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Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem (2001)

by Alina Carmen Cojocaru, M. Ram Murty
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Sato–Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height

by William D. Banks, Igor E. Shparlinski , 2008
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ANALYTIC PROBLEMS FOR ELLIPTIC CURVES

by E. Kowalski , 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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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 (This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv).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
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...s in somewhat attenuated form. More recently, since the first version of this paper was written, there have been quite interesting work by Cojocaru [Co], Cojocaru and Duke [CD] and Cojocaru and Murty =-=[CM]-=- on similar topics. 3.2. Analysis of the elliptic splitting problem on GRH. For fixed d � 1, the asymptotic behavior of πE(X; d, 1) is given by the Chebotarev Density Theorem. Under GRH, it can be sta...

On the exponent of the group of points on elliptic curves in extension fields

by Florian Luca, Igor E. Shparlinski - 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
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...satisfactorily answered by Vlădut¸ [28]. In the situation where E is defined over Q, the question about the cyclicity of the reduction E(Fp) when p runs over the primes appears to be much harder (see =-=[5, 6, 7]-=- for recent advances and surveys of other related results). In particular, this problem is closely related to the famous Lang-Trotter conjecture. One can also study an apparently easier question about...

Bounded gaps between primes with a given primitive root, II

by Roger C. Baker, Paul Pollack
"... 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 2-torsion 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.
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... ≤ x, as x→∞. Ram Murty [Mur83] showed that when E has CM, Serre’s asymptotic formula can be proved unconditionally; a simpler argument for the same conclusion has been given by Cojocaru [Coj03]. See =-=[CM04]-=- and [AM10] for investigations into the size of the error term in Serre’s formula. We prove the following bounded gaps result. Theorem 1.2. Let E/Q be an elliptic curve with an irrational 2-torsion po...

Small exponent point groups on elliptic curves

by Florian Luca, James Mckee, Igor E. Shparlinski - JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX 18 (2006), 471–476 , 2006
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On the average exponent of elliptic curves modulo p

by Tristan Freiberg, Pär Kurlberg
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...oting the Möbius function of k, c∗E = ∑∞ k=1 µ(k)/nLk . Furthermore, c ∗ E > 0 unless all 2-torsion points on E are defined over Q, an obvious obstruction1 to Ẽ(Fp) being cyclic. Cojocaru and Murty =-=[4]-=- obtained versions of (1.3) with effective error terms, and in the special case in which E has CM, Murty [13] was quite remarkably able to establish (1.3) unconditionally (the proofs were later signif...

A GEOMETRIC VARIANT OF TITCHMARSH DIVISOR PROBLEM

by Amir Akbary, Dragos Ghioca
"... 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.
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...t in any extension from a given family of Galois extensions. Our Theorem 1.4 also can be considered as a higher dimensional analogue of the cyclicity question for elliptic curves (see [20], [15], and =-=[4]-=-). Therefore, similar to the cyclicity question for elliptic curves, one may ask when is the density from our Theorem 1.4 positive, i.e. ∞∑ µ(m) δA := > 0? [Q(A[m]) : Q] m=1 It is clear that if A[2] ⊂...

AN ANALOGUE OF THE SIEGEL-WALFISZ THEOREM FOR THE CYCLICITY OF CM ELLIPTIC CURVES MOD p

by Amir Akbary, V. Kumar Murty
"... 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 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 Siegel-Walfisz theorem regarding the distribution of primes in arithmetic progressions. 1.

On the asymptotics for invariants of elliptic curves modulo p

by Adam Tyler Felix, M. Ram Murty , 2013
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...is, there are no zeroes of the Dedekind zeta functions of Q(E[k]) in the region ℜ(s) >3/4as k ranges over squarefree integers. In [6], she also simplified the unconditional proof of when E hasCM,andin=-=[8]-=-,sheandM.RamMurtyprovedthatif the GRH is assumed, then the relation becomes if E does not have CM and NE (x) = cEli(x) + ON (x 5/6 (log x) 2/3 ) NE (x) = cEli(x) + ON (x 3/4 (log x) 1/2 ) if E has CM....

Drinfeld modules with maximal Galois action on their torsion points

by David Zywina , 1110
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...nditionally. If cE > 0, then Murty and Gupta [GM90] showed that fE(x)≫E x/ log2 x, and the constant cE is positive if and only if Q(E[2]) 6= Q. For more background and progress on the conjecture, see =-=[CM04]-=-. The Drinfeld module analogue has been formulated and proven by W. Kuo and Y.-R. Liu [KL09]. We shall say that φFp is cyclic if it is isomorphic as an A-module to A/wA for some non-constant element w...

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