Titchmarsh divisor problem for abelian varieties of type I, II, III, and IV

Transcription

1 Titchmarsh divisor problem for abelian varieties of type I, II, III, and IV Cristian Virdol Department of Mathematics Yonsei University September 8, 05 Abstract We study Titchmarsh divisor problem in the context of abelian varieties. For abelian varieties of type I, II, III, and IV, under GRH, we obtain asymptotic formulas. 00 Mathematics subject classification: G0, G5. Keywords: Abelian varieties, Titchmarsh divisor problem. Introduction Let A be an abelian variety defined over a number field F, of conductor N, and of dimension r, where r is an integer. Let Σ F be the set of finite places of F, and let P A be the set of primes Σ F of good reduction for A, (i.e. (N F/Q, N F/Q N ) = ). For each positive integer m, let A[m] be the set of m-torsion points of A. Let a be a positive integer. For P A we define τ A,a,F ( ) = {m N (m, N F/Q ) =, σ acts on A[m] (Z/mZ) r through the scalar matrix a I r }, where σ denotes a Frobenius element at in Gal(F (A[m])/F ). Since F (ζ m ) F (A[m]), using the Weil pairing on the abelian variety A we get that the action of σ on F (ζ m ) is given by ζ m ζm a. Hence if m τ A,a,F ( ), then N F/Q a (mod m), and thus τ A,a,F ( ) < for all P A such that N F/Q > a (in order to simplify our notations, and also because we are interested in asymptotic estimates, we assume these kind of facts from now on). For x R we define f A,a,F (x) := τ A,a,F ( ). P A N F/Q x

2 In this paper we prove the following results (which are a geometric analog of Titchmarsh division problem in the context of abelian varieties (see [AG] for details)): Theorem.. Let A be an abelian variety over a number field F of dimension r. Let a be a positive integer. Assume that the Generalized Riemann Hypothesis (GRH) holds for the Dedekind zeta functions of the division fields for A. Then we have where f A,a,F (x) = c A,F li x + O(x 4r r+3 r r+ 4r r+4 (log x) r r+ ), c A,F = m= [F (A[m]) : F ]. Theorem.. Let A be an abelian variety over a number field F of dimension r, such that End F A Q =End F A Q. Assume that i) r = he, and End F A Q = E is a totally real number field, such that [E : Q] = e, or that ii) r = he, and End F A Q = M is a totally indefinite quaternion division algebra over a totally real number field E, such that [E : Q] = e, or that iii) r = he and End F A Q = M is a totally definite quaternion division algebra over a totally real number field E, such that [E : Q] = e, where h is an integer. Let a be a positive integer. Assume that GRH holds for the Dedekind zeta functions of the division fields for A. Then we have where f A,a,F (x) = c A,F li x + O(x 4h h+3 h h+ 4h h+4 (log x) h h+ ), c A,F = m= [F (A[m]) : F ]. Theorem.3. Let A be an abelian variety over a number field F of dimension r, such that End F A Q =End F A Q. Assume that r = hed, and End F A Q = M is a division algebra over a CM number field K, such that [K : Q] = e, and [M : K] = d, where d, e, and h are integers. Let a be a positive integer. Assume that GRH holds for the Dedekind zeta functions of the division fields for A. Then we have where f A,a,F (x) = c A,F li x + O(x 4h h+3 h h+ 4h h+4 (log x) h h+ ), c A,F = Moreover when e = we have m= [F (A[m]) : F ]. f A,a,F (x) = c A,F li x + O(x h +h h +h h +h (log x) h +h ).

3 Theorem.4. Let A be an abelian variety over a number field F of dimension r. Assume that i) r = he, and End F A Q = E is a totally real number field, such that [E : Q] = e and h is an odd integer, or that ii) r = he, and End F A Q = M is a totally indefinite quaternion division algebra over a totally real number field E, such that [E : Q] = e and h is an odd integer. Let a be a positive integer. Assume that GRH, Artin s Holomorphy Conjecture (AHC), and Pair Correlation Conjecture (PCC) hold for the Artin L-functions attached to the irreducible characters of Gal(F (A[m])/F ) for any integer m. Then, for every ɛ > 0, we have where f A,a,F (x) = c A,F li x + O(x r r+ c A,F = m= (r 4r r+) r+ 8r 4r+8 eh +eh e+ɛ ), [F (A[m]) : F ]. Theorem.5. Let A be an abelian variety over a number field F of dimension r. Assume that r = he and End F A Q = M is a totally definite quaternion division algebra over a totally real number field E, such that [E : Q] = e, where h is an odd integer. Let a be a positive integer. Assume that GRH, AHC and PCC hold for the Artin L-functions attached to the irreducible characters of Gal(F (A[m])/F ) for any integer m. Then, for every ɛ > 0, we have where f A,a,F (x) = c A,F li x + O(x r r+ c A,F = m= (r 4r r+) r+ 8r 4r+8 eh +eh+ɛ ), [F (A[m]) : F ]. We remark that in Theorems.4 and.5 one could impose, as we did in Theorems. and.3 above, the condition End F A Q =End F A Q and in this case one obtains better asymptotic formulas (but the computations are not so nice). All these results are generalizations of [AG] where the authors were able to prove only the very particular case when the abelian variety A is defined over Q and contains an abelian subvariety E of dimension one also defined over Q (actually one can formulate many generalizations of [AG], i.e. one can consider an abelian variety A defined over a number field F, of arbitrary dimension, which contains an abelian subvariety E defined over F of arbitrary dimension and of arbitrary type, etc.). We remark that the exact same methods used in this paper could be extended to Drindeld modules of arbitrary ranks. General abelian varieties For F a number field, we denote G F := Gal( F /F ). Let A be an abelian variety over F of dimension r, and of conductor N. Let Σ F be the set of finite 3

4 places of F, and for a prime of F, let F be the residue field at. Let P A be the set of primes Σ F of good reduction for A, (i.e. (N F/Q, N F/Q N ) = ). For P A, we denote by Ā the reduction of A at. For m an integer, let A[m] be the m-division points of A in F. Then A[m] (Z/mZ) r. If F (A[m]) is the field obtained by adjoining to F the elements of A[m], then we have a natural injection Φ m : Gal(F (A[m])/F ) Aut(A[m]) GL r (Z/mZ). We denote G m := Im Φ m (Gal(F (A[m])/F )). Define For a rational prime l, let n(m) := G m = [F (A[m]) : F ]. T l (A) = lim A[l n ], and V l (A) = T l (A) Q. The Galois group G F acts on T l (A) Z r l, where Z l is the l-adic completion of Z at l, and also on V l (A) Q r l, and we obtain a representation ρ A,l : G F Aut(T l (A)) GL r (Z l ) Aut(V l (A)) GL r (Q l ), which is unramified outside ln F/Q N. If P A, let σ be the Artin symbol of in G F, and let l be a rational prime satisfying (l, N F/Q ) =. We denote by P A, (X) = X r + b,a ( )X r b r,a ( )X + N F/Q r Z[X] the characteristic polynomial of σ on T l (A). Then P A, (X) is independent of l. One can identify T l (A) with T l (Ā), where Ā is the reduction of A at, and the action of σ on T l (A) is the same as the action of the Frobenius π of Ā on T l (Ā). For a an integer define P A,a, (X) = X r + b,a,a ( )X r b r,a,a ( )X + b r,a,a ( ) Z[X] by P A,a, (X) := P A, (X + a). We know (see for example [SI]): Lemma.. Let A be an abelian variety defined over a number field F, of dimension r, of conductor N, and let a and m be positive integers. Then. The extension F (A[m])/F is ramified only at places dividing mn,. F (ζ m ) F (A[m]). Hence if σ, for P A such that (N F/Q, m) =, acts on A[m] (Z/mZ) r through the scalar matrix a I r, from the Weil pairing we get that m N F/Q a. We know (see (3.) of [AG]): Lemma.. Let A be an abelian variety defined over a number field F. Let ɛ > 0. Then, with the same notations as above, we have G m m ɛ. 4

5 Lemma.3. Let A be an abelian variety over a number field F, of conductor N. Let P A, and let p be the rational prime below. Let m be an integer relatively prime to p, and let a be a positive integer. If σ acts on A[m] (Z/mZ) r through the scalar matrix a I r, then for any k =,..., r. m k b k,a,a ( ), Proof: Let l m be a rational prime, and let m(l) be the largest natural number such that l m(l) m. Let π : Ā( F ) Ā( F ), be the Frobenius endomorphism at. From the hypothesis we know that Ā( F )[l m(l) ] Ker(π a), and we get that ρ A,l (σ ) = ai r + l m(l) B l, where B l M r (Z l ). Thus X r + b,a,a ( )X r b r,a,a ( )X + b r,a,a ( ) = P A,a, (X) = P A, (X + a) = det((x + a)i r ρ A,l (σ )) = det(xi r l m(l) B l ), and we obtain that l m(l)k b k,a,a ( ), for any k =,..., r, and we conclude the proof of Lemma.3. Lemma.4. Let a be a positive integer. Then we have for any k =,..., r. b k,a,a ( ) ( + a) r N F/Q k, Proof: We know (Riemann Hypothesis) that P A, (X) = (X x, )... (X x r, ), where x i, = N F/Q. Hence X r + b,a,a ( )X r b r,a,a ( )X + b r,a,a ( ) = P A,,a (X) = P A, (X + a) = (X ( ) (x, a))... (X (x r, a)), r from which we deduce that b k,a,a ( ) (N k F/Q + a) k r ( + a) k N F/Q k ( + a) r N F/Q k, for any k =,..., r. Lemma.5. We have b r,a,a ( ) = N F/Q r + c b,a,a ( ) + c b,a,a ( ) c r b r,a,a ( ) + c r, where c,..., c r, are integers which depend only on r and a. Proof: From P A,,a (X) := P A, (X + a), we get ( ) r b,a,a ( ) = b,a ( ) + a, ( ) ( ) r b,a,a ( ) = b,a ( ) + b,a ( )a + a r, 5

6 b r,a,a ( ) = N F/Q r + b r,a ( )a ( ( ) a ) r r, r and by writing b,a ( ) in terms of b,a,a ( ); then b,a ( ) in terms of b,a,a ( ) and b,a,a ( );...; and N F/Q r in terms of b,a,a ( ),..., and b r,a,a ( ), we are done with the proof of Lemma.5. 3 Abelian varieties of type I, II and III Let A be an abelian variety of dimension r, defined over a number field F, such that End F A Q =EndA F Q. Assume that i) r = he, and End F A Q = E is a totally real number, such that [E : Q] = e, or that ii) r = he, and End F A Q = M is a totally indefinite quaternion division algebra over a totally real number field E, such that [E : Q] = e, or that iii) r = he and End F A Q = M is a totally definite quaternion division algebra over a totally real number field E, such that [E : Q] = e, where h is an integer. Let O E be the ring of integers of E, and let {σ,..., σ e } be the set of all embeddings of E into C. Let l be a rational prime. Since the actions of M :=End F A Q and G F on V l (A) commute we obtain a h-dimensional representation ρ l : G F Aut Ml V l (A) = GL h (E l ), where M l :=End F A Q l, and E l := E Q l. We denote by Q A, (X) = X h + c,a ( )X h c h,a ( )X + N F/Q h O E [X] the characteristic polynomial of σ on V l (A). For a Z define Q A,,a (X) = X h + c,a,a ( )X h c h,a,a ( )X + c h,a,a ( ) O E [X] by Q A,,a (X) := Q A, (X + a). Lemma 3.. Let A be an abelian variety over a number field F, as above. Let P A, and let m be a positive integer such that (N F/Q, m) =. Let a be a positive integer. If σ acts on A[m] (Z/mZ) r through the scalar matrix a I r, then m k c k,a,a ( ), for any k =,..., h. Proof: Let l m be a rational prime, and let m(l) be the largest natural number such that l m(l) m. Let π : Ā( F ) Ā( F ), be the Frobenius endomorphism at. From the hypothesis we know that Ā( F )[l m(l) ] Ker(π a), and we get that ρ A,l (σ ) = ai h +l m(l) B l, where B l M h (O E Z l ). Thus X h +c,a,a ( )X h +...+c h,a,a ( )X +c h,a,a ( ) = Q A,a, (X) = Q A, (X + a) = det((x + a)i h ρ A,l (σ )) = det(xi h l m(l) B l ), and we obtain that l m(l)k c k,a,a ( ), for any k =,..., h, and we conclude the proof of Lemma 3.. 6

7 Lemma 3.. Let a be a positive integer. Then we have σ i (c k,a,a ( )) ( + a) r N F/Q k, for any i =,..., e, and for any k =,..., h. Proof: We assume that σ i is the trivial embedding of E into C and prove Lemma 3. in this case. We know (Riemann Hypothesis) that Q A, (X) = (X y, )... (X y h, ), where y i, = N F/Q. Hence X h + c,a,a ( )X h c h,a,a ( )X + c h,a,a ( ) = Q A,,a (X) = Q A, (X + a) = (X (y, a))... (X (y h, a)), from which we deduce, as in the proof of Lemma.4 above, that c k,a,a ( ) ( + a) r N F/Q k, for any k =,..., r. When σ i is not the trivial embedding of E into C the proof of Lemma 3. is similar. Lemma 3.3. We have c h,a,a ( ) = N F/Q h +d,a c,a,a ( )+d,a c,a,a ( )+...+d h,a c h,a,a ( )+d h,a, where d,a,..., d h,a, are integers which depend only on h and a. Proof: The proof is similar to the proof of Lemma.5. Lemma 3.4. With the same notations as above, for any m N and any x R, we have π a (x, F (A[m])/F ) := { P A N F/Q x, (m, N F/Q ) =, σ acts on A[m] (Z/mZ) r through the scalar matrix a I r } x h h+ m +. h h+ Proof: Let P A be such that N F/Q x, (m, N F/Q ) =, and σ acts on A[m] (Z/mZ) r through the scalar matrix a I r. Then from Lemma 3.3 we know that c h,a,a ( ) = N F/Q h + d,a c,a,a ( ) + d,a c,a,a ( ) d h,a c h,a,a ( ) + d h,a, and hence e i= σ i(c h,a,a ( )) = en F/Q h + e d,a i= σ e i(c,a,a ( ))+...+d h,a i= σ i(c h,a,a ( ))+ed h,a. Also, from Lemma 3. we know that m k e i= σ i(c k,a,a ( )), for any k =,..., h, and from Lemma 3. we know that e i= σ i(c k,a,a ( )) e( + a) r N F/Q k x k, for any k =,..., h. Thus using the fact that e i= σ i(c k,a,a ( )), for k =,..., h, are integers, we get that the number of e i= σ i(c k,a,a ( )) is ( x k + ), for k =,..., h. Hence, since N m k F/Q x, and because m h en F/Q h e + d,a i= σ e i(c,a,a ( )) d h,a i= σ i(c h,a,a ( )) + 7

8 ed h,a, we get that (we remark also that there are at most [F : Q] of primes for which the numbers N F/Q are equal): π a (x, F (A[m])/F ) ( x h m + ) ( x k m k + ) k= ( x h h x + )( m m + ) h h x h h+ +. m h h+ Lemma 3.5. Let A be an abelian variety over a number field F of dimension r. Assume that i) r = he, and End F A Q = E is a totally real number field, such that [E : Q] = e and h is an odd integer, or that ii) r = he, and End F A Q = M is a totally indefinite quaternion division algebra over a totally real number field E, such that [E : Q] = e and h is an odd integer. Then for every ɛ > 0 we have m e(h +h)+ ɛ G m m e(h +h)+. Proof: By eventually replacing F by a finite extension, one can assume that End F A Q =End F A Q, and we do assume this from now on. Then, as above we have an injective map (see [BGK] for details) ρ m : G m GSp h (O E /mo E ). Since Sp h (O E /mo E ) m e(h +h) (we remark also that Sp h (O E /mo E ) is a monic polynomial in m of degree e(h +h); for us even ϕ(m) = m l m ( l ) is a monic polynomial of degree one; all we are interested in the proofs below is that there exist positive constants c and d such that m e(h +h) c ν(m) Sp h (O E /mo E ) m e(h +h) d ν(m), where ν(m) is the number of prime divisors of m), we get easily (see also the beginning of the proof of Theorem 7.4 of [BGK]) that G m m e(h +h)+, and we are done with the proof of the inequality on the right hand side of Lemma 3.5. Now we prove that there exists a positive constant C such that m e(h +h)+ C ν(m) < G m, for any positive integer m, where ν(m) is the number of distinct primes dividing m. From Theorem of [SE], after replacing F by a finite extension, we obtain that the function l d [F (A[l d ]) : F ] 8

9 is multiplicative in l, where l runs over the rational primes (d stands for arbitrary powers of l). Hence it is sufficient to prove that there exists a positive constant C such that for any rational prime l and any non-negative integer d we have l d(e(h +h)+) C < G l d. But this fact is known and it follows from the open image theorems of the l-adic representations associated our abelian varieties, i.e. the abelian varieties that appear in [BGK] (more exactly see Theorem E and Theorem 7.38, and also the beginning of the proof of Theorem 7.4 of [BGK]). Now, because for any ɛ > 0 we have C ν(m) m ɛ, from above we get that m e(h +h)+ ɛ G m, and we are done with the proof of Lemma 3.5. Corollary 3.6. Under the same assumptions as in Lemma 3.5 for every ɛ > 0 we have m e(h h+) ɛ G m G m m e(h h+)+ɛ, where G m denotes the set of conjugacy classes of G m. Proof: The set of conjugacy classes in Sp h (O E /mo E ) is a polynomial in m of degree he e. Then because of the open image theorems from [BGK] that we mentioned in the proof of Theorem 3.5, one can deduce easily Corollary 3.6. Lemma 3.7. Let A be an abelian variety over a number field F of dimension r. Assume that r = he and End F A Q = M is a totally definite quaternion division algebra over a totally real number field E, such that [E : Q] = e, where h is an odd integer. Then for every ɛ > 0 we have m eh + ɛ G m m eh +. Proof: By eventually replacing F by a finite extension, one can assume that End F A Q =End F A Q, and we do assume this from now on. As above we have an injective map (see [BGK] for details) ρ m : G m GO h (O E /mo E ). Since SO h (O E /mo E ) m eh (we remark also that SO h (O E /mo E ) is a monic polynomial in m of degree eh ), as in the proof of Lemma 3.5 above, we get that G m m eh +, and we are done with the proof of the inequality on the right hand side of Lemma 3.7. The rest of Lemma 3.7 could be proved in the same way as Lemma 3.5 by using this time Theorems 7. and 5.3 of [BGK] instead of Theorems E and 7.38 of [BGK]. 9

10 Corollary 3.8. Under the same assumptions as in Lemma 3.7 for every ɛ > 0 we have m eh +eh ɛ G m G m m eh +eh+ɛ. where G m denotes the set of conjugacy classes of G m. Proof: Since the set of conjugacy classes in SO h (O E /mo E ) is a polynomial in m of degree he, and SO h (O E /mo E ) is a monic polynomial in m of degree eh one can prove Corollary 3.8 by the same argument as in the proof of Corollary Abelian varieties of type IV Let A be an abelian variety of dimension r, defined over a number field F, such that End F A Q =EndA F Q. Assume that End F (A) Q = M, where M is a division algebra over a CM number field K. Then r = hed where [M : K] = d and [K : Q] = e, where d, e and h are integers (note that because we always have in the case of abelian varieties [M : Q] r, we get that d h (see [M])). Let σ,..., σ e, be the set of embeddings of K into C. Let O K be the ring of integers of K, and let E be the maximal totally real subfield of K. Then obviously [K : E] =. Let l be a rational prime. Since the actions of M :=End F A Q and G F on V l (A) commute we obtain a h-dimensional representation ρ l : G F Aut Ml V l (A) = GL h (K l ), where M l :=End F A Q l, and K l := K Q l. We denote by Q A, (X) = X h + c,a ( )X h c h,a ( )X + N F/Q h O K [X] the characteristic polynomial of σ on V l (A). For a Z define Q A,,a (X) = X h + c,a,a ( )X h c h,a,a ( )X + c h,a,a ( ) O K [X] by Q A,,a (X) := Q A, (X + a). Then one can prove similar results as in Lemmas 3., 3., and 3.3 for abelian varieties of type IV. We say that P A is a prime of supersingular good reduction for A if Q A, (X) = (x +N F/Q ) h ; otherwise we say that P A is a prime of ordinary good reduction for A. If P A is a prime of ordinary good reduction for A, then ( ) C 0 ρ(σ ) = 0 Ct where C GL h (O K Z l ) and C t is the complex conjugate transpose of C (the complex conjugation is taken with respect to K/E; see [M] and [SI] for details, and for the theory of l-adic representations associated to abelian varieties of type I, II, III see [BGK] and [BGK]). We denote by R A, (X) = X h + d,a ( )X h d h,a ( )X + d h,a ( ) O K [X] the characteristic polynomial of C. For a Z define R A,,a (X) = X h + d,a,a ( )X h d h,a,a ( )X + d h,a,a ( ) O K [X] by R A,,a (X) := R A, (X + a). 0

11 Lemma 4.. Let A be an abelian variety over a number field F, as above. Let P A be a prime of ordinary reduction for A, and let m be a positive integer such that (N F/Q, m) =. Let a be a positive integer. If σ acts on A[m] (Z/mZ) r through the scalar matrix a I r, then for any k =,..., h. m k d k,a,a ( ), Proof: The proof is similar to the proof of Lemma 3.. Lemma 4.. We have σ i (d k,a,a ( )) ( + a) r N F/Q k, for any i =,..., e, and for any k =,..., h. Proof: The proof is similar to the proof of Lemma 3.. Lemma 4.3. We have d h,a,a ( ) = d h,a ( )+e,a d,a,a ( )+e,a d,a,a ( )+...+e h,a d h,a,a ( )+e h,a, where e,a,..., e h,a, are integers which depend only on h and a. Proof: The proof is similar to the proof of Lemma.5. Lemma 4.4. With the same notations as above, for any m N and any x R, we have π a (x, F (A[m])/F ) := { P A N F/Q x, (m, N F/Q ) =, σ acts on A[m] (Z/mZ) r through the scalar matrix a I r } x h h+ m +. h h+ Proof: By using similar results as in Lemmas 3., 3., and 3.3 one can prove Lemma 4.4 in the same way as Lemma 3.4. Lemma 4.5. Assume that e =. Then, with the same notations as above, for any m N and any x R, we have π o a(x, F (A[m])/F ) := { P A N F/Q x, (m, N F/Q ) =, is ordinary, and σ acts on A[m] (Z/mZ) r through the scalar matrix a I r } x h +h m +. h +h

12 Proof: Since e =, we have that K = Q( D), where D is a positive integer, and O K d (Z + DZ), for some positive integer d. Let P A be such that N F/Q x, (m, N F/Q ) =, is ordinary, and σ acts on A[m] (Z/mZ) r through the scalar matrix a I r. Then from Lemma 4.3 we have that d h,a ( ) = d h,a,a ( ) e,a d,a,a ( ) e,a d,a,a ( )... e h,a d h,a,a ( ) e h,a. But from Lemma 4. we know that m k d k,a,a ( ), for any k =,..., h, and from Lemma 4. we know that d k,a,a ( ) ( + a) r N F/Q k, for any k =,..., h. Hence d h,a ( ) = m h d h md e h,a, where d,..., d h O K, such that d k x k m k, for k =,..., h. By writing d k = e k + Df k O K d (Z + DZ), where e k, f k d Z, we get that N F/Q h = det ρ l (σ ) = N K/Q d h,a ( ) = (m h e h me e h,a ) + D(m h f h mf ), and also that e k x k m k, and f k x k m k, for k =,..., h. Hence π o a(x, F (A[m])/F ) h ( x k m k + ) k= x h +h +. m h +h 5 Chebotarev density theorem Let L/F be a Galois extension of number fields, with Galois group G. We denote by n L and d L the degree and the discriminant of L/Q, and by d F the discriminant of F/Q. Let P(L/F ) be the set of rational primes p which lie below places of F which ramify in L/F. We know (see page 30 of [SE]): Lemma 5.. If L/F is Galois extension of number fields, then log d L G log d F + n L ( G ) p P(L/F ) log p + n L log G. Using the same assumptions as above, let C be a conjugacy class in G. For a positive real number x, let π C (x, L/F ) := { Σ F N F/Q x, unramified in L/F, σ C}, where σ is a Frobenius element at. The Chebotarev density theorem says that π C (x, L/F ) C C x li x G G log x, and moreover we know (see [SE] and [MM]):

13 Lemma 5.. Let L/F be a Galois extension of number fields. If the Dedekind zeta function of L satisfies the GRH, then π C (x, L/F ) C G li x C x log d L (log x + ), G where the implied O-constant depends only on F. Moreover, if the Artin L- functions attached to the irreducible characters of G satisfy GRH, AHC, and PCC, then π C (x, L/F ) C G li x C ( G G ) 4 x (log x( G ( p P(L/F ) p))), where G denotes the set of conjugacy classes of G. depends only on F. The implied O-constant 6 The proof of Theorem. We want to estimate the sum f A,a,F (x) := τ A,a,F ( ). P A N F/Q x If m τ A,a,F ( ), then m r P A, (a) < ( + a) r x r. Hence it is sufficient to consider only positive integers m satisfying m ( + a)x. Thus τ A,a,F ( ) = π a (x, F (A[m])/F ), P A N F/Q x m (+a)x where π a (x, F (A[m])/F ) := { P A N F/Q x, (m, N F/Q ) =, σ acts on A[m] (Z/mZ) r through the scalar matrix a I r }, If y = y(x) is a real number with y (+a)x (y will be chosen later), then f A,a,F (x) = π a (x, F (A[m])/F ) m (+a)x = π a (x, F (A[m])/F ) + π a (x, F (A[m])/F ) y<m (+a)x = main + error. (6.) 3

14 From Lemmas 5. and., under GRH, we get main = n(m) li x + = O(x log(mnf/q N x)) n(m) li x + O(yx log(nf/q N x)). (6.) Now we estimate the error. For each b = (b,..., b r ) Z r, with b k ( + a) r x k for k =,..., r, and for each positive integer m we define S b,a (m) := { P A N F/Q x, b k,a,a ( ) = b k for k =,..., r, σ acts on A[m] (Z/mZ) r through the scalar matrix a I r }. Then, because from Lemma.4 we know that b k,a,a ( ) ( + a) r x k, for k =,..., r, we obtain error S b,a (m). y<m (+a)x b Z r b k (+a) r x k, for k=,...,r From Lemma.3 we know that for each S b,a (m) we have m k b k,a,a ( ), for k =,..., r, and from Lemma.5 we know that b r,a,a ( ) = N F/Q r + c b,a,a ( ) + c b,a,a ( ) c r b r,a,a ( ) + c r, and therefore y<m (+a)x y<m (+a)x b Z r b k (+a) r x k, for k=,...,r S b,a (m) b Z r P A b k (+a) r x k N, for k=,...,r F/Q x m k b k,a ( )=b k, for k=,...,r b k, for k=,...,r m r b r,a ( )=N F/Q r +c b +...+c r b r +c r y<m (+a)x b Z r b k (+a) r x k, for k=,...,r m k b k, for k=,...,r y<m (+a)x y<m (+a)x ( x r m + ) ( x k m k + ) ( x m k= r r x + )( m + ) r r ( x m + ) 4

15 From above we get x r r+ y. r r+ r r+ f A,a,F (x) = n(m) li x + O(yx x log x) + O( y ). r r+ We choose y such that x y log x = x r r+, i.e. y r r+ x r r+ 4r r+4 y := (log x) r r+. Then f A,a,F (x) = 4r r+3 r r+ li x + O(x 4r r+4 (log x) r r+ ). n(m) From Lemma., with ɛ =, we obtain m>y n(m) m>y m 3 y Since y := r r+ x 4r r+4 (log x) r r+, we get f A,a,F (x) = c A,F li x + O(x 4r r+3 r r+ 4r r+4 (log x) r r+ ), and Theorem. is proved. 7 The proof of Theorem. As in 6 above we have f A,a,F (x) = main + error, (7.) and main = n(m) li x + O(yx log(nf/q N x)). (7.) From Lemma 3.4 we know that error y<m (+a)x π a (x, F (A[m])/F ) y<m (+a)x ( x h h+ m + ) h h+ 5

16 Hence x h h+ y. h h+ h h+ f A,a,F (x) = n(m) li x + O(yx x log x) + O( y ). h h+ We choose y such that x y log x = x h h+, i.e. y h h+ y := (log x) x h h+ 4h h+4 h h+. Then f A,a,F (x) = 4h h+3 h h+ li x + O(x 4h h+4 (log x) h h+ ). n(m) From Lemma., with ɛ =, we obtain m>y n(m) m>y m 3 y Since y = h h+ x 4h h+4 (log x) h h+, we get f A,a,F (x) = c A,F li x + O(x 4h h+3 h h+ 4h h+4 (log x) h h+ ), and Theorem. is proved. 8 The proof of Theorem.3 By using Lemma 4.4 one can prove the first part of Theorem.3 in the same way as Theorem.. Now we prove the second part of Theorem.3. As in 7 above we have f A,a,F (x) = main + error, (8.) and main = n(m) li x + O(yx log(nf/q N x)). (8.) Now we estimate the error. We have error πa(x, o F (A[m])/F ) + y<m (+a)x y<m (+a)x π s a(x, F (A[m])/F ), 6

17 where and π o a(x, F (A[m])/F ) := { P A N F/Q x, has ordinary reduction and σ acts on A[m] (Z/mZ) r through the scalar matrix a I r }, π s a(x, F (A[m])/F ) := { P A N F/Q x, has supersingular reduction and σ acts on A[m] (Z/mZ) r through the scalar matrix a I r }. From Lemma 4.5 we get that y<m (+a)x π o a(x, F (A[m])/F ) y<m (+a)x ( x h +h m + ) h +h x h +h y. h +h If σ acts on A[m] (Z/mZ) r through the scalar matrix a I r, then m r P A, (a) = (N F/Q + a ) r (see Lemma.3), and also m N F/Q a (see. of Lemma. above), and hence m a. Thus we choose y > a, and we get πa(x, s F (A[m])/F ) = 0. From above we obtain y<m (+a)x h +h f A,a,F (x) = n(m) li x + O(yx x log x) + O( y ). h +h We choose y such that x y log x = x h +h, i.e. y h +h Then f A,a,F (x) = x h +h h +h y := (log x) h +h. h +h h +h li x + O(x h +h (log x) h +h ). n(m) From Lemma., with ɛ =, we obtain n(m) m>y m>y m 3 y Since y = h +h x h +h (log x) h +h, we get f A,a,F (x) = c A,F li x + O(x h +h h +h h +h (log x) h +h ), and Theorem.3 is proved. 7

18 9 The proof of Theorem.4 As in 7 above we have and f A,a,F (x) = main + error, (9.) error = O( x r r+ ). (9.) y r r+ Now we estimate main. Fix ɛ > 0. Then from Lemmas 5., 5.,., and Corollary 3.6, under GRH, AHC, and PCC, we get main = n(m) li x + O(( G m G m ) 4 x log(mnf/q N x)) n(m) li x + O((m e(h h+)+ɛ ) 4 x log(mnf/q N x)) = From above we get f A,a,F (x) = eh li x + eh+e ɛ O(y 4 x log(nf/q N x)). (9.3) n(m) eh li x + eh+e ɛ O(y 4 x x r r+ log x) + O( n(m) y ). r r+ We choose y such that x y eh eh+e ɛ 4 log x = x r r+, i.e. y r r+ y := x 4r r+ 8r 4r+8 eh +eh e+ɛ 4 (log x) 8r 4r+8 eh +eh e+ɛ. Then f A,a,F (x) = n(m) li x +O(x r r+ (r r+) From Lemma 3.5, we obtain m>y 4r r+ 8r 4r+8 eh +eh e+ɛ (log x) n(m) m>y y h e+he ɛ = x (h m h e+he+ ɛ 8r 4r+4 8r 4r+8 eh +eh e+ɛ ) 4r e+he ɛ)( r+ 8r 4r+8 eh ) +eh e+ɛ (log x) 8 8h e+4he 4ɛ 8r 4r+8 eh +eh e+ɛ.

19 Hence, for ɛ > 0 sufficiently small, we get f A,a,F (x) = c A,F li x +O(x r r+ (r r+) 4r r+ 8r 4r+8 eh +eh e+ɛ (log x) 8r 4r+4 8r 4r+8 eh +eh e+ɛ ), and Theorem.4 is proved (because we can choose ɛ > 0 arbitrarily small). 0 The proof of Theorem.5 As in 7 above we have and f A,a,F (x) = main + error, (0.) error = O( x r r+ ). (0.) y r r+ Now we estimate main. Fix ɛ > 0. Then from Lemmas 5., 5.,., and Corollary 3.8, under GRH, AHC, and PCC, we get main = n(m) li x + O(( G m G m ) 4 x log(mnf/q N x)) n(m) li x + O((m eh +eh+ɛ ) 4 x log(mnf/q N x)) = From above we get f A,a,F (x) = eh li x + eh ɛ O(y 4 x log(nf/q N x)). (0.3) n(m) eh li x + eh ɛ O(y 4 x x r r+ log x) + O( n(m) y ). r r+ We choose y such that x y eh eh ɛ 4 log x = x r r+, i.e. y r r+ y := x 4r r+ 8r 4r+8 eh +eh+ɛ 4 (log x) 8r 4r+8 eh +eh+ɛ. Then f A,a,F (x) = n(m) li x +O(x r r+ (r r+) 4r r+ 8r 4r+8 eh +eh+ɛ (log x) 8r 4r+4 8r 4r+8 eh +eh+ɛ ) 9

20 From Lemma 3.5, we obtain m>y n(m) m>y y h e ɛ = x (h m h e+ ɛ 4r e ɛ)( r+ 8r 4r+8 eh ) +eh+ɛ (log x) Hence, for ɛ > 0 sufficiently small, we get 4h e 4ɛ 8r 4r+8 eh +eh+ɛ f A,a,F (x) = c A,F li x. +O(x r r+ (r r+) 4r r+ 8r 4r+8 eh +eh+ɛ (log x) 8r 4r+4 8r 4r+8 eh +eh+ɛ ), and Theorem.5 is proved (because we can choose ɛ > 0 arbitrarily small). References [AG] A. Akbary, D. Ghioca, A geometric variant of Titchmarsh divisor problem, Int. J. of Numb. Theory, 8 () (0), [BGK] G. Banaszak, W. Gajda, P. Krason, On the image of l-adic Galois representations for abelian varieties of type I and II, Doc. Math., Extra Volume: John H. Coates Sixtieth Birthday (006), [BGK] G. Banaszak, W. Gajda, P. Krason, On the image of Galois l-adic representations for abelian varieties of type III, Tohoku Math. J. 6 (00), [CM] A.C. Cojocaru and M.R. Murty, Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik s problem, Mathematische Annalen 330 (004), [M] D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5. Published for the Tata Institute of Fundamental Research, Bombay, by Oxford University Press. [SE] J.P. Serre, Quelques applications du theoreme de densite de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math., no. 54, (98), 3-0. [SE] J.P. Serre, Une critre d indpendance pour une famille de reprsentations l-adiques, Comentarii Mathematici Helvetici, 88 (03), [SH] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, 97. [SI] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 5. Springer, New York (994). 0