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Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider

Abstract. Let E be a CM elliptic curve defined over Q and of conductor N. We establish an asymptotic formula, uniform in N and with improved error term, for the counting function of primes p for which the reduction mod p of E is cyclic. Our result resembles the classical Siegel-Walfisz theorem regarding the distribution of primes in arithmetic progressions. 1.

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Consider A an abelian variety of dimension r, defined over a number field F, such that EndF A ⊗ Q =EndAF ¯ ⊗ Q, with endomorphism algebra of type I or II in the Albert classification. Assume that A has potential good reduction everywhere. For ℘ a finite prime of F, we denote by F℘ the residue field at ℘. If A has good reduction at ℘, let Ā be the reduction of A at ℘. In this paper, under GRH, for a large family of abelian varieties A of type I and II, we obtain an asymptotic formula for the number of primes ℘ of F, with NF/Q ℘ ≤ x, for which Ā(F℘) has at most 2r −1 cyclic components.

Abstract. We formulate a geometric analogue of the Titchmarsh Divisor Problem in the context of abelian varieties. For any abelian variety A defined over Q, we study the asymptotic distribution of the primes of Z which split completely in the division fields of A. For all abelian varieties which contain an elliptic curve we establish an asymptotic formula for such primes under the assumption of GRH. We explain how to derive an unconditional asymptotic formula in the case that the abelian variety is a CM elliptic curve. 1.

Consider A an abelian variety over a number field F of dimension r, where r ≥ 1 is an integer. Assume that EndF ¯ A ⊗ Q = K, where K is a CM-field such that [K: Q] = 2r. For ℘ a finite prime of F, we denote by F ℘ the residue field at ℘. If A has good reduction at ℘, let Ā be the reduction of A at ℘. In this paper, under GRH, we obtain an asymptotic formula for the number of primes ℘ of F, with NF/Q ℘ ≤ x, for which Ā(F℘) has at most 2r − 1 cyclic components.

Consider A an abelian variety of dimension r, defined over a number field F. For ℘ a finite prime of F, we denote by F ℘ the residue field at ℘. If A has good reduction at ℘, let Ā be the reduction of A at ℘. In this paper, under GRH, for a large family of abelian varieties A, we obtain an asymptotic formula for the number of primes ℘ of F, with NF/Q ℘ ≤ x, for which Ā(F℘) has at most 2r − 1 cyclic components.

Abstract not found

Abstract. Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, denote by ep the exponent of the reduction of E modulo p. Under GRH, we prove that there is a constant CE ∈ (0,1) such that 1 π(x) p�x ep = 1