such that every integer is a sum of s kth powers . We establish γ(k,pn)?[k/2]+2 and provided that k is not divisible by (p−1)/2. Next, let t=(p−1)/(p−1,k), and q be any positive integer. We show that if ?(t)?q then γ(k,pn)?c(q)k1/q for some constant c(q). These results generalize results known for the case of prime moduli.

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On conjugacy of homeomorphisms of the circle possessing periodic points

We give a necessary and sufficient condition for topological conjugacy of homeomorphisms of the circle having periodic points. As an application we get the following theorem on the representation of homeomorphisms. The homeomorphism has a periodic point of period n iff there exist a positive integer q<n relatively prime to n and a homeomorphism such that the lift of Φ⁻¹○F○Φ restricted to [0,1] has the form

     

The support problem for abelian varieties

Let A be an abelian variety over a number field K. If P and Q are K-rational points of A such that the order of the reduction of Q divides the order of the ) reduction of P for almost all prime ideals , then there exists a K-endomorphism φ of A and a positive integer k such that φ(P)=kQ.     

Finiteness theorems in the class field theory of varieties over local fields

We show that the geometrical part of the abelian étale fundamental group of a proper smooth variety over a local field is finitely generated over with finite torsion, and describe its rank by the special fiber of the Néron model of the Albanese variety. As an application, we complete the class field theory of curves over local fields developed by Bloch and Saito, in which the theorem concerning the p-primary part in the positive characteristic case has remained unproven.     

A refined counter-example to the support conjecture for abelian varieties

If A/K is an abelian variety over a number field and P and Q are rational points, the original support conjecture asserted that if the order of divides the order of for almost all primes p of K, then Q is obtained from P by applying an endomorphism of A. This is now known to be untrue. In this note we prove that it is not even true modulo the torsion of A.