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33
Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K su ..."
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Cited by 17 (1 self)
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Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K such that p = σ, satis
Algorithmic enumeration of ideal classes for quaternion orders
 SIAM J. Comput. (SICOMP
"... Abstract. We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of twosided ideal ..."
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Cited by 12 (7 self)
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Abstract. We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of twosided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2. Key words. quaternion algebras, maximal orders, ideal classes, number theory AMS subject classifications. 11R52 Since the very first calculations of Gauss for imaginary quadratic fields, the problem of computing the class group of a number field F has seen broad interest. Due to the evident close association between the class number and regulator (embodied in the Dirichlet class number formula), one often computes the class group and unit group in tandem as follows. Problem (ClassUnitGroup(ZF)). Given the ring of integers ZF of a number field F, compute the class group Cl ZF and unit group Z ∗ F.
Computing ray class groups, conductors and discriminants
 MATH. COMP
, 1998
"... We use the algorithmic computation of exact sequences of Abelian groups to compute the complete structure of (ZK/m) ∗ for an ideal m of a number field K, as well as ray class groups of number fields, and conductors and discriminants of the corresponding Abelian extensions. As an application we gi ..."
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Cited by 9 (1 self)
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We use the algorithmic computation of exact sequences of Abelian groups to compute the complete structure of (ZK/m) ∗ for an ideal m of a number field K, as well as ray class groups of number fields, and conductors and discriminants of the corresponding Abelian extensions. As an application we give several number fields with discriminants less than previously known ones.
FINITENESS OF ARITHMETIC HYPERBOLIC REFLECTION GROUPS
, 2006
"... Abstract. We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups. 1. ..."
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Cited by 9 (0 self)
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Abstract. We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups. 1.
Zeros of Dedekind zeta functions in the critical strip
 Math.Comp.66 (1997), 1295–1321. MR 98d:11140 Laboratoire d’Algorithmique Arithmétique, Université BordeauxI,351coursdela Libération, F33405 Talence Cedex France Email address: omar@math.ubordeaux.fr
"... Abstract. In this paper, we describe a computation which established the GRH to height 92 (resp. 40) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff ..."
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Cited by 6 (0 self)
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Abstract. In this paper, we describe a computation which established the GRH to height 92 (resp. 40) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff error to obtain results which are mathematically rigorous, and we generalize Turing’s criterion to prove that there is no zero off the critical line. We finally give results concerning the GRH for cubic and quartic fields, tables of low zeros for number fields of degree 5 and 6, and statistics about the smallest zero of a number field. 0. Introduction and notations The Riemann zeta function and its generalization to number fields, the Dedekind zeta function, have been for well over a hundred years one of the central tools in number theory. It is recognized that the deepest single open problem in mathematics is the settling of the Riemann Hypothesis, and number theorists know that its
Infinite Global Fields and the Generalized Brauer–Siegel Theorem
 Moscow Math. J
"... To our teacher Yu.I.Manin on the occasion of his 65th birthday Abstract. The paper has two purposes. First, we start to develop a theory of infinite global fields, i.e., of infinite algebraic extensions either of Q or of Fr(t). We produce a series of invariants of such fields, and we introduce and s ..."
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Cited by 6 (2 self)
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To our teacher Yu.I.Manin on the occasion of his 65th birthday Abstract. The paper has two purposes. First, we start to develop a theory of infinite global fields, i.e., of infinite algebraic extensions either of Q or of Fr(t). We produce a series of invariants of such fields, and we introduce and study a kind of zetafunction for them. Second, for sequences of number fields with growing discriminant we prove generalizations of the Odlyzko–Serre bounds and of the Brauer–Siegel theorem, taking into account nonarchimedean places. This leads to asymptotic bounds on the ratio log hR / log √ D  valid without the standard assumption n / log √ D  → 0, thus including, in particular, the case of unramified towers. Then we produce examples of class field towers, showing that this assumption is indeed necessary for the Brauer–Siegel theorem to hold. As an easy consequence we ameliorate on existing bounds for regulators. 2000 Math. Subj. Class. 11G20, 11R37, 11R42, 14G05, 14G15, 14H05 Key words and phrases. Global field, number field, curve over a finite field, class number, regulator, discriminant bound, explicit formulae, infinite global field, Brauer–Siegel theorem 1
On Powers as Sums of Two Cubes
, 2000
"... In a paper of Kraus, it is proved that x 3 + y 3 = z p for p 17 has only trivial primitive solutions, provided that p satisfies a relatively mild and easily tested condition. In this article we prove that the primitive solutions of x 3 + y 3 = z p with p = 4; 5; 7; 11; 13, correspond t ..."
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Cited by 5 (0 self)
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In a paper of Kraus, it is proved that x 3 + y 3 = z p for p 17 has only trivial primitive solutions, provided that p satisfies a relatively mild and easily tested condition. In this article we prove that the primitive solutions of x 3 + y 3 = z p with p = 4; 5; 7; 11; 13, correspond to rational points on hyperelliptic curves with Jacobians of relatively small rank. Consequently, Chabauty methods may be applied to try to find all rational points. We do this for p = 4; 5, thus proving that x 3 + y 3 = z 4 and x 3 + y 3 = z 5 have only trivial primitive solutions. In the process we meet a Jacobian of a curve that has more 6torsion at any prime of good reduction than it has globally. Furthermore, some pointers are given to computational aids for applying Chabauty methods.
Constructions of Codes from Number Fields
 IEEE Transactions on Information Theory
, 2003
"... We de ne numbertheoretic errorcorrecting codes based on algebraic number elds, thereby providing a generalization of Chinese Remainder Codes akin to the generalization of ReedSolomon codes to Algebraicgeometric codes. Our construction is very similar to (and in fact less general than) the on ..."
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Cited by 5 (0 self)
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We de ne numbertheoretic errorcorrecting codes based on algebraic number elds, thereby providing a generalization of Chinese Remainder Codes akin to the generalization of ReedSolomon codes to Algebraicgeometric codes. Our construction is very similar to (and in fact less general than) the one given by Lenstra [9], but the parallel with the function eld case is more apparent, since we only use the nonarchimedean places for the encoding. We prove that over an alphabet size as small as 19, there even exist asymptotically good number eld codes of the type we consider. This result is based on the existence of certain number elds that have an in nite class eld tower in which some primes of small norm split completely.
Toward verification of the Riemann hypothesis: Application of the Li criterion, to appear in
 Math. Phys., Analysis and Geometry (2005). 14 M. W. Coffey, New
, 2004
"... We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λk}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the seque ..."
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Cited by 4 (2 self)
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We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λk}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence ηj are valid. The constants ηj enter the Laurent expansion of the logarithmic derivative of the zeta function about s = 1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λn and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function. Key words and phrases Riemann zeta function, Riemann xi function, logarithmic derivatives, Riemann hypothesis,
SHIMURA CURVES OF GENUS AT MOST TWO
, 2008
"... Abstract. We enumerate all Shimura curves XD 0 (N) ofgenusatmosttwo: there are exactly 858 such curves, up to equivalence. The elliptic modular curve X0(N) is the quotient of the completed upper halfplane H ∗ by the congruence subgroup Γ0(N) of matrices in SL2(Z) that are upper triangular modulo N ∈ ..."
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Cited by 4 (3 self)
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Abstract. We enumerate all Shimura curves XD 0 (N) ofgenusatmosttwo: there are exactly 858 such curves, up to equivalence. The elliptic modular curve X0(N) is the quotient of the completed upper halfplane H ∗ by the congruence subgroup Γ0(N) of matrices in SL2(Z) that are upper triangular modulo N ∈ Z>0. The curve X0(N) forms a coarse moduli space for (generalized) elliptic curves equipped with a cyclic subgroup of order N. Modular curves have been studied in great detail due to their importance in many fields, especially arithmetic geometry and the theory of automorphic forms. The genus of X0(N) goes to infinity with N according to an explicit formula [6, Chapter 3], [5]; for example, X0(N) has genus at most two if and only if N ≤ 29 or