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25
Sato–Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
, 2008
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ANALYTIC PROBLEMS FOR ELLIPTIC CURVES
, 2005
"... Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the dist ..."
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Cited by 15 (0 self)
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Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an interesting dichotomy for the occurence of the possible groups. (This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv). Contents
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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Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider
Bounded gaps between primes with a given primitive root, II
"... Let m be a natural number, and letQ be a set containing at least exp(Cm) primes. We show that one can find infinitely many strings of m consecutive primes each of which has some q ∈ Q as a primitive root, all lying in an interval of length OQ(exp(C ′m)). This is a bounded gaps variant of a theorem o ..."
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Let m be a natural number, and letQ be a set containing at least exp(Cm) primes. We show that one can find infinitely many strings of m consecutive primes each of which has some q ∈ Q as a primitive root, all lying in an interval of length OQ(exp(C ′m)). This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin’s conjecture. Let E/Q be an elliptic curve with an irrational 2torsion point. Assume GRH. Then for every m, there are infinitely many strings of m consecutive primes p for which E(Fp) is cyclic, all lying an interval of length OE(exp(C ′′m)). If E has CM, then the GRH assumption can be removed. Here C, C ′, and C ′ ′ are absolute constants.
Small exponent point groups on elliptic curves
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX 18 (2006), 471–476
, 2006
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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.
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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Cited by 4 (2 self)
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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.
Drinfeld modules with maximal Galois action on their torsion points
, 1110
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