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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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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
The square sieve and the Lang–Trotter conjecture
 Canadian Journal of Mathematics
, 2001
"... 1 Let E be an elliptic curve defined over Q and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which Q(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, �under a ce ..."
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1 Let E be an elliptic curve defined over Q and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which Q(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, �under a certain generalized Riemann hypothesis we show that this number is OE x 17 18 log x, and unconditionally. We also prove that the number we show that this number is OE,K � 13 x(log log x) 12 (log x) 25 24 of imaginary quadratic fields K, with − disc K ≤ x and of the form K = Q(πp), is ≫E log log log x for x ≥ x0(E). These results represent progress towards a 1976 LangTrotter conjecture. 1
Extreme values of Artin Lfunctions and class numbers
 Compositio Math
"... Assuming the GRH and Artin conjecture for Artin Lfunctions, we prove that there exists a totally real number field of any fixed degree (> 1) with an arbitrarily large discriminant whose normal closure has the full symmetric group as Galois group and whose class number is essentially as large as ..."
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Assuming the GRH and Artin conjecture for Artin Lfunctions, we prove that there exists a totally real number field of any fixed degree (> 1) with an arbitrarily large discriminant whose normal closure has the full symmetric group as Galois group and whose class number is essentially as large as possible. One ingredient is an unconditional construction of totally real fields with small regulators. Another is the existence of Artin Lfunctions with large special values. Assuming the GRH and Artin conjecture it is shown that there exist an Artin Lfunctions with arbitrarily large conductor whose value at s = 1 is extremal and whose associated Galois representation has a fixed image, which is an arbitrary nontrivial finite irreducible subgroup of GL(n, C) with property GalT. 1 Class numbers of number fields. Let K be a number field whose group of ideal classes has size h, called the class number of K. As K ranges over some natural family, it is interesting to
On The Exceptional Zeros Of RankinSelberg LFunctions
, 2003
"... Introduction In this paper we study the possibility of real zeros near s = 1 for the RankinSelberg Lfunctions L(s, f g) and L(s, sym 2 (g) sym 2 (g)), where f, g are newforms, holomorphic or otherwise, on the upper half plane H, and sym 2 (g) denotes the automorphic form on GL(3)/Q asso ..."
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Introduction In this paper we study the possibility of real zeros near s = 1 for the RankinSelberg Lfunctions L(s, f g) and L(s, sym 2 (g) sym 2 (g)), where f, g are newforms, holomorphic or otherwise, on the upper half plane H, and sym 2 (g) denotes the automorphic form on GL(3)/Q associated to g by Gelbart and Jacquet ([GJ79]). We prove that the set of such zeros of these Lfunctions is the union of the corresponding sets for L(s, #) with # a quadratic Dirichlet character, which divide them. Such a divisibility does not occur in general, for example when f, g are of level 1. When f is a Maass form for SL(2,<
On Smooth Ideals in Number Fields
 J. Number Theory
, 1993
"... For y 2 IR ?0 an integral ideal of an algebraic number field F is called ysmooth if the norms of all of its prime ideal factors are bounded by y. Assuming the generalized Riemann hypothesis we prove a lower bound for the number /F (x; y) of integral ysmooth ideals in F whose norms are bounded ..."
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For y 2 IR ?0 an integral ideal of an algebraic number field F is called ysmooth if the norms of all of its prime ideal factors are bounded by y. Assuming the generalized Riemann hypothesis we prove a lower bound for the number /F (x; y) of integral ysmooth ideals in F whose norms are bounded by x 2 IR ?0 . Apart from x and y this bound only depends on the degree of F . 1 Introduction and result Let F be an algebraic number field of discriminant \Delta F . For y 2 IR ?0 an integral ideal of F is called ysmooth if the norms of all of its prime ideal factors are bounded by y. The number of integral ysmooth ideals of F whose norms do not exceed x 2 IR ?0 is denoted by /F (x; y). In this paper we prove the following result. 1. Theorem Assume that the Generalized Riemann Hypothesis (GRH) is correct. For any n 2 IN and for any ffl 2 IR ?0 there is an effectively computable constant x 0 (ffl; n) 2 IR ?0 such that for any x; y 2 IR ?0 with x ? x 0 (ffl; n) and for every algebra...
A generalization of the BarbanDavenportHalberstam Theorem to number fields
 J. Number Theory
"... Abstract. For a fixed number field K, we consider the mean square error in estimating the number of primes with norm congruent to a modulo q by the Chebotarëv Density Theorem when averaging over all q ≤ Q and all appropriate a. Using a large sieve inequality, we obtain an upper bound similar to the ..."
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Cited by 4 (4 self)
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Abstract. For a fixed number field K, we consider the mean square error in estimating the number of primes with norm congruent to a modulo q by the Chebotarëv Density Theorem when averaging over all q ≤ Q and all appropriate a. Using a large sieve inequality, we obtain an upper bound similar to the BarbanDavenportHalberstam Theorem. 1.
An analogue of Artin’s conjecture for abelian extensions
 Journal of Number Theory
, 1984
"... integer a # * 1, or a perfect square, there exist infinitely many primes p for which a is a primitive root, modulo p. Moreover, if N,(x) denotes the number of such primes up to x, he conjectured that for a certain constant A (a), N,(x) A(a)?flogs‘ as s + co. In 1967, Hooley [ 3] proved this conj ..."
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integer a # * 1, or a perfect square, there exist infinitely many primes p for which a is a primitive root, modulo p. Moreover, if N,(x) denotes the number of such primes up to x, he conjectured that for a certain constant A (a), N,(x) A(a)?flogs‘ as s + co. In 1967, Hooley [ 3] proved this conjecture assuming the Riemann hypothesis for a certain (infinite) set of Dedekind zeta functions. Later, Goldstein]2] formulated a general conjecture, a special case of which was Artin’s conjecture. His conjecture was as follows: for each prime q, let L, be an algebraic number field, normal and of finite degree over Q. For each squarefree k, set where L, is taken to primes which do not where L, = TTL,, qlk be Q. Let n(li) = IL, : U4]. Then, the set of rational split completely in any L, has a natural density 6, and p denotes the usual Mobius function. Simple ideas from algebraic number theory reveal that Artin’s conjecture is recaptured by the special case L, = O(&,, a”9), where c, is a primitive qth root of unity.
Almost prime values of the order of elliptic curves over finite fields
, 2008
"... Abstract. Let E be an elliptic curve over Q without complex multiplication, and which is not isogenous to a curve with nontrivial rational torsion. For each prime p of good reduction, let E(Fp)  be the order of the group of points of the reduced curve over Fp. We prove in this paper that, under t ..."
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Abstract. Let E be an elliptic curve over Q without complex multiplication, and which is not isogenous to a curve with nontrivial rational torsion. For each prime p of good reduction, let E(Fp)  be the order of the group of points of the reduced curve over Fp. We prove in this paper that, under the GRH, there are at least 2.778Ctwin E x/(log x)2 primes p such that E(Fp)  has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [18] and Murty & Miri [13]. This is also the first result where the dependence on the conjectural constant Ctwin E appearing in the twin prime conjecture for elliptic curves (also known as Koblitz’s conjecture) is made explicit. This is achieved by sieving a slightly different sequence than the one of [18] and [13]. By sieving the same sequence and using Selberg’s linear sieve, we can also improve the constant of Zywina [22] appearing in the upper bound for the number of primes p such that E(Fp)  is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH. 1.
Brauer–Siegel theorem for elliptic surfaces
 Int. Math. Res. Not. IMRN 2008
"... Abstract. We consider higherdimensional analogues of the classical BrauerSiegel theorem focusing on the case of abelian varieties over global function fields. We prove such an analogue in the case of constant families of elliptic curves. To our teachers V.E. Voskresenskiĭ and Yu.I. Manin to their ..."
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Abstract. We consider higherdimensional analogues of the classical BrauerSiegel theorem focusing on the case of abelian varieties over global function fields. We prove such an analogue in the case of constant families of elliptic curves. To our teachers V.E. Voskresenskiĭ and Yu.I. Manin to their 80th and 70th birthdays, respectively 1.