The distribution and density of cyclic groups of the reductions of an elliptic curve over a function field
aa r X i v : . [ m a t h . N T ] S e p The distribution and density of cyclic groups of thereductions of an elliptic curve over a function field
Márton Erdé[email protected]
Abstract
Let K be a global field of finite characteristic p ≥ , and let E/K be a non-isotrivial ellipticcurve. We give an asympotoic formula of the number of places ν for which the reduction of E at ν is a cyclic group. Moreover we determine when the Dirichlet density of those places is 0. Let K be a global field of characteristic p and genus g K , and let k = F q ⊂ K ( q = p f ) be thealgebraic closure of F p in K . We denote by V K the set of places of K . For ν ∈ V K , we denote by k ν the residue field of K at ν , and by deg( ν ) := [ k ν : F q ] the degree of ν . Let k be an algebraic closureof k . Denote φ : ( x x q ) ∈ Gal( k/k ) the q -Frobenius. Let k r | k be the unique degree r extensionin k .Let E/K be an elliptic curve over K with j -invariant j E / ∈ k , which we shall standardly callnon-isotrivial. We denote by V E/K the set of places of K for which the reduction E ν /k ν is smoothand | V E/K | = P ν / ∈ V E/K deg( ν ) . For n ∈ N \ { } let V E/K ( n ) = { ν ∈ V E/K | deg( ν ) = n } .From the theory of elliptic curves we know that for ν ∈ V E/K , E ν ( k ν ) ≃ Z /d ν Z × Z /d ν e ν Z fornonzero integers d ν , e ν , uniquely determined by E and ν . We call the integers d ν and d ν e ν theelementary divisors of E ν .The goal of this paper is to extend the results of [CT] about the distribution of the places ν ∈ V E/K for which E ν ( k ν ) is a cyclic group. Such questions have been investigated for thereductions of an elliptic curve defined over Q (e.g. in [BaSh], [Co1], [Co2], [CoMu], [GuMu], [Mu1],[Mu2], [Se2]), mainly in relation with the elliptic curve analogue of Artin’s primitive root conjectureformulated by Lang and Trotter in [LaTr]. This latter conjecture was investigated in the functionfield setting E/K by Clark and Kuwata [ClKu], and by Hall and Voloch [HaVo] (see also Voloch’swork on constant curves [Vo1], [Vo2]). In [ClKu], a particular emphasis was placed on the study ofthe cyclicity of E ν ( k ν ) .In this paper we obtain an explicit asymptotic formula for the number of places ν ∈ V E/K , offixed degree, for which E ν ( k ν ) is cyclic. Our result is a direct extension of the work of [CT] whichworked in finite characteristic p > . Theorem 1.
Let
E/K be a non-isotrivial elliptic curve. For all ε > there exists c = c ( K, E, ε ) such that for all n ∈ N we have (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) ν ∈ V E/K ( n ) | E ν ( k ν ) is cyclic (cid:1) − δ ( E/K, , n ) q n n (cid:12)(cid:12)(cid:12)(cid:12) ≤ c q n/ ε n , where δ ( E/K, , n ) = X m ≤ qn/ m | q n − µ ( m )ord m ( q ) | K ( E [ m ]) : K | , a µ is the Moebius function and ord m ( q ) denotes the multiplicative order of q modulo m for m ∈ N , ( m, q ) = 1 . δ refers to d ν = 1 . The exact same calculation for (cid:0) ν ∈ V E/K ( n ) | d ν = d (cid:1) yields a same result.We are also able to answer a previously unsolved question concerning the Dirichlet density δ ( E/K, of places ν such that E ν ( k ν ) is cyclic: we can characterize when this density is 0. Theorem 2.
Let
E/K be a non-isotrivial elliptic curve. Then δ ( E/K,
1) = 0 if and only if δ ( E/K, ,
1) = 0 . Surprisingly this can happen in the case, when the torsion subgroup of E ( K ) is cyclic, as well.We sketch the original proof of Theorem 1 in the following:With simple inclusion-exclusion principle we get (cid:0) ν ∈ V E/K ( n ) | E ν ( k ν ) is cyclic (cid:1) = X m µ ( m ) (cid:0) ν ∈ V E/K ( n ) | ( Z /m Z ) ≤ E ν ( k ν ) (cid:1) , moreover the sum has very few nonzero terms: if ( Z /m Z ) ≤ E ν ( k ν ) then • by Hasse’s theorem | E ν ( k ν ) | ≤ q n + 1 + 2 √ q n , thus m ≤ q n/ + 1 , • by the Weil-pairing the cyclotomic field F q ( ζ m ) ≤ k ν , thus m | q n − .By [CT] Corollary 10, ( Z /m Z ) ⊆ E ν ( k ν ) if and only if ν splits completely in K ( E [ m ]) /K ) orequivalently the conjugacy class of the Frobenius at ν in Gal( K ( E [ m ]) /K ) ≤ GL ( Z /m Z ) is the setconsisting of the identity element. Thus let c m be the integer for which the algebraic closure of k in K ( E [ m ]) is F q cm and π ( n, K ( E [ m ] /K ) = ν ∈ V E/K ( n ) | ν splits completely in K ( E [ m ]) /K )) .Note that c m = ord m ( q ) , which corresponds to the algebraic part of the field extension K ( E [ m ]) /K and Gal( K ( E [ m ]) /K F q cm ) ≤ SL ( Z /m Z ) describes the geometric part.This enables us to use an effective version of the Chebotarev density theorem ([MuSc] Theorem2): we obtain that if m, n ∈ N such that ( m, p ) = 1 and ord m ( q ) | n , then there exists ρ = ρ ( E, K, m ) such that (cid:12)(cid:12)(cid:12)(cid:12) π ( n, K ( E [ m ]) /K ) − ord m ( q ) · q n [ K ( E [ m ]) : K ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ (3 g K + ( ρ + 1) | V E/K | ) q n/ n + | V E/K | n ! + | V E/K | . If K ( E [ m ]) /K is at most tamely ramified for all n (consequently if p > ) then ρ = 0 . Thecontribution of the paper to the proof is that we handle wildly ramified field extensions to bound ρ independently from m . For this we need to do some local computation, this is contained in Section2. Now we can simply sum up these estimations and by standard arguments prove the theorem.In Section 3 we investigate the density δ ( E/K, .If K ( E [ ℓ ]) = K for some prime ℓ = p or equivalently if the torsion subgroup of E ( K ) is notcyclic, it is clear that for all ν ∈ V ( E/K ) the group of E ν ( k ν ) is not cyclic either. It is a naturalquestion to ask whether the converse is true. In [ CT ] is proven that for the special case K = F q ( j E ) the answer is affirmative.We start by showing an elliptic curve E/K with cyclic torsion subgroup such that for infinitelymany n ∈ N the value of δ ( E/K, , n ) is 0 (and for at least one n we have ν ∈ V E/K ( n ) | E ν ( k ν ) is cyclic ) = 0 .)Then we prove Theorem 2 and finally we construct an elliptic curve E/K with cyclic torsionsubgroup for which δ ( E/K, ,
1) = 0 and hence δ ( E/K, is also 0. Let L | K be a Galois extension of function fields with constant field k , and unramified away froma set of places S . Let | S | = P ν ∈ S deg( ν ) and let c be the integer such that the algebraic closure of2 in L is a degree c extension of k . Let π ( n, L/K ) be the number of places of degree n of K whichsplit completely in L/K . ρ L/K is an integer as defined in [Se3] 1.2 and [MuSc] 3 and which we willredefine and estimate thereafter.We will use the following version of Chebotarev density theorem for global function fields:
Theorem 3. ([MuSc], Theorem 2.) If c | n , then (cid:12)(cid:12)(cid:12)(cid:12) π ( n, L/K ) − c L : K ] | V K ( n ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) (3 g K + ( ρ L/K + 1) | S | ) q n/ n + | S | n (cid:19) + | S | . Otherwise π ( n, L/K ) = 0 . Recall the definition of ρ L/K :By the abuse of notation let first K denote a local field, with ring of integers o K , maximal ideal m K ⊳ o K and standard valuation val K : K → Z ∪ {∞} .Let L | K be a finite, totally ramified extension with ramification index e L/K , different ideal D L/K ⊳ o L and let us denote by val L ( D L/K ) the expontent of m L in D L/K : the integer n for which D L/K = m nL .We then have p ∤ e L/K ⇐⇒ val L ( D L/K ) = e L/K − ⇐⇒ L | K is at most tamely ramified.([Na], Theorem 4.8)Thus there exists an integer ≤ j < e L/K such that val L ( D L/K ) ≡ − j − e L/K ) . Then j = 0 ⇐⇒ L | K is at most tamely ramified. Finally let ρ L/K = ( , if L/K is at most tamely ramified , val L ( D L/K ) − ( e L/K − j − e L/K , if L/K is wildly ramified , Notice that this is really an integer and ρ L/K = l val L ( D L/K )+1 e L/K m − , where ⌈·⌉ is the ceilingfunction. Remark that if we would have Hensel’s bound val L ( D L/K ) ≤ e L/K − p ( e L/K ) (which fails in the function field case), then we would get ρ L/K = 1 if L | K is wildly ramified and 0otherwise.Also note that if L = K ( α ) with a separable Eisenstein polynomial f satisfying f ( α ) = 0 , then e L/K = deg( f ) and val L ( D L/K ) = val L ( f ′ ( α )) .Now, return to the original situation, that is, K denotes a global function field, and L/K is aGalois extension with constant field k . Then let ρ L/K = max ν ( ρ L ν /K ν ) . Lemma 4.
Let M | L | K be a tower of totally ramified Galois extensions with constant field k . Wethen have1. ρ L/K ≤ ρ M/K ≤ ρ L/K + l ρ M/L e L/K m .2. ρ M/K = ρ L/K if M | L is at most tamely ramified.Proof. Clearly it suffices to verify the statements locally. Let us once again denote by K = K ν , L = L ν and M = M ν . Then ρ M/K = (cid:24) val M ( D M/K ) + 1 e M/K (cid:25) − (cid:24) val M ( D M/L ) + val M ( D L/K ) + 1 e M/L · e L/K (cid:25) − (cid:24) e L/K · val M ( D M/L ) + 1 e M/L + val L ( D L/K ) + 1 e L/K − e L/K (cid:25) − . Using the trivial inequality ⌈ a + b ⌉ ≤ ⌈ a ⌉ + ⌈ b ⌉ we get the upper bound in 1. The lower boundis also clear since (val M ( D M/L ) + 1) /e M/L ≥ , hence e L/K (cid:18) val M ( D M/L ) + 1 e M/L − (cid:19) ≥ . Moreover equality holds if and only if M | L is at most tamely ramified, which proves 2.3 roposition 5. Let
E/K be a non-isotrivial elliptic curve over K and m, n ∈ N such that ( m, p ) = 1 and ord m ( q ) | n . Then there exists ρ independent from n such that (cid:12)(cid:12)(cid:12)(cid:12) π ( n, K ( E [ m ]) /K ) − ord m ( q ) · q n [ K ( E [ m ]) : K ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ (3 g K + ( ρ + 1) | V E/K | ) q n/ n + | V E/K | n ! + | V E/K | . Proof.
We use the fact that there exits a finite extension K ′ | K such that E/K ′ has either goodor split multiplicative reduction over K ′ ([Si1] Proposition VII.5.4.), hence for all m we have that K ( E [ m ]) /K is at most tamely ramified ([Si2] Theorem 10.2). Then by Lemma 4 we get that ρ K ( E [ m ]) /K ≤ ρ K ′ ( E [ m ]) /K = ρ K ′ /K =: ρ , which does not depend on m .Theorem 1 follows from this by standard arguments. Remarks.
1. Let k = F , K = k ( j ) and E/K be the elliptic curve with j -invariant j definedby the equation y + xy + x + j − = 0 . Here K ( E ) /K is wildly ramified at ∞ . How-ever if we consider the Deuring normal form E ′ : y + txy + y + x = 0 , some computationwith Tate’s algorithm show that there is no more wild ramification. The j -invariant of E ′ is t / ( t + 1) . Hence we can set L = K ( t ) = k ( t ) with t satisfying f ( t ) = t + jt + j = 0 -doing that we adjoin the coordinates of 2 points of the 3-torsion. Here we have e L/K = 12 and val L ( D L/K ) = val L ( f ′ ( t )) = 14 since f is an Eisenstein polynomial. Thus ρ L/K = l val L ( D L/K )+1 e L/K m − .We need one more field extension, since E and E ′ are not isomorphic over L , only over M = L ( s ) with s + ts + t = 0 . Here we have e M/L = 2 and val M ( D M/L ) = 4 . Hence as inthe proof of 4 ρ M/K = (cid:24) val M ( D M/L ) + e M/L · val L ( D L/K ) + 1 e M/L · e L/K (cid:25) − (cid:24) (cid:25) − . Hence in this case we have (cid:12)(cid:12)(cid:12)(cid:12) π ( n, K ( E [ m ]) /K ) − n | SL ( Z m ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ · n/ + 1 n + 2 . For a ∈ F ∗ n let E ( a ) / F n : y + xy + x + a = 0 and let E (0) / F n : y + y + x = 0 . Denote f ( n ) = a ∈ F n | E ( a ) is cyclic) = P d | n d ν ∈ V E/K ( d ) | d ν = 1) . Here only the term d = n is relevant, thus f ( n ) ≃ g ( n )+ O (2 n/ ) , where g ( n ) = 2 n P m ≤ n/ m | n − | SL ( Z /m Z ) | . We computedsome values of f ( n ) and g ( n ) , and the following tables illustrate the result. n f ( n ) g ( n ) f ( n ) − g ( n )1 2 2 02 3 3 . − .
833 8 8 04 15 15 . − .
25 32 32 06 60 61 . − .
147 128 128 08 246 243 .
22 2 . n f ( n ) g ( n ) f ( n ) − g ( n )9 512 510 .
48 1 . . − . . − . . − . . − . .
37 3 . .
91 28 . Note that if n − > is a Mersenne prime, then our esimate is sharp: there is no nontrivial m such that m | n − , thus δ ( E/K, , n ) = 1 and also E ν ( k ν ) is cyclic for all ν ∈ V E/K ( n ) .2. Let k = F and K = k ( j ) and E : y + xy − x + j − = 0 the elliptic curve with j -invariant j over K . We have wild ramification at ∞ again.Now set L = K ( µ ) with µ − j (( µ − = 0 . E/L is isomorphic with E ′ : y = x ( x − x − µ + 1) - this is a bit varied version of the Legendre normal form4omposed with a quadratic extension. Here is no more wild ramification and again we have ρ L/K = 1 .The Chebotarev density theorem in this case gives: (cid:12)(cid:12)(cid:12)(cid:12) π ( n, K ( E [ m ]) /K ) − n | SL ( Z m ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ · n/ + 1 n + 2 . Let n ∈ N \ { } . Recall that δ ( E/K, , n ) = X m ≥ m | q n − µ ( m )ord m ( q ) | K ( E [ m ]) : K | . It is mentioned in [CT] Remark 17, that if there exists a prime ℓ = p such that K ( E [ ℓ ]) = K (orequivalently the torsion subgroup of E ( K ) is not cyclic), then for all n we have δ ( E/K, , n ) = 0 .Moreover if K = F q ( j E ) , then the converse holds.We show that it is not true in general: Proposition 6.
There exists an elliptic curve
E/K with cyclic torsion subgroup such that forinfinitely many n ∈ N we have δ ( E/K, , n ) = 0 .Proof. Let K = F ( t ) . We construct a curve E such that the extension K ( E [3]) | K is algebraic andof degree 2. Then since ord (5) = 2 , and if | n we have δ ( E/K, , n ) = X m ≥ m | n − µ ( m )ord m (5) | K ( E [ m ]) : K | = X ∤ m ≤ m | n − (cid:18) µ ( m )ord m (5) | K ( E [ m ]) : K | − µ ( m )ord m (5) | K ( E [3 m ]) : K | (cid:19) . Then either we have | ord m (5) = ord m (5) and K ( E [3 m ]) = K ( E [3]) K ( E [ m ]) = K ( E [ m ]) , since K ( E [ m ]) contains the cyclotomic field K ( ζ m ) ≥ K ( ζ ) = K ( E [3]) . Or ∤ ord m (5) and hence ord m (5) = 2 · ord m (5) , moreover K ( E [3]) (cid:2) K ( E [ m ]) since (ord m ( q ) , ord ( q )) = 1 , consequently | K ( E [3 m ]) : K ( E [ m ]) | = 2 . Thus all terms on the right-hand side are 0, and δ ( E/K, , n ) = 0 .To realize an explicit example set p ( t ) = t − t + 2 t , q ( t ) = 2 t + t + t − and E : y = x + p ( t ) x + q ( t ) over K . Then ∆ E = − p ( t ) + 27 q ( t ) ) = 0 and j E = 1728 · p ( t ) ∆ E / ∈ F q ,thus E is non-isotrivial.Moreover E : y = x − has 6 points over F , hence the torsion subgroup of E ( K ) is cyclic.The third division polynomial is ψ ( x ) = 3( x + 2 p ( t ) x − q ( t ) x − p ( t ) ) = 3( x − (1 + 2 t ) x − ( t − t − t − x + 2 t )( x + 1) , where on the right-hand side the first term is irreducible over F [ t ] . However if wedenote L = K ( √ , the following points are in E ( L )[3] : ( − , ±√ t − t + 2 t − , ( − t , ± t − t + 2 t + 1)) , hence E ( L )[3] is the whole 3-torsion. Thus K ( E [3]) = L and | K ( E [3]) : K | = 2 , indeed. Remarks.
1. For n = 2 it is easy to verify, that there is no place ν ∈ V E/K (2) such that E ν ’sgroup is cyclic. Thus ν ∈ V E/K ( n ) | d ν = 1) = 0 = ⇒ ∃ ℓ = p : K ( E [ ℓ ]) = K .2. The same can be carried out for q = 5 , q ≡ . For example if q = 2 we can choose K = F ( t )( u ) , where u + ( t + 1) u + ( t + 1) = 0 and E/K : y + xy = x + ( t + t + t + t ) .If q , then we shall use a different prime ℓ instead of 3 such that ord ℓ ( q ) > .5ow we shall turn to a slightly different question. Whether the same phenomenon can arise ifwe consider all places ν ∈ V E/K at once. Recall that by the definition of Dirichlet density we have δ ( E/K,
1) = lim s → P ν ∈ VE/K d ν =1 q − s deg( ν ) P ν ∈ V E/K q − s deg( ν ) . Of course, if the torsion subgroup of E ( K ) is not cyclic, then by definition δ ( E/K,
1) = 0 . Ourgoal is to determine when δ ( E/K,
1) = 0 in general.Recall that for all but finitely many primes ℓ we have Gal( K ( E [ ℓ ]) /K F q ord ℓ ( q ) ) ≃ SL ( Z ℓ ) ([Ig]Theorem 4, [Se1], [CT] Theorem 6) Let M ( E/K ) be the torsion conductor of E/K - the productof the finitely many exceptional primes ℓ i and N ( E/K ) be the least common multiple of ord ℓ i ( q ) .Moreover if m , m ∈ N such that ( m , p ) = ( m , p ) = ( m , M ) = 1 and m is composed of primesdividing M ( E/K ) , then K ( E [ m ]) ∩ K ( E [ m ]) = K F q (ord m q ) , ord m q )) . ([CT] Corollary 8)Now we are ready to prove Theorem 2: Proof.
First assume that δ ( E/K, , > . Let N = N ( E/K ) and M = M ( E/K ) . If ( n, N ) = 1 ,then in the definition of δ ( E/K, , n ) all m | q n − can be written in the form m = m m ′ with ( m , M ) = 1 and m ′ | q − . Note that ord m ( q ) = ord m ( q ) . Using the previously mentioned factswe get | K ( E [ m ]) : K | = | K ( E [ m ]) : K | · | K ( E [ m ′ ]) : K | and we can proceed as in [CT] Remark17: δ ( E/K, , n ) = X m | qn − ( m ,q − X m ′ | q − µ ( m m ′ )ord m m ′ ( q ) | K ( E [ m m ′ ]) : K | == X m | qn − ( m ,q − µ ( m ) | K ( E [ m ]) : K F ord m ( q ) q | X m ′ | q − µ ( m ′ ) | K ( E [ m ′ ]) : K | == X m | qn − ( m ,q − µ ( m ) | SL ( Z /m Z ) | · δ ( E/K, , > δ ( E/K, , Y ℓ ∤ pM (cid:18) − ℓ ( ℓ − (cid:19) = ε > . Now from the definition of δ ( E/K, we have δ ( E/K,
1) = lim s → P n>n P ν ∈ VE/K ( n ) d ν =1 q − sn P n>n | V E/K ( n ) | q − sn ≥ P n ≡ N ) n>n ν ∈ V E/K ( n ) | d ν = 1) q − sn P n ≡ N ) n>n N | V E/K ( n ) | q − sn ≥≥ P n ≡ N ) n>n ε/ · | V E/K ( n ) | q − sn P n ≡ N ) n>n N | V E/K ( n ) | q − sn = ε N > . Here we used the fact that | ν ∈ V E/K ( n ) | d ν = 1) − δ ( E/K, , n ) ·| V E/K ( n ) || < c ( K, E ) q n/ < q n / for a fixed c ( K, E ) > and for n > n ( K, E ) depending only on K, E (cf [CT] Theorem 1.1 andthe asymptotic formula for | V E/K ( n ) | ).To prove the converse statement assume that δ ( E/K, ,
1) = 0 . We will show that for all n ∈ N we have δ ( E/K, , n ) = 0 . Then there exists C > such that | V E/K ( n ) | ≥ Cq n /n and by Theorem1 we have some c = c ( K, E, such that ν ∈ V E/K ( n ) | d ν = 1) ≥ cq n/ /n . So by definition δ ( E/K,
1) = lim s → P n ν ∈ V E/K ( n ) | d ν = 1) q − ns P n | V E/K ( n ) | q − ns ≤ lim s → P n cq − n ( s − / /n P n Cq − n ( s − /n = 0 , and we are done. 6e shall examine in detail when is δ ( E/K, ,
1) = 0 . δ ( E/K, ,
1) = X m | q − µ ( m ) | K ( E [ m ]) : K | = 1 | K ( E [ q − K | X m | q − | K ( E [ q − K ( E [ m ]) | . Let H = Gal( K ( E [ q − /K ) ≤ SL ( Z / ( q − Z ) and for primes ℓ | q − denote H ℓ = H ∩ Ker( π ℓ ) ,where π ℓ : SL ( Z / ( q − Z ) → SL ( Z /ℓ Z )) is the modulo ℓ reduction. So by the inclusion exclusionprinciple δ ( E/K, ,
1) = 0 if and only if we have H = S ℓ H ℓ .Now let n be an arbitrary integer. We can refine our computation of δ ( E/K, , n ) as follows: δ ( E/K, , n ) = X m | qn − ( m ,q − X m ′ | q − µ ( m m ′ ) ord m m ′ ( q ) | K ( E [ m m ′ ]) : K | = X m | qn − ( m ,q − µ ( m )ord m ( q ) | K ( E [ m ( q − K | S ( m ) , where S ( m ) = P m ′ | q − µ ( m ′ ) | K ( E [ m ( q − K ( E [ m m ′ ]) | .We claim that S ( m ) is 0 for all m . For this let H ( m ) = Gal( K ( E [ m ( q − K ( E [ m ])) and for primes ℓ | q − let H ( m ) ℓ = Gal( K ( E [ m ( q − /K ( E [ m ℓ ])) . As before, we have S ( m ) = 0 ⇐⇒ H ( m ) = S ℓ H ( m ) ℓ .Let σ ∈ H ( m ) . We can view it as an element of Gal( K ( E [ m ( q − /K , and thus σ = σ | K ( E [ q − ∈ H . Since δ ( E/K, ,
1) = 0 there exists ℓ | q − such that σ ∈ H ℓ . But this meansthat σ fixes K ( E [ ℓ ]) . Moreover by definition σ fixes K ( E [ m ]) , thus also K ( E [ m ℓ ]) = K ( E [ ℓ ]) · K ( E [ m ]) . Hence σ ∈ H ( m ) ℓ = Gal( K ( E [ m ( q − /K ( E [ m ℓ ])) andas we can do that for any σ , we got H ( m ) = S ℓ H ( m ) ℓ . Corollary 7.
The proof shows that if q − has at most 2 prime factors ℓ and ℓ , then δ ( E/K, > if and only if E ( K ) has cyclic torsion subgroup.Proof. In this case H = H ℓ ∪ H ℓ - the union of two proper subgroup can not be the wholegroup.By the first glimpse one would expect that δ ( E/K, ,
1) = 0 if and only if the torsion subgroupof E ( K ) is not cyclic. This is not true, in the following we construct counterexamples. Proposition 8. If q − has at least 3 distinct prime divisors, there exists an elliptic curve E/K with cyclic torsion subgroup for which δ ( E/K,
1) = 0 .Proof. If q − has at least 3 distinct prime divisors, we can construct some subgroups H ≤ SL ( Z / ( q − Z ) such that H = S ℓ H ℓ .In the case of p = 2 we can write q − q q q with q i > and pairwise relatively prime. Let H contain the central elements diag(1) , diag( x ) , diag( x ) and diag( x ) of SL ( Z / ( q − Z ) , wherewe have x i ≡ q j ) if i = j and x i ≡ − q j ) if i = j . Then H ≃ ( Z / Z ) and the H ℓ -sare the nontrivial subgroups of H .In the case of p > we can write q − α q q with α ≥ , q i odd and ( q , q ) = 1 . Thereexists r ∈ Z / ( q − Z such that r ≡ α ) , r ≡ q ) , r ≡ − q ) . Let H = (cid:26)(cid:18) (cid:19) , (cid:18) − − (cid:19) , (cid:18) r α − q q r (cid:19) , (cid:18) − r α − q q − r (cid:19)(cid:27) . As above we have H ≃ ( Z / Z ) and the H ℓ -s are the nontrivial subgroups of H .For example the smallest q is q = 31 , then we have H = (cid:26)(cid:18) (cid:19) , (cid:18)
11 150 11 (cid:19) , (cid:18)
19 150 19 (cid:19) , (cid:18)
29 00 29 (cid:19)(cid:27) ≃ ( Z / Z ) ≤ SL ( Z / Z ) . Now our task is to find an elliptic curve
E/K such that the algebraic closure of the primefield in K has q elements and Gal( K ( E [ q − /K ) = H . Let E/ F q ( t ) be a curve with j -invariant t . Then by Igusa’s results ([Ig], Theorem 3) we have N ( E/ F q ( t )) = 1 . We have7 = Gal( F q ( t )( E [ q − / F q ( t )) ≃ SL ( Z / ( q − Z ) , hence we can identify G with the speciallinear group. Let H ≤ G the subgroup we constructed above and K = ( F q ( t )( E [ q − H and con-sider E/K . It is clear that the constant field of K has size q and that Gal( K ( E [ q − /K ) = H ≤ G .Moreover the only exceptional primes are the primes dividing q − , since the geometric part of theextensions F q ( t )( E [ ℓ ]) are disjoint. Remark.
It does not follow from the statement that for no ν ∈ V E/K is E ν ( k ν ) cyclic. However we haveproven that for only a few ν -s this is the case. References [BaSh] W. D. Banks and I. E. Shparlinski. Sato-Tate, cyclicity, and divisibility statistics onaverage for elliptic curves of small height.
Israel J. Math , 173:253–277, 2009.[ClKu] D.A. Clark and M. Kuwata. Generalized Artin’s conjecture for primitive roots and cyclicitymod p of elliptic curves over function fields.
Canad. Math. Bull. , 38 (2):167–173, 1995.[CoMu] A. C. Cojocaru and M. R. Murty. Cyclicity of elliptic curves modulo p and elliptic curveanalogues of Linnik’s problem.
Math. Ann. 330 , 330:601–625, 2004.[Co1] A. C. Cojocaru. On the cyclicity of the group of Fp-rational points of non-CM ellipticcurves.
Journal of Number Theory , 96 (2):335–350, 2002.[Co2] A. C. Cojocaru. Cyclicity of CM elliptic curves modulo p.
Trans. Amer. Math. Soc. , 335(7):2651–2662, 2003.[CT] A. C. Cojocaru and Á. Tóth. The distribution and growth of the elementary divisorsof the reductions of an elliptic curve over a function field.
Journal of Number Theory ,132:953–965, 2012.[GuMu] R. Gupta and M. R. Murty. Cyclicity and generation of points mod p on elliptic curves.
Invent. Math. , 101 (1):225–235, 1990.[HaVo] C. Hall and J. F. Voloch. Towards Lang-Trotter for elliptic curves over function fields(part 1).
Pure Appl. Math. Q. , 2 (1):163–178, 2006.[Ig] J.-I. Igusa. Fibre systems of Jacobian varieties. III. Fibre systems of elliptic curves.
Amer.J. Math. , 81:453–476, 1959.[LaTr] S. Lang and H. Trotter. Primitive points on elliptic curves.
Bull. Amer. Math. Soc. , 83(2):289–292, 1977.[MuSc] V. K. Murty and J. Scherk. Effective versions of the Chebotarev density theorem forfunction fields.
C. R. Acad. Sci. Paris , 319, Série I:523–528, 1994.[Mu1] M. R. Murty. On Artin’s conjecture.
Journal of Number Theory , 16 (2):147–168, 1983.[Mu2] M. R. Murty. On the supersingular reduction of elliptic curves.
Proc. Indian Acad. Sci. ,97 (1-3):247–250, 1987.[Na] W. Narkiewicz.
Elementary and analytic theory of algebraic numbers , volume 57. Mono-grafie Matematyczne, Warsawa, 1974.[Se1] J.-P. Serre. Propriétés galoisiennes des points d’ordre fini des courbes elliptiques.
Invent.Math. , 15 (4):259–331, 1972.[Se2] J.-P. Serre.
Summaries of courses of the 1977-78 academic year , volume 54. Collège deFrance, Paris, 1978. 8Se3] J.-P. Serre. Quelques applications du théorème de densité de Chebotarev.
Publ. Math.Inst. Hautes Études Sci. , 54:123–201, 1981.[Si1] J.-H. Silverman.
The Arithmetic of Elliptic Curves . Springer-Verlag, 1986.[Si2] J.-H. Silverman.
Advanced Topics in the Arithmetic of Elliptic Curves . Springer-Verlag,1995.[Vo1] J. F. Voloch. A note on elliptic curves over finite fields.
Bull. Soc. Math. France , 116(4):455–458, 1988.[Vo2] J. F. Voloch. Primitive points on constant elliptic curves over function fields.