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Sato–Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
, 2008
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On the exponent of the group of points on elliptic curves in extension fields
 Intern. Math. Research Notices
"... Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider ..."
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Cited by 6 (3 self)
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Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider
AN ANALOGUE OF THE SIEGELWALFISZ THEOREM FOR THE CYCLICITY OF CM ELLIPTIC CURVES MOD p
"... 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 SiegelWalfisz theorem regar ..."
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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 SiegelWalfisz theorem regarding the distribution of primes in arithmetic progressions. 1.
Cyclicity of finite abelian varieties of type III, submitted. Available here: www2.math.kyushuu.ac.jp/ virdol
"... 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 ..."
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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.
A GEOMETRIC VARIANT OF TITCHMARSH DIVISOR PROBLEM
"... 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 con ..."
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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.
Cyclicity of finite CM abelian varieties
, 2012
"... 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 CMfield 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 ..."
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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 CMfield 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.
Cyclicity of finite abelian varieties
, 2012
"... 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 asymptoti ..."
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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.
THE AVERAGE EXPONENT OF ELLIPTIC CURVES MODULO p
"... 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 ..."
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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