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Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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The primes contain arbitrarily long polynomial progressions
 Acta Math
"... Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε suc ..."
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Cited by 30 (4 self)
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Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m),..., x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties. Contents
DEFINING INTEGRALITY AT PRIME SETS OF HIGH DENSITY IN NUMBER FIELDS
 VOL. 101, NO. 1 DUKE MATHEMATICAL JOURNAL
, 2000
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Constructing nonresidues in finite fields and the extended Riemann hypothesis
 Math. Comp
, 1991
"... Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in pol ..."
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Cited by 8 (0 self)
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Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomialtime bound holds even if k is exponentially large. More generally, assuming the ERH, in time (n log p) O(n) we can construct a set of elements
Analogies between group actions on 3manifolds and number fields
 Comment. Math. Helv
"... Abstract. Let a cyclic group G act either on a number field L or on a 3manifold M. Let sL be the number of ramified primes in the extension L G ⊂ L and sM be the number of components of the branching set of the branched covering M → M/G. In this paper, we prove several formulas relating sL and sM t ..."
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Abstract. Let a cyclic group G act either on a number field L or on a 3manifold M. Let sL be the number of ramified primes in the extension L G ⊂ L and sM be the number of components of the branching set of the branched covering M → M/G. In this paper, we prove several formulas relating sL and sM to the induced Gaction on H1(M) and Cl(L), respectively. We observe that the formulas for 3manifolds and number fields are almost identical, and therefore, they provide new evidence for the correspondence between 3manifolds and number fields postulated in arithmetic topology. One of our results extends the Gauss formula for the number of ramified primes in quadratic extensions of Q to all cyclic extensions of Q.
Projective planes in algebraically closed fields
 Proc. London Math. Soc
, 1991
"... We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. This is applied to give new examples of algebraic characteristic sets of matroids. ..."
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We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. This is applied to give new examples of algebraic characteristic sets of matroids. The main technique used, which is motivated by classical projective geometry, is that a particular configuration of four lines and six points in the geometry indicates the presence of a connected onedimensional algebraic group.
A question of Stark
 Pacific J. of Math
, 1997
"... One of the programs of Stark’s conjectures is to find as many connections as possible between the values that Artin L–functions or their derivatives take (especially at s =0) and arithmetic information associated to algebraic number fields. The most refined of Stark’s conjectures involves the values ..."
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One of the programs of Stark’s conjectures is to find as many connections as possible between the values that Artin L–functions or their derivatives take (especially at s =0) and arithmetic information associated to algebraic number fields. The most refined of Stark’s conjectures involves the values of first derivatives of L–functions at s =0.It was recognized early on that the conjecture should be extended to cover cases where the order of vanishing of the L–functions at s =0is greater than one. In 1980, Stark posed a question along these lines that we will consider in detail here. In particular, we will study his question for relative quadratic extensions and prove that an affirmative answer to his question exists for all cases considered. 1. Introduction. Our aim in this section is to state Stark’s question and see how it is related
Densities of 4ranks of K2(O
 Acta Arith
"... Abstract. In [1], the authors established a method of determining the structure of the 2Sylow subgroup of the tame kernel K2(O) for certain quadratic number fields. Specifically, the 4rank for these fields was characterized in terms of positive definite binary quadratic forms. Numerical calculatio ..."
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Abstract. In [1], the authors established a method of determining the structure of the 2Sylow subgroup of the tame kernel K2(O) for certain quadratic number fields. Specifically, the 4rank for these fields was characterized in terms of positive definite binary quadratic forms. Numerical calculations led to questions concerning possible density results of the 4rank of tame kernels. In this paper, we succeed in giving affirmative answers to these questions. 1.
Algebras with Uniformly Distributed Invariants
 J. Algebra 4__4
, 1977
"... Let K be a finite abelian extension of the rational field Q. If A is a central simple algebra over K then we let [A] denote the class of A in the Brauer ..."
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Let K be a finite abelian extension of the rational field Q. If A is a central simple algebra over K then we let [A] denote the class of A in the Brauer
NILPOTENT EXTENSIONS OF NUMBER FIELDS WITH BOUNDED RAMIFICATION
"... We study a variant ofthe inverse problem ofGalois theory and Abhyankar’s conjecture. If p is an odd rational prime and G is a finite pgroup generated by s elements, s minimal, does there exist a normal extension L/Q such that Gal (L/Q) ∼ = G with at most s rational primes that ramify in L? Given ..."
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We study a variant ofthe inverse problem ofGalois theory and Abhyankar’s conjecture. If p is an odd rational prime and G is a finite pgroup generated by s elements, s minimal, does there exist a normal extension L/Q such that Gal (L/Q) ∼ = G with at most s rational primes that ramify in L? Given a nilpotent group ofodd order G with s generators, we obtain a Galois extension L/Q with precisely s prime divisors of Q ramified. Furthermore if K is a number field satisfying K ∩ Q(ζ p n i