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73
Twisted Lfunctions over number fields and Hilbert’s eleventh problem
"... Let K be a totally real number field, π an irreducible cuspidal representation of GL2(K) \ GL2(AK) with unitary central character, and χ a Hecke character of conductor q. Then L(1/2, π ⊗ χ) ≪ (N q) 1 2 − 1 8 (1−2θ)+ε, where 0 � θ � 1/2 is any exponent towards the Ramanujan– Petersson conjecture (θ ..."
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Cited by 20 (5 self)
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Let K be a totally real number field, π an irreducible cuspidal representation of GL2(K) \ GL2(AK) with unitary central character, and χ a Hecke character of conductor q. Then L(1/2, π ⊗ χ) ≪ (N q) 1 2 − 1 8 (1−2θ)+ε, where 0 � θ � 1/2 is any exponent towards the Ramanujan– Petersson conjecture (θ = 1/9 is admissible). The proof is based on a spectral decomposition of shifted convolution sums and a generalized Kuznetsov formula.
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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Cited by 16 (4 self)
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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,<
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 15 (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
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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Cited by 12 (4 self)
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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
Siegel Zeros and Cusp Forms
"... Given a Dirichlet series L(s) with Euler product in {(s)> 1}, admitting a meromorphic continuation to the whole splane and a functional equation relating s to 1 − s, a fundamental problem is to know if L(s) has a Siegel zero, i.e., a real zero in (1 − δ,1) for a small δ. (See §2 for a precise d ..."
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Cited by 9 (1 self)
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Given a Dirichlet series L(s) with Euler product in {(s)> 1}, admitting a meromorphic continuation to the whole splane and a functional equation relating s to 1 − s, a fundamental problem is to know if L(s) has a Siegel zero, i.e., a real zero in (1 − δ,1) for a small δ. (See §2 for a precise definition.)
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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Cited by 8 (1 self)
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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 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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Cited by 7 (0 self)
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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...
DISTRIBUTION OF PERIODIC TORUS ORBITS AND DUKE’S THEOREM FOR CUBIC FIELDS
"... We study periodic torus orbits on space of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits; for rank 3 lattices, we show that the equivalence classes become uniformly distributed. This is a cubic analogue of D ..."
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We study periodic torus orbits on space of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits; for rank 3 lattices, we show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke’s theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular 5fold SL3(Z)\SL3(R)/SO3. In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL3(Z)\SL3(R)/SO3 of volume ≤ V becomes equidistributed as V → ∞. The proof combines subconvexity estimates, measure classification, and local harmonic analysis.
On the symmetric powers of cusp forms on GL(2) of icosahedral type
 Int. Math. Res. Not
"... In this Note, we prove three theorems. Throughout, F will denote a number field with absolute Galois group GF = Gal ( ¯ F/F). When ρ is an irreducible continuous 2–dimensional C representation of GF, one says that it is icosahedral, resp. octahedral, resp. tetrahedral, resp. ..."
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Cited by 7 (1 self)
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In this Note, we prove three theorems. Throughout, F will denote a number field with absolute Galois group GF = Gal ( ¯ F/F). When ρ is an irreducible continuous 2–dimensional C representation of GF, one says that it is icosahedral, resp. octahedral, resp. tetrahedral, resp.