AVERAGE RECIPROCALS OF THE ORDER OF a MODULO n

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1 AVERAGE RECIPROCALS OF THE ORDER OF a MODULO n KIM, SUNGJIN Abstract Let a > be an integer Denote by l an the multiplicative order of a modulo integers n We prove that l = an Oa ep 2 + o log log, n,n,a= which is an improvement over [8, Theorem 5] Further, we obtain several applications toward number fields and 2-dimensional abelian varieties of CMtype Introduction Define l a n by the multiplicative order of a modulo n In [6], Kurlberg and Rudnick showed that there eist a δ > 0 such that l a n n ep log n δ for all but o integers n In [7], Kurlberg and Pomerance obtained the following result by applying Fouvry s result For some γ > 0, l a n n /2+γ for positive proportion of n On the other hand, Zelinsky [8] proved that n,n,a= n,n,a= ϕn l a n = O a 2 log α for any 0 < α < 3 Indeed, this result can be interpreted as l a n = O a log α for any 0 < α < 3 Furthermore, he was able to generalize to number fields Let K be a number field, and assume that U K its group of units is infinite Let R K be the ring of integers in K For integral ideal I, denote by NI the norm of I, and ϕi the Euler s totient function of I, which is defined by: ϕi = NI p I Np Denote by U K I the subgroup of U K formed by elements which are modulo I He obtained that ϕi 2 [U K : U K I] = O K log α This also can be interpreted as NI NI In the author s work [5, Theorem 23], it is shown that NI [U K : U K I] = O K log α [U K : U K I] log log 2/5 2 for all but O ep 2 5 log 3/5 integal ideals NI This implies that [U K : U K I] = O K ep 25 log 2/5

2 2 KIM, SUNGJIN The same idea as in [5, Theorem 23] also applies to n,n,a= l a n = O a ep We show that the same idea in [5, Theorem 23] further leads to NI 25 log 2/5 [U K : U K I] = O K ep c log, also n,n,a= l a n = O a ep c log for some positive constant c Adopting an idea from Pomerance [2], we further improve these: Theorem Let l a n be the multiplicative order of a modulo n Then l a n = O a ep 2 + o log n,n,a= log Furthermore, let K be a number field, and assume that U K its group of units is infinite For integral ideal I, denote by NI the norm of I Denote by U K I the subgroup of U K formed by elements which are modulo I Then NI [U K : U K I] = O K ep 2 + o log It is possible to apply this to improve on [5, Theorem 8] m< log Theorem 2 Let A be an absolutely simple abelian variety of dimension 2 defined over a degree 4 CM-field with CM-type K, Φ, a Suppose that the refle type K, Φ, a satisfies K = K Then we have tm = O K ep 4 + o log log The significance in this theorem is that this opens up a possibility of proving a special case g = 2 of the author s Conjecture in [5] unconditionally If we are able to prove π A ; mf, a i K log B for some positive absolute constant B in the case Nmf > /2, then this would provide an unconditional proof of [5, Conjecture ]: Conjecture Let A be an absolutely simple abelian variety of dimension 2 defined over a degree 4 CM-field with CM-type K, Φ, a Suppose that the refle type K, Φ, a satisfies K = K Then we have d p = C A Li + O A,B log B, Np where C A = m= ϕm [ka[m] : k]

3 AVERAGE RECIPROCALS OF THE ORDER OF a MODULO n 3 2 Backgrounds and Proofs 2 Smooth Numbers Let ψ, y be the number of positive integers n whose prime divisors p y For any U > 0, it is well known that ψ, /u = ρu + O log uniformly for u U The function ρu is called the Dickman function, and it satisfies ρu = for 0 < u, uρ u = ρu for u > This function also satisfies the following asymptotic formula see []: log log u ρu = ep u log u + log log u + log u log u + O log log u 2 log u 2 From the upper bound of de Bruijn [], and lower bound of Hildebrand [4], we have Theorem 2 Let ɛ > 0, we have ψ, /u = ρu ep uniformly for u ɛ log / For a fied positive c, let u = log c Then we have O ɛ u ep log u 3/5 ɛ Corollary 2 For 0 c, we have ψ, ep c log = ep 2c log + o For a given number field K, define ψ K, y to be the number of integral ideals I with NI such that Np y for any prime ideal p I Then the above theorem and corollary have their analogue see [3, Section 3]: Theorem 22 Let ɛ > 0, we have ψ K, /u = ψ K, ρu ep uniformly for u ɛ log / As before, for a fied positive c, let u = O ɛ u ep log u 3/5 ɛ log c Then we have Corollary 22 For 0 c, we have ψ K, ep c log = ψ K, ep 2c log + o Let a > be an integer For some z > 0, it is clear that l a n < z implies n i<z ai Since the number of prime factors of i<z ai is O a z 2 / log z, the number of integers n such that l a n < z is O a ψ, c a z 2 Therefore, by taking z = ep c log, we establish the following: Lemma 2 Let a > be an integer Then there is c a > 0 such that l a n ep c a log for all but O a ep ca log integers n

4 4 KIM, SUNGJIN Using the lower bound ep c a log for most n, and the trivial lower bound for the eceptional set of n, it follows that l a n = O a ep c a log n,n,a= for some positive constant c a Furthermore, let K be a number field, and assume that U K its group of units is infinite For integral ideal I, denote by NI the norm of I Denote by U K I the subgroup of U K formed by elements which are modulo I Let a U K be a unit of infinite order We use the notation l a I for the order of a modulo I Then we have [U K : U K I] l a I The same idea as above applies, and we obtain for some c K > 0, [U K : U K I] = O K ep c K log, NI To prove Theorem, we adopt an idea of Pomerance [2, Theorem ]: Theorem 23 Let a > be an integer There is an 0 a such that if 0 a, then ep 2 + o log log m,l am=n We may assume that n < Similarly as in [, Section 3], Pomerance applies Rankin s method Then for any c > 0, c m c c m c = c p c = c A m,l am=n l am=n p m l ap n Then the optimal choice for c is c = 4 + log /2 with a requirement log A = olog / Here, A is the Euler product p c l ap n Taking the sum of the LHS of Theorem 23 for n < z = ep log 4 log, we obtain a strengthened version of Lemma 2 Lemma 22 Let a > be an integer Then there is c a > 0 such that l a n ep log log 4 for all but O a ep 4 + o log log integers n l ap n However, we do not use the lemma to prove Theorem 2 Instead, observe the following: l a m = n n< n< m Applying Theorem 23 directly, we obtain that n m,l am=n n< n ep = O a ep m,l am=n 2 + o log log log log 2 + o This proves the first part of Theorem The statement for the number field follows from a modified version of Theorem 23

5 AVERAGE RECIPROCALS OF THE ORDER OF a MODULO n 5 Theorem 24 Let a be an integral element of K which is not a root of unity There is an 0 K, a such that if 0 K, a, then ep 2 + o log log NI,l ai=n The proof is almost identical, with only difference in the Euler product: c NI c c NI c = c Np c = c A NI,l ai=n l ai=n p I l ap n As in the proof of [2, Theorem ], we may assume that > n otherwise there are no I satisfying NI together with l a I = n The Euler product A is treated by log A = Np c + O[K : Q] l ap n Np c + O[K : Q] = d n l ap=d The prime ideals p with l a p = d all divide the principal ideal a d Then the number of prime ideals p dividing a d is O [K : Q] d log a logd+ where a is a conjugate of a with maimal a Let q,, q t be all prime divisors of a d Note that for a given norm, there are at most [K : Q] prime ideals of the same norm Each prime divisor q i of a d satisfies Nq i mod d Then we have t Np c = Nq c i [K : Q] dj + c [K : Q]d c c d log a c l ap=d i= Following the rest of the proof, we obtain that j d log a l ap n log A [K : Q] log a log log /2 + O[K : Q], which yields log A = olog / This completes the proof Applying Theorem 24, we obtain the second part of Theorem We need a principal ideal version of Theorem 24 to prove corresponding result on 2-dimensional abelian varieties with CM type Theorem 25 Let a be an integral element of K which is not a root of unity There is an 0 K, a such that if 0 K, a, then ep 2 + o log log m,l am=n The proof is almost identical, with only difference in the Euler product: c m c c m c = c m,l am=n l am=n p m l ap n l ap n p c = c A As in the proof of [2, Theorem ], we may assume that [K:Q] > n otherwise there are no m satisfying m together with l a m = n The Euler product A is treated by log A = p c + O l ap n p c + O = d n l ap=d The primes p with l a p = d all divide the principal ideal a d Then prime p dividing a d also divides the integer Na d The number of such p is O [K : Q] d log a logd+ where a is a conjugate of a with maimal a Let q,, q t be all prime divisors of Na d Each prime divisor q i of Na d satisfies q i mod d Then we have t p c = dj + c d c c [K : Q]d log a c l ap=d i= q c i j [K:Q]d log a

6 6 KIM, SUNGJIN Following the rest of the proof, we obtain that log A [K : Q] log a log log /2 + O, which yields log A = olog / This completes the proof We insert an etra factor R wm where wm is the number of distinct prime divisors of m, yet the upper bound still holds Theorem 26 Let a be an integral element of K which is not a root of unity Let R > 0 There is an 0 K, a, R such that if 0 K, a, R, then R wm ep 2 + o log log m,l am=n In this one, the Euler product behaves likes Rth power of the previous one In fact, R wm c R wm m c c R wm m c m,l am=n Euler product A is treated by log A = l am=n = c l ap n l ap n Rp c + OR = d n p m l ap n + Rp c + Rp 2c + = c A l ap=d Rp c + OR Following the rest of the proof, we obtain that log A R[K : Q] log a log log /2 + OR, which yields log A = olog / This completes the proof Corollary 23 Let a be an integral element of K which is not a root of unity Let R > 0 Then R wm l a m ep 2 + o log log m This is an easy consequence of Theorem 26 We write R wm l a m = R wm n m n< [K:Q] m,l am=n n ep 2 + o log log n< [K:Q] = ep 2 + o log log 22 Abelian Varieties with CM-type We give necessary definitions and theorems that are required to state Theorem 3 For more details, one can refer to [5], also [8] The CM theory for elliptic curves see [6], [4], also [7] can be generalized to abelian varieties The endomorphism rings of abelian varieties are far more comple than those of elliptic curves However, their center as an algebra can be described via CM-fields see [8, p6, Theorem 3]: Definition 2 A CM-field is a totally imaginary quadratic etension of a totally real number field Theorem 27 Let A be an abelian variety Then the center K of End Q A := EndA Q is either a totally real field or a CM field Furthermore, we have by the following proposition see [5, p36, Proposition ] that the degree of K in above theorem is bounded by 2dimA

7 AVERAGE RECIPROCALS OF THE ORDER OF a MODULO n 7 Proposition 2 Let A be an abelian variety of dimension g and S a commutative semi-simple subalgebra of End Q A Then we have [S : Q] 2g In particular, K S, which gives [K : Q] [S : Q] 2g We are interested in the case that [K : Q] = 2g, and K is a CM field The following definition generalizes comple multiplication of elliptic curves to abelian varieties see [5, p4, Theorem 2], also [8, p72] Theorem 28 Let A be an abelian variety of dimension g Suppose that the center of End Q A is K, and K is a CM field of degree 2g over Q We say that A admits comple multiplication In this case, there is an ordered set Φ = {φ,, φ g } of g distinct isomorphisms of K into C such that no two of them is conjugate We call this pair K, Φ the CM-type Furthermore, there eists a lattice a in K such that there is an analytic isomorphism θ : C g /Φa AC We write K, Φ, a to indicate a is a lattice in K with respect to θ In short, we say that A is of typecm-type K, Φ, a with respect to θ Under the inclusion i : K End Q A, we have that is an order in K This gives rise to the following composition: O = {τ K iτ EndA} = {τ K τa a} Corollary 24 Let A be an abelian variety of dimension g with CM-type K, Φ, a with respect to θ Then θ Φ maps K/a to A tor, i e Φ K/a C g θ /Φa A tor Proof This is clear from noticing that a Q = K Also, Φ is Q-linear, and Φa Q is a torsion subgroup of C g /Φa We define a refle-type of a given CM-type see [5, p59-62] Let K be a CM-field of degree 2g, Φ = {φ,, φ g } a set of g embeddings of K into C so that K, Φ is a CM-type Let L be a Galois etension of Q containing K, and G the Galois group of L over Q Let ρ be an element of G that induces comple conjugation on K Let S be the set of all elements of G that induce φ i for some i =,, g A CM-type is called primitive if any abelian variety with the type is simple The following proposition gives a criterion for primitiveness of CM-type see [5, p6, Proposition 26] Proposition 22 Let K, Φ be a CM-type Let L, G, ρ, S as above, and H the subgroup of G corresponding to K Put H S = {γ G γs = S} Then K, Φ is primitive if and only if H = H S The following proposition relates a CM-type K, Φ and a primitive CM-type K, Φ see [5, p62, Proposition 28] Proposition 23 Let L, G, ρ, S as above Put S = {σ σ S}, H S = {γ G γs = S } Let K be the subfield of L corresponding to H S, and let Φ = {ψ,, ψ g } be a set of g embeddings of K to C so that no two of them are conjugate Then K, Φ is a primitive CM-type We call K, Φ the refle of CM-type K, Φ We define a type norm for a given CM-type The following map is well defined on K : N K,Φ : K K, σ σ Φ Then this map allows an etension to N K,Φ : A K A K This etension is called the type norm It can be seen that N K,Φ is a continuous homomorphism on A K see [5, p24] The field of definition k

8 8 KIM, SUNGJIN of an abelian variety A with CM-type K, Φ contains the refle K In brief, k K Thus, we can also define the type norm on the field of definition: N Φ k = N K,Φ N k K where N k K is the standard norm map of ideles Note that if g = elliptic curves then K = K An analogue of [9, p 62, Lemma 4] can be obtained from applying the Main Theorem of Comple Multiplication see [8, Theorem, p84] The idea of the proof is the same as in [9, p 62, Lemma 4], but we need a modification due to type norm factor in the Main Theorem of Comple Multiplication Lemma 23 Let A, K, Φ, K, Φ, k be the same notations as before Let m 2 be an integer Then there eists a nonzero rational integer f such that ka[m] k mf, where k mf is the ray class field corresponding to the principal ideal mf k Let K be a number field of degree n = r + 2r 2 with ring of integers O K and r the number of distinct real embeddings of K, and let m be an integral ideal of K Define a m-ideal class group by an abelian group of equivalence classes of ideals in the following relation: a b mod m, if ab = α, α K, α mod m, and α is totally positive Let α, β K Denote by α β mod* m if v p m v p α β for all primes p and αβ is totally positive Then we can rewrite the equivalence relation by ab P m K = {α : α mod* m} The m-ideal class group coincides with our definition C m K = JK m/p K m in the previous chapter Denote by hm the cardinality of JK m/p K m, and h by the class number of K We have a formula that relates hm and the class number h of K This follows from an eact sequence: UK O K /mo K {±} r C m K CK Denote by T m the cardinality of the image of the unit group UK in O K /mo K {±} r Then we have hm = 2r hϕm T m where ϕm = O K /mo K A direct corollary of Lemma 23 is the following: Lemma 24 Let A, K, Φ, K, Φ, k be the same notations as before Suppose also that p k is a prime of good reduction for A, and p m Let f be the nonzero integer as in Lemma 23 Given m, there are tm ideal classes modulo mf k such that p splits completely in ka[m] if and only if p a,, p a tm Furthermore, tm satisfies the following identity by class field theory, tm hmf = [ka[m] : k] By Lemma 23 below, there is an absolute positive constant R depending only on A such that tm = hmf [ka[m] : k] where N = NA is an integer depending only on A m2l ν T mf Rwm, The last inequality can be obtained from applying the following theorem on etension degree of division fields along with a formula for hmf see [3, Theorem ], also []

9 AVERAGE RECIPROCALS OF THE ORDER OF a MODULO n 9 Lemma 25 Let A be an abelian variety of CM type K, Φ, a of dimension g defined over a number field k Then for some c, c 2 > 0, n m = [ka[m] : k] satisfies m ν c wm n m m ν c wm 2, where wm is the number of distinct prime factors of m, ν is an integer defined by RankΦ, K, and 2 + log 2 g ν g + if A is absolutely simple Since the refle type Φ, K is always simple and RankΦ, K = RankΦ, K, we also have that 2 + log 2 g ν g + if [K : Q] = g Thus, we have ma2 + log 2 g, 2 + log 2 g ν ming +, g + On the assumptions for Theorem 2, g = 2 gives the only choice for ν = g + = 3 Then Lemma 24 gives m tm T mf Rwm Taking the sum over m above, we have by Corollary 23 in section 2, R wm tm K T mf K ep 4 + o log log m m This completes the proof of Theorem 2 References [] N G de Bruijn, On the number of positive integers and free of prime factors > y, Nederl Akad Wetensch Proc Ser A 69=Indag Math , pp [2] E Fouvry, Theoreme de Brun-Titchmarsh; application au theoreme de Fermat, Invent Math, 79, 985 pp [3] A Granville, Smooth numbers: computational number theory and beyond, Algorithmic Number Theory, MSRI Publications, Volume 44, 2008 [4] A J Hildebrand, On the Number of Positive Integers and Free of Prime Factors > y, Journal of Number Theory 22, 986 pp [5] S Kim, Average of the First Invariant Factor of the Reductions of Abelian Varieties of CM Type, accepted for publication [6] P Kurlberg, Z Rudnick, On Quantum Ergodicity for Linear Maps of the Torus, Comm Math Phys, 222:20-227, 200 [7] P Kurlberg, C Pomerance, On the period of linear congruential and power generators, Acta Arith , pp [8] S Lang, Comple Multiplication, Springer-Verlag, 983 [9] M R Murty, On Artin s Conjecture, Journal of Number Theory, Vol 6, no2, April 983 [0] H Montgomery, R Vaughan, Multiplicative Number Theory I, Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press 2007 [] J Neukirch, Algebraic Number Theory, Springer 999 [2] C Pomerance, On the distribution of pseudoprimes, Math Comp 37 98, pp [3] K Ribet, Division Fields of Abelian Varieties with Comple Multiplication, Memoires de la S M F 2e serie, tome 2 980, pp [4] K Rubin, Elliptic Curves with Comple Multiplication and the Conjecture of Birch and Swinnerton-Dyer, available at [5] G Shimura, Abelian Varieties with Comple Multiplications and Modular Functions, Princeton University Press, 998 [6] J Silverman, The Arithmetic of Elliptic Curves, 2nd Edition, Graduate Tets in Mathematics 06, Springer [7] J Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Tets in Mathematics, Springer [8] J Zelinsky, Upper bounds for the number of primitive ray class characters with conductor below a given bound, arxiv preprint arxiv: