aa r X i v : . [ m a t h . N T ] J a n Average Cyclicity for Elliptic Curves in Torsion Families
Luke FredericksJanuary 18, 2021
Abstract
We prove asymptotic formulas for cyclicity of reductions of elliptic curves overthe rationals in a family of curves having specified torsion. These results agree withestablished conditional results and with average results taken over larger families. Asa key tool, we prove an analogue of a result of Vl˘adut¸ that estimates the number ofelliptic curves over a finite field with a specified torsion point and cyclic group structure.
Let E be an elliptic curve over Q . Then E is defined by a Weierstrass equation E : y + a xy + a y = x + a x + a x + a , a i ∈ Q . By an admissible changes of variable and after clearing denominators, E can be given bya short Weierstrass equation E ( a,b ) : y = x + ax + b, a, b ∈ Z . It is well-known that given an elliptic curve E over a field K , the set E ( K ) of K -rationalpoints of E has the structure of an abelian group. When K is a number field, the Mordell-Weil theorem says that E ( K ) is finitely generated, and Mazur’s torsion theorem classifiesthe possibilities for E ( Q ) tors . If K = F q is the finite field of q elements, then E ( F q ) is a finiteabelian group of rank at most 2; that is, E ( F q ) ∼ = Z /n Z × Z /n Z where n | n . Denote by a q ( E ) = E ( F q ) − q −
1; then the Hasse bound states that | a q ( E ) |≤ √ q. For all but finitely many primes p , we obtain an elliptic curve E p / F p by reducing thecoefficients of a Weierstrass model of E modulo p . It is natural to ask how how the propertiesof E p vary with p . Given E/ Q , denote by π tE ( x ) = { p ≤ x : a p ( E ) = t } cycE ( x ) = { p ≤ x : E p ( F p ) is cyclic } . The asymptotic growth of these functions are the subjects of the Lang-Trotter conjectureand the cyclicity conjecture, respectively.
Conjecture 1 (Lang-Trotter) . Let E/ Q be an elliptic curve, and let r ∈ Z with r = 0 if E has complex multiplication. Then π rE ( x ) ∼ C E,r √ x log x , (1)where C E,r is an explicit constant depending only on E and r .Denote by Q ( E [ k ]) the field obtained by adjoining to Q the coordinates of all points inthe k -torsion subgroup E [ k ] of E , and by li( x ) the logarithmic integral. Conjecture 2 (Cyclicity) . Let E/ Q be a non-CM elliptic curve. Then π cycE ( x ) ∼ X k ≥ µ ( k )[ Q ( E [ k ]) : Q )] ! li( x ) . (2)The study of π cycE goes back to the work of Borosh, Moreno and Porta [3] who suggestedthat for certain chosen examples of E/ Q , E p ( F p ) is cyclic for infinitely many p . Serreformulated and proved the cyclicity conjecture conditional on the Generalized RiemannHypothesis for the division fields of E [15]. The best result to date is the following conditionaltheorem of Cojocaru and Murty. Theorem 1. [5, Theorem 1.1] Let E be a non-CM elliptic curve defined over Q of conductor N . Assuming GRH for the Dedekind zeta functions of the division fields of E , we have that π cycE ( x ) = f E li( x ) + O N (cid:0) x / (log x ) / (cid:1) (3)where f E = X k ≥ µ ( k )[ Q ( E [ k ]) : Q )] . (4)Unconditionally, we cannot prove that the asymptotic in (1) or (2) holds for a singleelliptic curve E . However, starting with the work of Fouvry and Murty [7], average versions ofConjectures 1 and 2 have been obtained. These average results provide strong unconditionalevidence for the corresponding conjectures.Let π / ( X ) = Z X dt √ t log t ∼ √ X log X . (5)In the case of the Lang-Trotter conjecture we have2 heorem 2. [6, Corollary 1.3] Let E ( a, b ) : y = x + ax + b , and let ǫ >
0. If
A, B > X ǫ then we have as X → ∞ AB X | a |≤ A | b |≤ B π rE ( a,b ) ( X ) ∼ D r π / ( X ) (6)where D r = 2 π Y ℓ ∤ r ℓ ( ℓ − ℓ − ℓ + 1)( ℓ − Y ℓ | r ℓ ℓ − . (7)Here and throughout the paper, ℓ denotes a prime number, and a product over ℓ is takenover all primes satisfying the given conditions.In the case of the cyclicity conjecture, Banks and Shparlinski proved Theorem 3. [1, Theorem 17] Let ǫ >
K > A and B satisfying AB ≥ x ǫ , A, B ≤ x − ǫ , we have14 AB X | a |≤ A X | b |≤ B π cycE ( a,b ) ( x ) = C cyc π ( x ) + O ( π ( x ) / (log x ) K ) , where C cyc = Y ℓ prime (cid:18) − ℓ ( ℓ − ℓ − (cid:19) , and the constant implied by O depends only on ǫ and K .The average asymptotics described above provide strong evidence for the conjecturesin each case. In particular, the constants f E and C cyc and C E,r and D r are clearly closelyrelated. We view C cyc and D r as idealized constants where the variation of individual curvesare averaged out. Furthermore, Jones [11] proved that the average of the constants predictedby the respective conjectures is indeed the constant seen in the average results.Jones proof leveraged the fact (also due to Jones) that almost all elliptic curves are whatare known as Serre curves [12]. However, there are interesting families which consist entirelyof elliptic curves that are not
Serre curves. These curves are essentially invisible in the prioraverage results cited above; it is therefore of interest to study the averages of the functions π tE and π cycE as E varies over such a family.One class of such families are the torsion families – the family of elliptic curves somespecified torsion structure. It follows from Mazur’s theorem that if E has a point of order m ≥
2, then m ∈ { , , , , , , , , , } . The elliptic curves E/ Q that have a rationalpoint of order m ≥ E m ( a )4 y + xy − ay = x − ax y + (1 − a ) xy − ay = x − ax y + (1 − a ) xy − ( a + a ) y = x − ( a + a ) x y + (1 + a − a ) xy + ( a − a ) y = x + ( a − a ) x y + (cid:16) − a +4 a − a (cid:17) xy + ( − a + 3 a − y = x + ( − a + 3 a − x y + (1 + a − a ) xy + ( a − a + 2 a − a ) y = x + ( a − a + 2 a − a ) x y + (cid:18) a − a − a + 1 a − a + 1 (cid:19) xy + (cid:18) − a + 3 a − a a − a + 11 a − a + 1 (cid:19) y = x + (cid:18) − a + 3 a − a a − a + 11 a − a + 1 (cid:19) x y + (cid:18) a − a + 2 a + 2 a − a − a + 3 a − (cid:19) xy + (cid:18) − a + 30 a − a + 21 a − a + aa − a + 6 a − a + 1 (cid:19) y = x + (cid:18) − a + 30 a − a + 21 a − a + aa − a + 6 a − a + 1 (cid:19) x Table 1: Parameterizations for elliptic curves with m -torsion.The discriminant ∆ m ( a ) of the curve with m -torsion given above is∆ ( a ) = (16 a + 1) a ∆ ( a ) = a ( a − a − ( a ) = (9 a + 1)( a + 1) a ∆ ( a ) = ( a − a ( a − a + 5 a + 1)∆ ( a ) = a − (2 a − ( a − (8 a − a + 1)∆ ( a ) = ( a − a ( a − a + 1) ( a − a + 3 a + 1)∆ ( a ) = (2 a − ( a − a ( a − a + 1) − (4 a − a − ( a ) = ( a − − (2 a − a (6 a − a + 1)(2 a − a + 1) (3 a − a + 1) . James [10] gave the first results in this direction when he obtained an asymptotic forthe Lang-Trotter conjecture on average over the family of curves with a point of order 3.Battista, Bayless, Ivnaov, and James [2] extended this investigation over the family of ellipticcurves which possess a Q -rational point of order m for m = 5 , Theorem 4. [2, Theorem 3] Let E m ( a ) be the parameterization of elliptic curves which havea rational point of order m ∈ { , , } . Then for any c >
0, we have1 C ( N ) X ′| a |≤ N π rE m ( a ) ( X ) = 2 π C r,m π / ( X ) + O X / N + √ X log c X ! , where P ′ represents the sum over non-singular curves, C ( N ) represents the number of curves4n the sum, and C r,m = C r ( m ) Y ℓ ∤ mℓ ∤ r ℓ ( ℓ − ℓ − ℓ + 1)( ℓ − Y ℓ ∤ mℓ | r ℓ ℓ − , where C r ( m ) = / m = 5 and r ≡ , , , / m = 7 and r ≡ , , , , , , / m = 9 and r ≡ , , . The main result in this paper is to establish an average cyclicity result over torsionfamilies of elliptic curves. More precisely, we prove
Theorem 5.
Let ǫ > A > x ǫ . Let E m ( a ) denote the parameterization of elliptic curvesover Q which have a rational m -torsion point for m = 2 ,
3. Then12 A X ′| a |≤ A π cycE m ( a ) ( x ) = C m Y ℓ ∤ m (cid:18) − ℓ ( ℓ − ℓ − (cid:19) π ( x ) + O (cid:18) x − ǫ log x (cid:19) where C m summarized in Table 2. m C m
12 1920 512 4142 12 56 1940 512
Table 2:
Remark 1.
The effect of the presence of m -torsion is apparent; we interpret the constant C cyc as a product of local factors, each of which is the probability that E p has cyclic ℓ -torsion.The presence of a point of order m should have some influence on these probabilities for ℓ | m .Indeed, a curve with a point of order ℓ over Q need only acquire a single linearly independentpoint of order ℓ for cyclicity of the reduction to fail. Compared to the generic case of a curvewithout a point of order ℓ , we expect it to be much less likely that the reductions of thesecurves are cyclic. Our result quantifies this heuristic reasoning. Remark 2.
The family of curves with a point of order m for m = 4 or 8 includes curveswith full 2-torsion. Since the torsion points of E ( Q ) injects into E p ( F p ), the reductionsof these curves never have cyclic group of F p -rational points. It would be interesting toobtain a similar result where the average is taken only over curves with cyclic 2-torsion.Furthermore, we note that we have not treated all torsion families; instead we have focusedon curves in one-parameter families. The two-parameter families (curves with 2-torsion or3-torsion points) will be the subject of future work.5 emark 3. The proof broadly follows the steps of [1]; however, we note that an importantfeature of that work was the use of character sums to reduce the size of the family over whichthe average is taken. The torsion families over which we average in the current paper areeach a one-parameter family, and it is unclear how to adapt the character sum estimates tothis context. Consequently, we average over curves in a larger family than we would prefer.Reducing the size of the family will be the subject of subsequent work.A key ingredient for the argument in [1] was the fixed-field count of Vl˘adut¸ [17] which, fora finite field of q elements, estimates the number of E/ F q which have cyclic group structure.Our proof requires the following analogous fixed field count which takes into account theadditional torsion data.It is frequently convenient to express counts of elliptic curves over finite fields as weightedcardinalities where we weight each curve by the size of its automorphism group. We indicateweighted cardinalities by ′ .For a prime number ℓ , denote by v ℓ ( n ) the ℓ -adic valuation of n . Concretely, any positiveinteger n can be written n = ℓ e m where ℓ ∤ m . Then v ℓ ( n ) = e . Theorem 6.
Denote by C q ( m ) = { E/ F q : E ( F q ) is cyclic and contains a point of order m } / ∼ = F q . Then ′ C q ( m ) = q Y ℓ | mq ≡ ℓ ) ℓ v ℓ ( m ) Y ℓ | mq ℓ ) ϕ ( ℓ v ℓ ( m ) ) Y ℓ ∤ mq ≡ ℓ ) (cid:18) − ℓ ( ℓ − (cid:19) + O (cid:0) q / (cid:1) . The structure of this paper is as follows. In Section 2, we give the proof of Theorem 6.In Section 3, we describe the isomorphism classes of E/ F p in the torsion families describedabove. In Section 4, we give the proof of Theorem 5. There has been significant recent interest in counting problems for elliptic curves over a fixedfinite field F q with specified conditions on their group of F q -rational points. See for exampleHowe, [8], Vl˘adut¸ [17], Castryck and Hubrechts [4] and Kaplan and Petrow [13].Our goal is to generalize Vl˘adut¸’s result giving the number of elliptic curves E/ F q suchthat E ( F q ) is cyclic to obtain a count of the number of E/ F q such that for some fixed m ∈ Z , E ( F q ) has a point of order m and is cyclic. We begin by recalling the various fixed fieldcounts we will require.Denote by ϕ the Euler totient function, and define ψ ( n ) = n Q l | n (1 + 1 /l ). For a | b ,denote by W ( a, b ) = { E/ F q : E [ b ]( F q ) ∼ = ( Z /a Z ) × ( Z /b Z ) } / ∼ = F q . Estimates for the size of W ( a, b ) are given by Howe [8]. Howe shows that | W ( a, b ) − ˆ w ( a, b ) | < Cq / C whereˆ w ( a, b ) = qψ ( b/a ) aϕ ( b ) ψ ( b ) Y l prime l | gcd( b,q − /b (cid:18) − l (cid:19) . It will be convenient to define ˜ w ( a, b ) = ˆ w ( a, b ) /q . Howe notes that ˜ w ( a, b ) is a multiplicativefunction of both arguments simultaneously.Vl˘adut¸ observes the ‘obvious’ cyclicity condition: E ( F q ) is cyclic if and only if for anyprime l | q − E W ( l, l ). Assuming that our curve E has a point of order m , we observethat E ( F q ) is cyclic if and only if for all l | q − ( E W ( l, m ) (in the case where l | m ) ,E W ( l, lm ) (in the case where l ∤ m ) . For q = p n , define r ′ q ( m ) by the following conditions:(a) r ′ q is multiplicative.(b) For l prime, l = p , we have r ′ q ( l n ) = ( / ( l n − l n − ) if v ≥ n, ( l v +1 + 1) / ( l n +2 v − ( l − v < n, where v = v l ( q − r ′ q ( p e ) = 1 / ( p e − p e − ).Denote by T q ( m ) = { E/ F q : E has a point of order m } / ∼ = F q . It follows from Theorem 3 of[4] that | T q ( m ) − qr ′ ( m ) |≤ Cm log log( m ) q / for an absolute and explicitly computable C ∈ R > .Applying these estimates and the inclusion/exclusion principle, we obtain the estimate ′ C q ( m ) = qr ′ q ( m ) − X d | md> ˆ w ( d, m ) + X t | q − t,m )=1 µ ( t ) ˆ w ( t, mt ) + O (cid:0) q / (cid:1) . (8)Note that we have˜ w (1 , m ) = Y l | m ˜ w (1 , l v l ( m ) ) = Y l | mv l ( q − ϕ ( l v l ( m ) ) Y l | mv l ( q − > l v l ( m ) . q , we see that it is sufficient to show that r ′ q ( m ) − X d | md> ˜ w ( d, m ) + X t | q − t,m )=1 µ ( t ) ˜ w ( t, mt ) = ˜ w (1 , m ) Y ℓ | q − ℓ ∤ m (cid:18) − ℓ ( ℓ − (cid:19) + O (cid:0) q − / (cid:1) . Furthermore, since ˜ w ( t, mt ) = ˜ w ( t, t ) ˜ w (1 , m ) for t relatively prime to m , it suffices to showthat r ′ q ( m ) = X d | m ˜ w ( d, m ) . (9)Indeed, in this case, we have C q ( m ) = qr ′ q ( m ) − X d | md> ˆ w ( d, m ) + X t | q − t,m )=1 µ ( t ) ˆ w ( t, mt ) + O ( q / )= qr ′ q ( m ) − q X d | md> ˜ w ( d, m ) + q X t | q − t,m )=1 µ ( t ) ˜ w ( t, t ) ˜ w (1 , m ) + O ( q / ))= q ˜ w (1 , m ) X t | q − t,m )=1 µttϕ ( t ) ψ ( t ) + O ( q / )= q Y l | mv l ( q − ϕ ( l v l ( m ) ) Y l | mv l ( q − > l v l ( m ) Y l | q − m,l )=1 (cid:18) − l ( l − (cid:19) + O ( q / ) . We will prove (9) by induction on the number of prime factors of m . The following lemmaprovides the base of induction. Lemma 1.
Denote by v the ℓ -adic valuation of q −
1. Then r ′ q ( ℓ e ) = t X k =0 ˜ w ( ℓ k , ℓ e ) (10)where t = ( v if v < ee if v ≥ e. Proof.
Suppose that v = 0. Then v X k =0 ˜ w ( ℓ k , ℓ e ) = ˜ w (1 , ℓ e ) = ψ ( ℓ e ) ϕ ( ℓ e ) ψ ( ℓ e ) = r ′ q ( ℓ e ) . < v < e . Then we have v X k =0 ˜ w ( ℓ k , ℓ e ) = ψ ( ℓ e − v ) ℓ v ϕ ( ℓ e ) ψ ( ℓ e ) + v − X k =0 ψ ( ℓ e − k )( ℓ − ℓ k +1 ϕ ( ℓ e ) ψ ( ℓ e )= ℓ e − v − ( ℓ + 1) ℓ v ϕ ( ℓ e ) ψ ( ℓ e ) + v − X k =0 ℓ e − k − ( ℓ − ℓ k +1 ϕ ( ℓ e ) ψ ( ℓ e )= 1 ℓ e ϕ ( ℓ e ) ψ ( ℓ e ) ℓ e − v + ℓ e − v − + v − X k =0 ( ℓ e − k − ℓ e − k − ) ! . The sum above telescopes to ℓ e − ℓ e − v , and we are left with ℓ e + ℓ e − v − ℓ e ϕ ( ℓ e ) ψ ( ℓ e ) = ℓ e − v − ( ℓ v +1 + 1) ℓ e − ( ℓ − ℓ v +1 + 1 ℓ e − v − ( ℓ −
1) = r ′ q ( ℓ e ) , as needed.Finally, suppose v ≥ e . we have e X k =0 ˜ w ( ℓ k , ℓ e ) = 1 ℓ e ϕ ( ℓ e ) ψ ( ℓ e ) + e − X k =0 ψ ( ℓ e − k )( ℓ − ℓ k +1 ϕ ( ℓ e ) ψ ( ℓ e )= 1 ℓ e ϕ ( ℓ e ) ψ ( ℓ e ) + e − X k =0 ℓ e − k − ( ℓ − ℓ k +1 ϕ ( ℓ e ) ψ ( ℓ e )= 1 ℓ e ϕ ( ℓ e ) ψ ( ℓ e ) e − X k =0 ( ℓ e − k − ℓ e − k − ) ! . As above, this sum telescopes, leaving ℓ e −
1. We are left with ℓ e ℓ e ϕ ( ℓ e ) ψ ( ℓ e ) = ℓ e ℓ e − ( ℓ −
1) = 1 ℓ e − ℓ e − = r ′ q ( ℓ e ) . Proof of Theorem 6.
Assume for induction that for some m > r ′ q ( m ) = X d | m ˜ w ( d, m ) . Let ℓ be a prime that does not divide m and let e ≥
1. Let t be as in the statement ofLemma 1. We have 9 d | mℓ e ˜ w ( d, mℓ e ) = t X k =0 X d | m ˜ w ( ℓ k d, ℓ e m ) = t X k =0 ˜ w ( ℓ k , ℓ e ) X d | m ˜ w ( d, m ) = r ′ q ( ℓ e ) r ′ q ( m ) = r ′ q ( ℓ e m )by Lemma 1 and the induction hypothesis. m -torsion Curves The parameterizations given in Table 1 were derived by Kubert by studying the modularcurve X ( m ). A point of X ( m ) corresponds to an elliptic curve E together with a point oforder m up to action by automorphisms of E . For example, if ( E, P ) represents a point of X ( m ) and E ) = 2, then ( E, − P ) represents the same point. We will be concernedwith which parameter values yield isomorphic curves over F q where q = p n and p >
3. Withat most 10 exceptions, an isomorphism class of elliptic curves Over F q consists of curveswhose automorphism group has cardinality 2. If E ) = 2 and E has cyclic m -torsion,then there are ϕ ( m ) / F q that yield an isomorphic curve. These correspondto the ϕ ( m ) points of order m on E m ( a ), up to the action of of Aut( E m ( a )).If E ) >
2, the number of parameters yielding an isomorphic curve will varydepending on the size of the automorpism group and the number of m -torsion points of E ;in general, the number of parameters which yield an isomorphic curve will not be ϕ ( m ) / O (1) isomorphism classes can be absorbed into the ‘unweighted’ version ofTheorem 6. Corollary 1 (to Theorem 6) . C q ( m ) = 2 q Y ℓ | mq ≡ ℓ ) ℓ v ℓ ( m ) Y ℓ | mq ℓ ) ϕ ( ℓ v ℓ ( m ) ) Y ℓ ∤ mq ≡ ℓ ) (cid:18) − ℓ ( ℓ − (cid:19) + O (cid:0) q / (cid:1) . Proof.
Denote by C q ( m, n ) = { E ∈ C q ( m ) : E ) = n } . Then C q ( m ) = C q ( m,
2) + C q ( m,
4) + C q ( m, . We then have C q ( m, / ′ C q ( m ) − C q ( m, / C q ( m, / . Multiplying by 2 and observing that C q ( m, / C q ( m, / O (1), we have C q ( m,
2) = 2 q Y ℓ | mq ≡ ℓ ) ℓ v ℓ ( m ) Y ℓ | mq ℓ ) ϕ ( ℓ v ℓ ( m ) ) Y ℓ ∤ mq ≡ ℓ ) (cid:18) − ℓ ( ℓ − (cid:19) + O (cid:0) q / (cid:1) as required. 10iven an m -torsion curve E m ( a ), we can perform a change of variables to obtain ashort Weierstrass equation y = x + Ax + B . In order for E m ( a ) to have more than twoautomorphisms, the j -invariant must be 0 or 1728. In terms of the short Weierstass equation,this means that A = 0 or B = 0, respectively. The coefficients A and B will be polynomialsor rational functions in the parameter a . Since such functions have finitely many zeros, wededuce the following Lemma 2.
Each torsion family E m ( a ) contains finitely many curves with j -invariant 0 or1728. The parameters which yield curves with these j -invariants are roots of a polynomialthat depends only on m .Using explicit change of variables, it is possible to specify precisely which parametervalues yield isomorphic curves. We summarize this below for curves with E ) = 2. m Parameters4 a a − a − a a (1 − a ) a − − (1 − a ) − a − a + 19 a ( a − a − − ( a − − a ( a − a − − a − a + 1Table 3: Parameters yielding isomorphic curves. The fixed field counts of E/ F p which have an m -torsion point and cyclic group of F p -pointsdepends on the value of p modulo the prime divisors of m . For one-parameter torsion familiesover Q , we are concerned with m ∈ { , , , , , , , } . In this case, m has at most twoprime divisors, and if m is not a prime power, then one of its prime factors is 2. Since all oddprimes are 1 (mod 2), the number of curves we are counting varies according to the value of p (mod ℓ ) where ℓ is the unique odd prime divisor of m .Let m ∈ { , , , , , , , } , and denote by E m ( A ) = { E m ( a ) : − A ≤ a ≤ A } the family of elliptic curves over Q with an m -torsion point given above. Write the primefactorization of m as m = 2 k ℓ n where we take n = ℓ = 1 if m is a power of 2. Let A ≥ x ǫ for x, ǫ ≥
0. 11e have X | a |≤ A π cycE m ( a ) ( x ) = X p ≤ x X b ∈ F p ∆ m ( b ) =0 E m ( b )( F p ) cyclic { a ∈ [ − A, A ] : E m ( a ) p ∼ = E m ( b ) } . There are 2
A/p + O (1) values of a which yield a particular Weierstrass model modulo p .Thus the expression above becomes X p ≤ x (cid:18) Ap + O (1) (cid:19) ϕ ( m )2 X b ∈ F p ∆ m ( b ) =0 E m ( b )( F p ) cyclic E m ( b ))=2 ϕ ( m )4 X b ∈ F p ∆ m ( b ) =0 E m ( b )( F p ) cyclic E m ( b ))=4 ϕ ( m )6 X b ∈ F p ∆ m ( b ) =0 E m ( b )( F p ) cyclic E m ( b ))=6 . Applying the estimate obtained in Corollary 1 and Lemma 2, this is equal to X p ≤ xp ≡ ℓ ) (cid:18) Ap + O (1) (cid:19) ϕ ( m )2 2 pm Y ℓ | p − ℓ,m )=1 (cid:18) − ℓ ( ℓ − (cid:19) + O (cid:0) p / (cid:1) + X p ≤ xp ℓ ) (cid:18) Ap + O (1) (cid:19) ϕ ( m )2 2 p k ϕ ( ℓ n ) Y ℓ | p − ℓ,m )=1 (cid:18) − ℓ ( ℓ − (cid:19) + O (cid:0) p / (cid:1) . Simplifying and applying the trivial estimate ϕ ( m ) p k ϕ ( ℓ n ) Y ℓ | p − ℓ,m )=1 (cid:18) − ℓ ( ℓ − (cid:19) < p, we have X p ≤ xp ≡ ℓ ) Aϕ ( m ) m Y ℓ | p − ℓ,m )=1 (cid:18) − ℓ ( ℓ − (cid:19) + X p ≤ xp ℓ ) Aϕ ( m )2 k ϕ ( ℓ n ) Y ℓ | p − (cid:18) − ℓ ( ℓ − (cid:19) + O X p ≤ x p ! . (11)Note that according to Lemma 3.4 of [16], we have X p ≤ x p = O (cid:18) x x (cid:19) .
12e analyze these two sums individually following [1]. A main input to this analysis is atheorem on averages of multiplicative functions due to [9, Theorem 3].Assume first that m is not a power of 2, so that ℓ >
1. For any integer n , define χ ℓ ( n ) = ( ℓ ∤ n ℓ | n,F ( n ) = Y ℓ | nℓ ∤ m (cid:18) − ℓ ( ℓ − (cid:19) , and F ′ ( n ) = Y ℓ | nℓ ∤ m (cid:18) − ℓ ( ℓ − (cid:19) χ ℓ h ( n ) . Note that χ ℓ and F are multiplicative (whence F ′ is multiplicative as well). We compute X p ≤ xp ≡ ℓ ) Y ℓ | p − ℓ,m )=1 (cid:18) − ℓ ( ℓ − (cid:19) = X p ≤ x Y ℓ | p − ℓ,m )=1 (cid:18) − ℓ ( ℓ − (cid:19) − X p ≤ xp ℓ ) Y ℓ | p − ℓ,m )=1 (cid:18) − q ( q − (cid:19) = X p ≤ x Y ℓ | p − ℓ,m )=1 (cid:18) − ℓ ( ℓ − (cid:19) − X p ≤ x Y ℓ | p − ℓ,m )=1 (cid:18) − ℓ ( ℓ − (cid:19) χ ℓ ( p − X p ≤ x F ( p −
1) + X p ≤ x F ′ ( p − . Denote by G and G ′ the Dirichlet convolution of the M¨obius µ function with F , F ′ , respec-tively. Explicitly, G and G ′ are multiplicative functions defined on prime powers by G ( ℓ k ) = − ℓ ( ℓ − if ℓ ∤ m, k = 10 if ℓ | m, k = 10 if k > , and G ′ ( ℓ k ) = − ℓ ( ℓ − if ℓ ∤ m, k = 1 − ℓ = ℓ , k = 10 if ℓ = 2 | m, k = 10 if k > . F, G and F ′ , G ′ satisfy the hypotheses of [9, Theorem 3]. It followsthat 1 π ( x ) X p ≤ x F ( p −
1) = ∞ X d =1 G ( d ) ϕ ( d ) + O B (log − B x )holds for any B >
0, and similarly for F ′ , G ′ . Now we have ∞ X d =1 G ( d ) ϕ ( d ) = Y ℓ ∤ m (cid:18) − ℓ ( ℓ − ℓ − (cid:19) , and ∞ X d =1 G ′ ( d ) ϕ ( d ) = ℓ − ℓ − Y ℓ ∤ m (cid:18) − ℓ ( ℓ − ℓ − (cid:19) . Thus, X p ≤ xp ≡ ℓ ) Aϕ ( m ) m Y ℓ | p − ℓ,m )=1 (cid:18) − ℓ ( ℓ − (cid:19) = 2 Aϕ ( m ) m ℓ − Y ℓ ∤ m (cid:18) − ℓ ( ℓ − ℓ − (cid:19) π ( x ) + O (cid:18) A x log B +1 (cid:19) . (12)Similarly, we write X p ≤ xp ℓ ) Y ℓ | p − (cid:18) − ℓ ( ℓ − (cid:19) = X p ≤ x Y ℓ | p − (cid:18) − ℓ ( ℓ − (cid:19) χ ℓ ( p −
1) = X p ≤ x F ′ ( p − π ( x ) X p ≤ x F ′ ( p −
1) = ∞ X d =1 G ′ ( d ) ϕ ( d ) + O B (log − B x )= ℓ − ℓ − Y ℓ ∤ ℓ (cid:18) − ℓ ( ℓ − ℓ − (cid:19) + O B (log − B x )holds for any B >
0. Thus, X p ≤ xp ℓ ) Aϕ ( m )2 k ϕ ( ℓ n ) Y ℓ | p − (cid:18) − ℓ ( ℓ − (cid:19) = 2 Aϕ ( m )2 k ϕ ( ℓ n ) ℓ − ℓ − Y ℓ ∤ m (cid:18) − ℓ ( ℓ − ℓ − (cid:19) π ( x ) + O (cid:18) A x log B +1 (cid:19) . (13)14ombining (11), (12), and (13), we have X | a |≤ A π cycE ( a ) ( x ) = (cid:18) Aϕ ( m )2 k ϕ ( ℓ n ) ℓ − ℓ − Aϕ ( m ) m ℓ − (cid:19) Y ℓ ∤ m (cid:18) − ℓ ( ℓ − ℓ − (cid:19) π ( x )+ O (cid:18) A x log B +1 (cid:19) + O (cid:18) x x (cid:19) . If m is a power of 2, the sum over p ℓ ) is empty. In this case, ϕ ( m ) /m = 1 / X p ≤ x Y ℓ | p − ℓ,m )=1 (cid:18) − ℓ ( ℓ − (cid:19) . An argument analogous to the above shows X | a |≤ A π cycE ( a ) ( x ) = A Y ℓ =2 (cid:18) − ℓ ( ℓ − ℓ − (cid:19) π ( x ) + O (cid:18) A x log B +1 (cid:19) + O (cid:18) x x (cid:19) . Computing C m = (cid:18) Aϕ ( m )2 k ϕ ( ℓ n ) ℓ − ℓ − Aϕ ( m ) m ℓ − (cid:19) for m ∈ { , , , , , , , } ,we complete the proof. References [1] William D. Banks and Igor E. Shparlinski. Sato-Tate, cyclicity, and divisibility statisticson average for elliptic curves of small height.
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