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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...

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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...

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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...

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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...

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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] ⊂...

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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....

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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...