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Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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Cited by 28 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Matrices of quaternions
 Pacific J. Math
, 1951
"... 1. Introduction, In this note, some theorems which concern matrices of complex numbers are generalized to matrices over real quaternions. First it is proved that every matrix of quaternions has a characteristic root. Next, there exist n ~ ~ 1 mutually orthogonal unit ^vectors all orthogonal to a gi ..."
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Cited by 22 (0 self)
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1. Introduction, In this note, some theorems which concern matrices of complex numbers are generalized to matrices over real quaternions. First it is proved that every matrix of quaternions has a characteristic root. Next, there exist n ~ ~ 1 mutually orthogonal unit ^vectors all orthogonal to a given vector. It is shown that Schur's lemma holds for matrices of quarternions: every matrix can be trans
Congruence subgroups and generalized FrobeniusSchur indicators
"... Abstract. We define generalized FrobeniusSchur indicators for objects in a linear pivotal category C. An equivariant indicator of an object is defined as a functional on the Grothendieck algebra of the quantum double Z(C) of C using the values of the generalized FrobeniusSchur indicators. In a sph ..."
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Cited by 17 (3 self)
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Abstract. We define generalized FrobeniusSchur indicators for objects in a linear pivotal category C. An equivariant indicator of an object is defined as a functional on the Grothendieck algebra of the quantum double Z(C) of C using the values of the generalized FrobeniusSchur indicators. In a spherical fusion category C with FrobeniusSchur exponent N, we prove that the set of all equivariant indicators admits a natural action of the modular group, and the kernel of the canonical modular representation of Z(C) is a congruence subgroup of level N. Moreover, if C is modular, then the kernel of the projective modular representation of C is also a congruence subgroup of level N, and every modular representation of C has a finite image.
Operatoralgebraic superridigity for SLn(Z), n ≥ 3
, 2008
"... For n ≥ 3, let Γ = SLn(Z). We prove the following superridigity result for Γ in the context of operator algebras. Let L(Γ) be the von Neumann algebra generated by the left regular representation of Γ. Let M be a factor and let U(M) be its unitary group. Let π: Γ → U(M) be a group homomorphism such t ..."
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Cited by 6 (0 self)
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For n ≥ 3, let Γ = SLn(Z). We prove the following superridigity result for Γ in the context of operator algebras. Let L(Γ) be the von Neumann algebra generated by the left regular representation of Γ. Let M be a factor and let U(M) be its unitary group. Let π: Γ → U(M) be a group homomorphism such that π(Γ) ′ ′ = M. Then either (i) M is finite dimensional, or (ii) there exists a subgroup of finite index Λ of Γ such that πΛ extends to a homomorphism U(L(Λ)) → U(M). This answers, in the special case of SLn(Z), a question of A. Connes discussed in [Jone00, Page 86]. The result is deduced from a complete description of the tracial states on the full C ∗ –algebra of Γ. As another application, we show that the full C ∗ –algebra of Γ has no faithful tracial state, thus answering a question of E. Kirchberg. 1
Periodic orbits of linear endomorphisms on the 2torus and its lattices, Nonlinearity 21
, 2008
"... Abstract. Counting periodic orbits of endomorphisms on the 2torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The situation on the lattices, up to local conjugacy, is com ..."
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Cited by 6 (4 self)
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Abstract. Counting periodic orbits of endomorphisms on the 2torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The situation on the lattices, up to local conjugacy, is completely determined by the determinant, the trace and a third invariant of the matrix defining the toral endomorphism. 1.
Online at stacks.iop.org/Non/21/2427
, 2008
"... Counting periodic orbits of endomorphisms on the 2torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The situation on the lattices, up to local conjugacy, is completely de ..."
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Counting periodic orbits of endomorphisms on the 2torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The situation on the lattices, up to local conjugacy, is completely determined by the determinant, the trace and a third invariant of the matrix defining the toral endomorphism. Mathematics Subject Classification: 37E30, 37A35, 37A20 1.