The Lang-Trotter Conjecture on Average

Transcription

1 arxiv:math/ v [math.nt] Se 2006 The Lang-Trotter Conjecture on Average Stehan Baier July, 208 Abstract For an ellitic curve E over Q and an integer r let πe r (x) be the number of rimes x of good reduction such that the trace of the FrobeniusmorhismofE/F equalsr. Weconsiderthequantityπ E r (x) on average over certain sets of ellitic curves. More in articular, we establish the following: If A,B > x /2+ε and AB > x 3/2+ε, then the arithmetic mean of πe r (x) over all ellitic curves E : y2 = x 3 +ax+b with a,b Z, a A and b B is C r x/logx, where Cr is some constant deending on r. This imroves a result of C. David and F. Paalardi. Moreover, we establish an almost-all result on πe r (x). Mathematics Subject Classification (2000): G05 Keywords: Lang-Trotter conjecture, average Frobenius distribution, character sums Introduction and main results Let E be anellitic curve over Q. For any rime number of goodreduction, let a (E) be the trace of the Frobenius morhism of E/F. Then the number of oints on the reduced curve modulo equals #E(F ) = + a (E). Furthermore, by Hasse s theorem, a (E) 2. For a fixed integer r, let π r E (x) := #{ x : a (E) = r}. If r = 0 and E has comlex multilication, Deuring [2] showed that

2 2 Lang-Trotter on average πe 0 π(x) (x) as x. 2 Primes with a = 0 are known as suersingular rimes. Lang and Trotter [7] conjectured that for all other cases an asymtotic estimate of the form x πe r (x) C E,r as x logx with a well-defined constant C E,r 0 holds. They used a robabilistic model to give an exlicit descrition of the constant C E,r. The constant can be 0, and the asymtotic estimate is then interreted to mean that there is only a finite number of rimes such that a (E) = r. A concise account of Lang- Trotter s robabilistic model and an exression of C E,r as an Euler roduct can be found in []. Fouvry and Murty [5] obtained average estimates related to the Lang- Trotter conjecture for the suersingular case r = 0. Their result was later generalized by David and Paalardi [] to any r Z. In this aer, we shall imrove the results of David and Paalardi. As in [], we define and a constant C r by (.) C r := 2 π Our first result is π /2 (x) := l r x 2 dt 2 tlogt ( l 2 ) l r x logx l(l 2 l ) (l )(l 2 ). Theorem : Let r be a fixed integer and A,B. Then, for every c > 0, we have AB a A b B = C r π /2 (x)+o π r E(a,b) (( A + B ) xlogx+ x5/ log 3 x AB + ) x log c, x

3 S.Baier 3 where the imlied O-constant deends only on c and r. David and Paalardi [] obtained the above result with (/A+/B)x 3/2 in lace of (/A+/B)xlogx and x 5/2 /(AB) in lace of x 5/ log 3 x/ AB in the O-term. From Theorem, we immediately obtain the following Lang-Trotter tye estimate on average. Theorem 2: Let ε > 0. If A,B > x /2+ε and AB > x 3/2+ε, we have as x, x (.2) πe(a,b) r AB C r logx. a A b B In [], (.2) was roved under the stronger condition A,B > x +ε. David and Paalardi asked if (.2) is consistent with the Lang-Trotter conjecture in the sense that (.3) AB a A b B C E(a,b),r C r as A,B. N. Jones [6] roved that this average estimate holds if the summation is restricted to a,b such that E(a,b) is a Serre curve. An ellitic curve is called a Serre curve if φ E (Gal(Q/Q)) is an index two subgrou in GL 2 (Ẑ), where φ E : Gal(Q/Q) GL 2 (Ẑ) denotes the Galois reresentation associated to E. By a result of Serre [8], φ E is never surjective, so in other words, E is a Serre curve if its Galois reresentation has image as large as ossible. Moreover, extending a result of W.D. Duke [3], Jones roved that, according to height, almost all ellitic curves over Q are Serre curves. This gives some evidence that (.3) really holds. Furthermore, David and Paalardi roved that π r E(a,b) (x) C r x/logx holds for almost all curves E(a,b) with a A and b B if A,B > x 2+ε (Theorem.3. in []). Here we show that this almost-all result holds for considerably smaller A, B-ranges.

4 Lang-Trotter on average Theorem 3: Let ε > 0 and fix c > 0. If A,B > x +ε and x 3+ε < AB < ex(ex( x/log c x)), then for all d > 2c and for all ellitic curves E(a,b) with a A and b B with at most O(AB/log d z) excetions, we have the inequality x πe(a,b) r (x) C rπ /2 (x) log c x. We shall establish the following more general estimate from which Theorem 3 can be derived by the Turán normal order method (c.f. []). Theorem : Let ε > 0. If A,B > x /2+ε and AB > x 3/2+ε, then for every c > 0, we have (.) = O AB a A b B (( A + B π r E(a,b) (x) C r π /2 (x) 2 ) x 2 + x5/2 log 3 x AB + x log c x +x/2 loglog(0ab) where the imlied O-constant deends only on c and r. ), 2 The work of David-Paalardi The following observations are the starting oint of David-Paalardi s work in []. Lemma : For r 2, the number of F -isomorhism classes of ellitic curves over F with + r oints is the total number of ideal classes of the ring Z[(D + D)/2], where D = r 2 is a negative integer which is congruent to 0 or modulo. This number is the Kronecker class number H(r 2 ). In the following, we set H r, = H(r 2 ). Lemma 2: Suose that 2,3. Then any ellitic curve over F has a model E : Y 2 = X 3 +ax +b

5 S.Baier 5 with a,b F. The ellitic curves E (a,b ) over, which are F -isomorhic to E, are given by all the choices a = µ a and b = µ 6 b with µ F. The number of such E is ( )/6, if a = 0 and mod 3; ( )/, if b = 0 and mod ; ( )/2, otherwise. The above Lemmas and 2 imly that the number of curves E(a,b) with a,b Z, 0 a,b < and a (E(a,b)) = r is (2.) H r, 2 Now David and Paalardi [] write (2.2) = AB AB a A b B π r E(a,b)(x) +O(). { a A, b B : a (E(a,b)) = r}, where B(r) = max{3,r,r 2 /}. Using (2.), the term on the right-hand side is ( )( )( ) 2A 2B (2.3) AB +O() +O() Hr, +O(). 2 This asymtotic estimate was used by David and Paalardi to rove their main theorem on the average Frobenius distribution of ellitic curves (Theorem in []). For the main term in (2.3) David and Paalardi roved the following. Lemma 3: Let r be a fixed integer. Then, for every c > 0, we have ( ) H r, x 2 = C rπ /2 (x)+o log c, x

6 6 Lang-Trotter on average where the constant C r is defined as in (.) and the imlied O-constant deends only on r and c. In this aer we shall sharen the error term in (2.3). 3 Preliminaries We first characterize the ellitic curves lying in a fixed F -isomorhism class, where is a rime 2,3. In the following, for z Z let z be the reduction of z mod. Furthermore, let z be a multilicative inverse mod, that is, zz mod. Lemma : Let a,b,c,d Z, abcd and E, E 2 be ellitic curves over F given by E : Y 2 = X 3 +ax +b. and E 2 : Y 2 = X 3 +cx +d. (i) If mod, then E and E 2 are F -isomorhic if and only if ca is a biquadratic residue mod and c 3 a 3 d 2 b 2 mod. (ii): If 3 mod, then E and E 2 are F -isomorhic if and only if ca and db are quadratic residues mod and c 3 a 3 d 2 b 2 mod. Proof: By Lemma 2, the curves E and E 2 are F -isomorhic if and only if there exists an integer m such that m and (3.) c m a mod and d m 6 b mod. (i) Suose that mod. If (3.) is satisfied, then it follows that ca is a biquadratic residue mod and c 3 a 3 m 2 d 2 b 2 mod. Assume, conversely, that ca is a biquadratic residue mod and (3.2) c 3 a 3 d 2 b 2 mod. Since mod, there exist two solutions m,m 2 of the congruence c m a mod such that m 2 2 m 2 mod, and (3.2) imlies that d 2 b 2 m 2 j mod for j =,2. From this it follows that db m 6 mod or db

7 S.Baier 7 m 6 m6 2 mod. Hence, the system (3.) is soluble for m. This comletes the roof of (i). (ii) Suose that 3 mod. If (3.) is satisfied, then it follows that ca and db are quadratic residues mod and c 3 a 3 m 2 d 2 b 2 mod. Assume, conversely, that ca and db are quadratic residues mod and (3.2) is satisfied. Then, since 3 mod, ca is also a biqadratic residue. Hence, there exists a solution m of the congruence c m a mod. Further, (3.2) imlies that d 2 b 2 m 2 mod. From this it follows that that db m 6 mod or db m 6 mod. But m 6 is a quadratic non-residue mod since 3 mod. Thus db m 6 mod since db is suosed to be a quadratic residue mod. Hence, we have db m 6 mod, and so the system (3.) is soluble for m. This comletes the roof of (ii). We shall detect ellitic curves lying in a fixed F -isomorhism class by using Dirichlet characters. For the estimation of certain error terms we then need the following results on character sums. Lemma 5: Let q,n Æ and (a n ) be any sequence of comlex numbers. Then 2 2 q a n χ(n) = ϕ(q) a n, χ mod q n N a= n N (a,q)= n a mod q where the outer sum on the left-hand side runs over all Dirichlet characters mod q. Proof: This is a consequence of the orthogonality relations for Dirichlet characters. Lemma 6: Let q,n Æ, q 2. Then χ(n) χ χ 0 n N N 2 qlog 6 q, where the outer sum on the left-hand side runs over all non-rincial Dirichlet characters mod q.

8 8 Lang-Trotter on average Proof: This is Lemma 3 in []. Lemma 7: Let q,n Æ, q 2 and χ be any non-rincial character mod q. Then χ(n) qlogq. n N Proof: This is the well-known inequality of Polya-Vinogradov. Furthermore, we shall need the following estimates for sums over H r,. Lemma 8: We have H /2 r, x5/, H r, x, H r, x and H r, 2. (3.3) Proof: By (26) in [], we have H r, x 3/2. Using the Cauchy-Schwarz inequality, we obtain H /2 r, x /2 H r, /2 x 5/ from (3.3). The remaining three estimates in Lemma 8 can be derived from (3.3) by artial summation. Finally, we shall need the following bound. Lemma 9: The number of F -isomorhism classes of ellitic curves containing curves E : Y 2 = X 3 +ax +b

9 S.Baier 9 over F with a = 0 or b = 0 is bounded by 0. Proof: By Lemma 2, the number of F -isomorhism classes containing curves E(0,b) with b F is 6, and the number of F -isomorhism classes containing curves E(a,0) with a F is. Proof of Theorem Let I r, be the number of F -isomorhism classes of ellitic curves E : Y 2 = X 3 +cx +d over F with + r oints such that c,d 0. Let (u,j,v,j ), j =,...,I r, be airs of integers such that the curves E(u,j,v,j ) form a system of reresentatives of these isomohism classes. We now write and { a A, b B : a (E(a,b)) = r} = { a A, b B : ab, a (E(a,b)) = r}+ ( ) AB O +A+B (.) = { a A, b B : ab, a (E(a,b)) = r} I r, { a A, b B : E(a,b) = E(u,j,v,j )}, j= where the symbol = stands for F -isomorhic. We rewrite the term on the right-hand side of (.) as a character sum. If mod, then, by Lemma (i) and the character relations, this term equals (.2) ϕ() I r, j= a A b B k= ( ) au k,j χ mod χ(a 3 u 3,j b 2 v 2,j), where ( /) is the biquadratic residue symbol. If 3 mod, then, by Lemma (ii) and the character relations, the term on the right-hand side of

10 0 Lang-Trotter on average (.) equals ϕ() χ mod I r, j= a A b B ( χ 0 (a)+ χ(a 3 u 3,j b 2 v 2,j ), ( ))( au,j χ 0 (b)+ ( )) bv,j where ( /) is the Legendre symbol and χ 0 is the rincial character. In the following, we consider only the case mod. The case 3 mod can be treated in a similar way. The exression in (.2) equals ϕ() k= χ mod j= I r, ( ) k u,j χ 3 (u,j )χ 2 (v,j ) a A We slit this exression into 3 arts M,E,E 2, where (i) M = contribution of k,χ with ( /) k χ3 = χ 0, χ 2 = χ 0 ; (ii) E = contribution of k,χ with ( /) k χ3 χ 0, χ 2 = χ 0 or ( /) k χ3 = χ 0, χ 2 χ 0 ; (iii) E 2 = contribution of k,χ with ( /) k χ3 χ 0, χ 2 χ 0. ( ) k a χ 3 (a) χ 2 (b). b B As one may exect, M shall turn out to be the main term and E, E 2 to be the error terms. Estimation of M. The only cases in which ( /) k χ3 = χ 0 and χ 2 = χ 0 are k = 0, χ = χ 0 and k = 2, χ = ( /). Now, by a short calculation, we obtain (.3) M = ABI r, 2 ( +O ( )). By Lemma 9, we have H r, I r, 0. Combining this with (.3), we obtain M = ABH ( r, AB +O 2 + ABH ) r,. 2

11 S.Baier Estimation of E. The number of solutions (k,χ) with k =,..., of ( /) k χ3 = χ 0 is bounded by 2, and χ 2 = χ 0 has recisely 2 solutions χ. Thus E is the sum of at most 2+ 2 = 20 terms of the form ϕ() I r, χ (u,j )χ 2 (v,j ) χ (a) χ 2 (b), j= a A b B where exactly one of the characters χ, χ 2 is the rincial character χ 0. Therefore, Lemma 7 imlies that E I r,(a+b) log. Estimation of E 2. Given k Z and a character χ mod, the number ( ) k of solutions χ of χ 3 = χ is 3, and the number of solutions χ of χ 2 = χ is 2. Thus, using the Cauchy-Schwarz inequality, we deduce that E 2 I r, ( ) k u,j 2 χ(u 3,j v2,j ) (.) /2 k= χ j= / χ(a) / χ(b). χ χ0 χ χ0 a A b B By Lemma (i), the number of j s such that u 3,j v2,j lie in a fixed residue class mod is bounded by. Using this, Lemma 5 and Lemma 6, the exression on the right-hand side of (.) is dominated by (I r, AB) /2 log 3. The final estimate. Combining all contributions, and using I r, H r,, we obtain (.5) { a A, b B : a (E(a,b)) = r} = ABH ( r, AB +O 2 + ABH r, +A+B +(H 2 r, AB) /2 log 3 + ) H r, (A+B) log The result of Theorem now follows from (2.2), (.5), Lemma 3 and Lemma 8.

12 2 Lang-Trotter on average 5 Proof of Theorem As in [], we set µ := AB a A b B π r E(a,b)(x). Fix any c > 0. Using Theorem and following the arguments in [], if A,B > x /2+ε and AB > x 3/2+ε, then ( ) x (5.) µ = C r π /2 (x)+o log c, x and the left-hand side of (.) is (5.2) a A b B {,q x : q, a (E(a,b)) = r = a q (E(a,b))} µ 2 + µ+ x log 2c x, where, q denote rimes. Similarly as in the receeding section, we have (5.3) = a A b B B(r)<,q x q {,q x : q, a (E(a,b)) = r = a q (E(a,b))} { a A, b B : a (E(a,b)) = r = a q (E(a,b))}.

13 S.Baier 3 Using Theorem and { : ab} = ω( ab ) loglog(0 ab ) if ab 0, we deduce (5.) = = B(r)<,q x q B(r)<,q x q +O x B(r)<,q x q { a A, b B : a (E(a,b)) = r = a q (E(a,b))} { a A, b B :,q ab, a (E(a,b)) = r = a q (E(a,b))} a A, b B ab πe(a,b) r (x) { a A, b B :,q ab, a (E(a,b)) = r = a q (E(a,b))} +O ( ABx /2 loglog(0ab)+(a+b)x 3/2). Now we fix,q with q. In the following, we confine ourselves to the case when q mod. The remaining cases q mod and q 3 mod can be treated in a similar way. Similarly as in the receeding section, we can exress the term { a A, b B :,q ab, a (E(a,b)) = r = a q (E(a,b))} as a character sum I r, I r,q 6ϕ()ϕ(q) ( au q,j q l= i= j= a A b B k= ) l χ mod q ( ) au k,i χ (a 3 u 3 q,j b 2 v 2 q,j ). χ mod χ(a 3 u 3,i b 2 v 2,i)

14 Lang-Trotter on average This sum equals (5.5) 6ϕ()ϕ(q) k= l= ( Ir,q ( ) l uq,j χ mod χ mod q χ q 3 (u q,j )χ 2 (v q,j ) j= ( ) χχ 2 (b). b B ( Ir, ( ) k u,i i= ) a A ( ) k a χ 3 (u,i )χ 2 (v,i ) ( ) l a q ) (χχ ) 3 (a) Let χ 0 be the rincial character mod and χ 0 be the rincial character mod q. Then χ 0 χ 0 is the rincial character mod q. As reviously, we slit the exression in (5.5) into 3 arts M,E,E 2, where (i) M = contribution of k,l,χ,χ with ( /) k ( /q)l (χχ ) 3 = χ 0 χ 0, (χχ ) 2 = χ 0 χ 0 ; (ii) E = contribution of k,l,χ,χ with ( /) k ( /q) l (χχ ) 3 χ 0 χ 0, (χχ ) 2 = χ 0 χ 0 or ( /) k ( /q)l (χχ ) 3 = χ 0 χ 0, (χχ ) 2 χ 0 χ 0 ; (iii) E 2 = contribution of k,l,χ,χ with ( /) k ( /q)l (χχ ) 3 χ 0 χ 0, (χχ ) 2 χ 0 χ 0. Estimation of M. The only cases in which ( /) k ( /q)l (χχ ) 3 = χ 0 χ 0, (χχ ) 2 = χ 0 χ 0 are: (a) k = l = 0, χ = χ 0, χ = χ 0; (b) k = l = 2, χ = ( /), χ = ( /q); (c) k = 0, l = 2, χ = χ 0, χ = ( /q); (d) k = 2, l = 0, χ = ( /), χ = χ 0. Now, by a short calculation, we obtain (5.6) M = ABI r,i r,q q ( +O ( + )). q

15 S.Baier 5 By Lemma 9, we have H r, I r, 0 and H r,q I r,q 0. Combining this with (5.6), we obtain M = ABH r,h r,q q ( ( AB(Hr, +H r,q ) +O +ABH r, H r,q q 2 q + )). q 2 Estimation of E. The number of solutions (k,l,χ,χ ) with k,l =,..., of ( /) k ( /q)l (χχ ) 3 χ 0 χ 0 is bounded by 22, and (χχ ) 2 = χ 0 χ 0 has recisely solutions (χ,χ ). Thus E is the sum of at most +6 = 228 terms of the form 6ϕ()ϕ(q) χ (a) I r, I r,q χ 2 (b) χ 3 (u,i )χ (v,i ) χ 3(u q,j )χ (v q,j ), a A b B i= where χ, χ 2 are characters mod q such that exactly one of them is the rincial character, χ 3, χ are characters mod, and χ 3, χ are characters mod q. Here the characters χ 3,, χ 3, deend on the characters χ,2. Now Lemma 7 imlies that E I r,i r,q (A+B) q logq. Estimation of E 2. Given k,l Z and a character χ mod q, the number of characters χ mod q such that ( /) k ( /q)l (χχ ) 3 = χ is 9, and the number of χ mod q such that χ 2 = χ is. Thus, using the Cauchy-Schwarz inequality, we deduce that (5.7) E 2 q k= χ l= I r,q χ j= χ 2 χ 0 χ 0 I r, ( ) l uq,j q i= ( ) k u,i χ (u 3 / χ(b) b B q,j v2 q,j), χ(u 3,i v2,i) 2 /2 j= 2 /2 χ χ 0 χ 0 χ (a) a A /

16 6 Lang-Trotter on average where χ runs over all characters mod, χ runs over all characters mod q, and χ,χ 2 run over all non-rincial characters mod q. By Lemma (i), the number of i s such that u 3,i v2,i lie in a fixed residue class mod is bounded by. The same is true for the number of j s such that u 3 q,j v2 q,j lie in a fixed residue class mod q. Using this, Lemma 5 and Lemma 6, the exression on the right-hand side of (5.7) is dominated by (I r, I r,q AB) /2 log 3 q. The final estimate. Combining all contributions, and using I r, H r,, we obtain (5.8) { a A, b B :,q ab, a (E(a,b)) = r = a q (E(a,b))} = ABH ( ( r,h r,q AB(Hr, +H r,q ) +O +ABH r, H r,q q q 2 q + ) q 2 +(H r, H r,q AB) /2 log 3 q + H ) r,h r,q (A+B) logq. q Wehaverovedthisestimateonlyfordistinct rimes,q with q mod, but the same estimate can be roved for q mod and q 3 mod in a similar way. Now, from (5.), (5.8), Lemma 3 and Lemma 8, we obtain (5.9) AB B(r)<,q x q = (C r π /2 (x)) 2 +O { a A, b B : a (E(a,b)) = r = a q (E(a,b))} ( ( + x5/2 log 3 x+ AB A + B H 2 r, 2 + x log c x +x/2 loglog(0ab) )x 2 ) From (23) in [] and h(d) d, we obtain H r, /2+ε which imlies that (5.0). H 2 r, 2 x ε.

17 S.Baier 7 The result of Theorem now follows from (5.), (5.2), (5.3), (5.9) and (5.0). Acknowledgement. This aer was written when the author held a ostdoctoral fellowshi at the Queen s University in Kingston, Canada. The author wishes to thank this institution for financial suort. He would further like to thank Prof. Ram Murty for his useful comments. References [] C. David, F. Paalardi, Average Frobenius Distributions of Ellitic Curves, Int. Math. Res. Not. (999) [2] M. Deuring, Die Tyen der Multilikatorenringe ellitischer Funktionenkörer, Abh. Math. Sem. Hansischen Univ. (9) [3] W.D. Duke, Ellitic curves with no excetional rimes, C. R. Acad. Sci., Paris, Sr. I, Math. 325 (997) [] J. Friedlander, H. Iwaniec, The divisor roblem for arithmetic rogressions, Acta Arith. 5 (985) [5] E. Fouvry, M.R. Murty, On the distribution of suersingular rimes, Canad. J. Math. 8 (996) 8-0. [6] N. Jones, The constants in the Lang-Trotter conjecture, rerint (2006). [7] S. Lang, H. Trotter, Frobenius Distributions in GL 2 extensions, Lecture Notes in Math. 50 (976) Sringer-Verlag, Berlin. [8] J. P. Serre, Proriétés galoisiennes des oints d ordre fini des courbes ellitiques, Invent. Math. 5 (972) Address of the author: Stehan Baier Queen s University

18 8 Lang-Trotter on average Jeffery Hall University Ave Kingston, ON K7L3N6 Canada