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24
On fusion categories
 Annals of Mathematics
"... Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Gr ..."
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Cited by 76 (17 self)
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Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger. Dedicated to Leonid Vainerman on the occasion of his 60th birthday 1. introduction The theory of fusion categories arises in many areas of mathematics such as representation theory, quantum groups, operator algebras and topology. The representation categories of semisimple (quasi) Hopf algebras are important examples of fusion categories. Fusion categories have been studied extensively in the literature,
Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Cited by 40 (3 self)
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
Cycles of quadratic polynomials and rational points on a genus 2 curve
, 1996
"... Abstract. It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1 ..."
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Cited by 32 (13 self)
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Abstract. It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), whose rational points had been previously computed. We prove there are none with N = 5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal(Q/Q)stable 5cycles, and show that there exist Gal(Q/Q)stable Ncycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N = 6. 1.
Classification Theory for Abstract Elementary Classes
 In Logic and Algebra, Yi Zhang editor, Contemporary Mathematics 302, AMS,(2002), 165–203
, 2002
"... In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the firstorder case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. S ..."
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Cited by 18 (4 self)
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In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the firstorder case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah's categoricity conjecture for L# 1 ,# . While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galoisstability.
On Kubota’s Dirichlet series
 J. Reine Angew. Math
, 2006
"... Kubota [19] showed how the theory of Eisenstein series on the higher metaplectic covers of SL2 (which he discovered) can be used to study the analytic properties of Dirichlet series formed with nth order Gauss sums. In this paper we will prove a functional equation for such Dirichlet series in the ..."
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Cited by 12 (9 self)
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Kubota [19] showed how the theory of Eisenstein series on the higher metaplectic covers of SL2 (which he discovered) can be used to study the analytic properties of Dirichlet series formed with nth order Gauss sums. In this paper we will prove a functional equation for such Dirichlet series in the precise form required by the companion paper [2]. Closely related results are in Eckhardt and Patterson [10]. The Kubota Dirichlet series are the entry point to a fascinating universe. Their residues, for example, are mysterious if n> 3, though there is tantalizing evidence that these residues exhibit a rich structure that can only be partially glimpsed at this time. When n = 4 the residues are the Fourier coefficients of the biquadratic theta function that were studied by Suzuki [23]. Suzuki found that he could only determine some of the coefficients. This failure to determine all the coefficients was explained in terms of the failure of uniqueness of Whittaker models for the generalized theta series by Deligne [9] and by Kazhdan and Patterson [15]. On the other hand, Patterson [22] conjectured that the mysterious coefficients are essentially square roots of Gauss sums. Evidence for Patterson’s conjecture is discussed in Bump
The Galois theory of periodic points of polynomial maps
 Proc. London Math. Soc. 68
, 1994
"... There is a growing body of evidence that problems associated with iterated maps can yield interesting insights in number theory and algebra. There are the papers of Narkiewicz [1922], Lewis [14], Liardet [15], and more recently, Silverman [27], dealing with the diophantine aspects of iterated maps, ..."
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Cited by 11 (0 self)
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There is a growing body of evidence that problems associated with iterated maps can yield interesting insights in number theory and algebra. There are the papers of Narkiewicz [1922], Lewis [14], Liardet [15], and more recently, Silverman [27], dealing with the diophantine aspects of iterated maps, as well as the papers
A generalised Skolem–Mahler–Lech theorem for affine varieties
 J. London Math. Soc
"... The Skolem–Mahler–Lech theorem states that if f(n) is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that f(m) is equal to 0 is the union of a finite number of arithmetic progressions in m � 0 and a finite set. We prove that if X is a subvariety of a ..."
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Cited by 10 (0 self)
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The Skolem–Mahler–Lech theorem states that if f(n) is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that f(m) is equal to 0 is the union of a finite number of arithmetic progressions in m � 0 and a finite set. We prove that if X is a subvariety of an affine variety Y over a field of characteristic 0 and q is a point in Y,andσ is an automorphism of Y, then the set of m such that σ m (q) lies in X is a union of a finite number of complete doublyinfinite arithmetic progressions and a finite set. We show that this is a generalisation of the Skolem–Mahler–Lech theorem. 1.
NéronSeveri groups under specialization
, 2009
"... Abstract. André used Hodgetheoretic methods to show that in a smooth proper family X → B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely diff ..."
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Cited by 5 (1 self)
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Abstract. André used Hodgetheoretic methods to show that in a smooth proper family X → B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to André’s theorem, which also proves the following refinement: in a family of varieties with good reduction at p, the locus on the base where the Picard number jumps is nowhere padically dense. Our proof uses the “padic Lefschetz (1, 1) theorem ” of Berthelot and Ogus, combined with an analysis of padic power series. We prove analogous statements for cycles of higher codimension, assuming a padic analogue of the variational Hodge conjecture, and prove that this analogue implies the usual variational Hodge conjecture. Applications are given to abelian schemes and to proper families of projective varieties. 1.
Ergodicity and mixing of noncommuting epimorphisms
 Proc. Lond. Math. Soc
"... Abstract. We study mixing properties of epimorphisms of a compact connected finitedimensional abelian group X. In particular, we show that a set F, F > dim X, of epimorphisms of X is mixing iff every subset of F of cardinality (dim X)+1 is mixing. We also construct examples of free nonabelian gro ..."
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Cited by 4 (1 self)
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Abstract. We study mixing properties of epimorphisms of a compact connected finitedimensional abelian group X. In particular, we show that a set F, F > dim X, of epimorphisms of X is mixing iff every subset of F of cardinality (dim X)+1 is mixing. We also construct examples of free nonabelian groups of automorphisms of tori which are mixing, but not mixing of order 3, and show that, under some irreducibility assumptions, ergodic groups of automorphisms contain mixing subgroups and free nonabelian mixing subsemigroups. 1.
Small gaps in coefficients of Lfunctions and Bfree numbers in short intervals
, 2005
"... We discuss questions related to the nonexistence of gaps in the series defining modular forms and other arithmetic functions of various types, and improve results of Serre, Balog & Ono and Alkan using new results about exponential sums and the distribution of Bfree numbers. ..."
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Cited by 4 (3 self)
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We discuss questions related to the nonexistence of gaps in the series defining modular forms and other arithmetic functions of various types, and improve results of Serre, Balog & Ono and Alkan using new results about exponential sums and the distribution of Bfree numbers.