Average Frobenius Distributions for Elliptic Curves over Abelian Extensions

25 Pages·2010·0.44 MB·English

AVERAGE FROBENIUS DISTRIBUTIONS FOR ELLIPTIC CURVES
OVER ABELIAN EXTENSIONS
NEIL CALKIN, BRYAN FAULKNER, KEVIN JAMES, MATT KING, AND DAVID PENNISTON
This paper is dedicated to the memory of our coauthor Matt King
Abstract. Let E be an elliptic curve defined over an abelian number field K of degree m.
ForaprimeidealpofO ofgoodreductionweconsiderE overthefinitefieldO /pandlet
K K
a (E) be the trace of the Frobenius morphism. If E does not have complex multiplication,
p
a generalization of the Lang-Trotter Conjecture asserts that given r,f ∈Z with f >0 and
f |m, there exists a constant C ≥0 such that
E,r,f
 √
x if f =1,
logx
#{p:N(p)≤x, deg (p)=f and a (E)=r}∼C · loglogx if f =2,
K p E,r,f
1 if f ≥3.
We prove that this conjecture holds on average in certain families of elliptic curves defined
over K.
1. Introduction and statement of results
Let E be an elliptic curve defined over a number field K. Set [K : Q] = m and denote
by O the ring of integers of K. Let p ⊂ O be a prime ideal which lies above the rational
K K
prime p ∈ Z, and denote by deg (p) the degree of p. If E has good reduction at p, then we
K
may consider E over the finite field O /p. If we denote by a (E) the trace of the Frobenius
K p
morphism, then the number of points on E over O /p is
K
#E(O /p) = N(p)+1−a (E),
K p
where N(p) = pdegK(p) is the number of elements of OK/p. Moreover, we have the Hasse
bound
(cid:112)
|a (E)| ≤ 2 N(p).
p
Let r,f ∈ Z with f > 0. Define
πr,f(x) := #{p : N(p) ≤ x, deg (p) = f and a (E) = r},
E K p
and let
(cid:90) x dt
π (x) := √ .
1/2
2 tlogt
2
In the case that K = Q, Lang and Trotter [12] made the following conjecture.
1This material is based upon work supported by the National Science Foundation under Grant No.
0552799.
Date: August 27, 2010.
1
Conjecture 1.1. If E/Q does not have complex multiplication, or if r (cid:54)= 0, then there is a
constant C such that
E,r
√
x
πr,1(x) ∼ C ·π (x) ∼ C · as x → ∞.
E E,r 1/2 E,r logx
Although the Lang-Trotter Conjecture remains open, there are many partial results. For
example, Elkies [7] proved that for any elliptic curve E/Q there are infinitely many primes p
such that a (E) = 0. Moreover, there are several results which verify that the conjecture is
p
true in an average sense over families of elliptic curves defined over Q (see [1], [3], [4], [8], [9],
[10]). For K (cid:54)= Q, less is known. In [5] David and Pappalardi proved the following result.
Theorem 1.2. Let K = Q(i), and let S denote the set of elliptic curves E : Y2 = X3 +
x
αX+β with α = a +a i,β = b +b i ∈ Z[i] and max{|a |,|a |,|b |,|b |} ≤ xlogx. If r (cid:54)= 0,
1 2 1 2 1 2 1 2
then
1 (cid:88)
πr,2(x) ∼ c loglogx,
|S | E r
x
E∈Sx
where
(cid:16) (cid:16) (cid:17)(cid:17)
q q −1− −r2
1 (cid:89) q
c = · .
r (cid:16) (cid:16) (cid:17)(cid:17)
3π (q −1) q − −1
q prime q
q>2
If r = 0, then
1 (cid:88)
π0,2(x) = O(1).
|S | E
x
E∈Sx
In this paper we generalize David and Pappalardi’s result as follows. Let {α ,...,α } be
1 m
an integral basis for O . Given (cid:126)v = (v ,...,v ) ∈ Zm, put (cid:107)(cid:126)v(cid:107) := max |v | and define
K 1 m 1≤i≤m i
R((cid:126)v) := (cid:80)m v α . For (cid:126)v,w(cid:126) ∈ Zm we write E for the curve
i=1 i i (cid:126)v,w(cid:126)
(1) E : y2 = x3 +R((cid:126)v)x+R(w(cid:126))
(cid:126)v,w(cid:126)
with discriminant ∆ = −16[4R((cid:126)v)3 +27R(w(cid:126))2], and for t ∈ R we let
(cid:126)v,w(cid:126) >0
C := {((cid:126)v,w(cid:126)) ∈ (Zm)2 : (cid:107)(cid:126)v(cid:107),(cid:107)w(cid:126)(cid:107) ≤ t and ∆ (cid:54)= 0}.
t (cid:126)v,w(cid:126)
In this paper we prove the following theorem (the case of even r can be handled in a similar
way).
Theorem 1.3. Suppose K is an abelian number field and r is an odd integer. Then there
are explicit constants D and D (see Section 2 for details) such that for any (cid:15) > 0,
r,1,K r,2,K
(cid:40)
1 (cid:88) D ·π (x) if f = 1 and t (cid:29) xlog2+(cid:15)x,
πr,f (x) ∼ r,1,K 1/2 √
|C | E(cid:126)v,w(cid:126) D loglogx if f = 2, m is even and t (cid:29) xlogx.
t r,2,K
((cid:126)v,w(cid:126))∈Ct
Moreover, if f ≥ 3, f | m and t ≥ x1/f, then
1 (cid:88)
πr,f (x) = O(1).
|C | E(cid:126)v,w(cid:126)
t
((cid:126)v,w(cid:126))∈Ct
Remark 1.4. While we have not pursued this, it would be interesting to have an explicit
expression for the error term in the f ≥ 3 case of Theorem 1.3.
2
The organization of the rest of this paper is as follows. In Section 2 we state Theorem 2.2,
which is a more precise version of Theorem 1.3, and show that it follows from three key
lemmas. Lemma 2.3, which is proved in Section 4, relates the desired average of the main
theorem to a weighted sum of special values of L-series. Lemma 2.4, which is proved in
Section 6, gives estimates for this weighted sum. Finally, Lemma 2.5, which is proved
in Section 8, gives an Euler product representation for one of the constants appearing in
Lemma 2.4. Sections 3, 5 and 7 contain various technical results which are essential to our
proofs of the three key lemmas.
2. Proof of Main Theorem
In this section we prove a more precise version of Theorem 1.3. Let r,f,A,B ∈ Z with
r odd, f,A,B > 0 and (A,B) = 1. Define ∆r,A,f := r2 − 4Af, and for q prime let ∆ =
q
(cid:16) (cid:17)
∆r,A,f := ord (∆r,A,f), B := ord (B) and γ = γ := ∆r,A,f/q∆q . Put
q q q q q r,A,f,q q
Q< := {q prime : q | B, q (cid:45) 2r and 0 < ∆ < B } and
r,A,B,f q q
Q≥ := {q prime : q | B, q (cid:45) 2r and ∆ ≥ B },
r,A,B,f q q
(cid:40)
γ if ∆ is even,
and for q ∈ Q< let Γ := q q
r,A,B,f q 0 if ∆ is odd.
q
We define constants for use in the cases f = 1 and f = 2 respectively as follows:
(cid:32) (cid:33)
(cid:89) q(q +γ ) (cid:89) Γ (qΓ +1) Γ2(q∆q/2−1 −1)
q q q q
k := 1+ +
r,A,B q2 −1 q∆q/2−1(q2 −1) q∆q/2−1(q −1)
q|B q∈Q<
r,A,B,1
q(cid:45)2r∆r,A,1
(cid:89) (cid:18) q(cid:98)(Bq+1)/2(cid:99) −1 qBq+2 (cid:19)(cid:89) q(q +γ )
q
· + ,
q(cid:98)(Bq−1)/2(cid:99)(q −1) q3(cid:98)(Bq+1)/2(cid:99)(q2 −1) q2 −1
q∈Q≥ q|B
r,A,B,1 q|r
(cid:32) (cid:33)
(cid:89) q2Bq+1(q −γq)+γqBq+1(q −1) (cid:89) (cid:18) q(cid:98)∆q/2(cid:99)+1 −1 (cid:19)
c :=
r,A,B q2Bq−1(q −γq)(q2 −γq) q(cid:98)∆q/2(cid:99)(q −1)
q|B q∈Q<
r,A,B
q(cid:45)2r∆r,A,2 2(cid:45)∆q
(cid:32) (cid:33)
(cid:89) q∆q/2 −1 q2(Bq−∆q)(q −γq)(q∆q+2 −γqq∆q +γq)+γqBq+1q(q −1)
· +
q∆q/2−1(q −1) q2Bq−∆q/2(q −γq)(q2 −γq)
q∈Q<
r,A,B
2|∆q
(cid:89) (cid:18)q2(cid:100)Bq/2(cid:101)+1(q +1)(q(cid:100)Bq/2(cid:101) −1)+qBq+2(cid:19)
·
q3(cid:100)Bq/2(cid:101)(q2 −1)
q∈Q≥
r,A,B
(cid:32) (cid:33)
(cid:89) q2Bq+1(q −γq)+γqBq+1(q −1)
· .
q2Bq−1(q −γq)(q2 −γq)
q|B
q|r
Nextwerecallaclassicalresultwhichgivesausefulcharacterizationofabeliannumberfields.
3
Fact 2.1. Let K/Q be a Galois number field. Then K is abelian if and only if there exists
an integer B such that deg (p) depends only on the residue class of p modulo B .
K K K
Suppose from now on that K is abelian, and for f | m fix a ,...,a ,B ∈ Z so that
f,1 f,(cid:96) K
f
a prime p ∈ Z splits into degree f primes in O if and only if p ≡ a ,...,a or a
K f,1 f,(cid:96) −1 f,(cid:96)
f f
(mod B ). Define
K
4m (cid:89) (cid:18) q2 (cid:19) (cid:89) q(q2 −q −1) (cid:88)(cid:96)1
D := · k ,
r,1,K 3πφ(B ) q2 −1 (q +1)(q −1)2 r,a1,i,BK
K q(cid:45)BK q(cid:45)2rBK i=1
q|r
and if m is even let
   (cid:16) (cid:16) (cid:17)(cid:17)
m (cid:89) q (cid:89) q q2 −q −1− −q1 (cid:88)(cid:96)2
Dr,2,K := 3πφ(BK) q(cid:45)BKq −(cid:16)−q1(cid:17)q(cid:45)2rBK (q −1)(q2 −1) · i=1 cr,a2,i,BK.
q|r
We now state our main result.
Theorem 2.2. Let K be an abelian number field, and suppose t ≥ x1/f. Then
√
1 (cid:88) (cid:18) x x3/2logx(cid:19)
πr,1 (x) = D ·π (x)+O +
|C | E(cid:126)v,w(cid:126) r,1,K 1/2 logc+1x t
t
((cid:126)v,w(cid:126))∈Ct
for any c > 0, and if m is even,
√
(cid:18) (cid:19)
1 (cid:88) xlogx
πr,2 (x) = D loglogx+O 1+ .
|C | E(cid:126)v,w(cid:126) r,2,K t
t
((cid:126)v,w(cid:126))∈Ct
Moreover, if f ≥ 3 and f | m, then
1 (cid:88)
πr,f (x) = O(1).
|C | E(cid:126)v,w(cid:126)
t
((cid:126)v,w(cid:126))∈Ct
Recall that if χ is a Dirichlet character, we have the Dirichlet L-series
∞
(cid:88) χ(n)
L(s,χ) = .
ns
n=1
(cid:0) (cid:1)
Given an integer d we let χ be the Kronecker character χ (•) = d . Set
d d •
B(r) := max{3,r2/4,∆ },
K
where ∆ is the discriminant of K, and for k any positive integer let
K
(cid:40)
(r2 −4pf)/k2 if k2 | r2 −4pf,
d (p) = d (p) :=
k k,r,f
0 otherwise.
We utilize the following three lemmas in our proof of Theorem 2.2.
4
Lemma 2.3. If f | m and t ≥ x1/f, then
(cid:34) (cid:96)
1 (cid:88) m 1 (cid:88)f (cid:88) 1 (cid:88)
πr,f (x) = √ L(1,χ )logp
|Ct| E(cid:126)v,w(cid:126) π xlogx √ k dk(p)
((cid:126)v,w(cid:126))∈Ct i=1 k≤2 x B(r)<p≤x1/f
(k,2r)=1 p≡af,i (modBK)
k2|r2−4pf
(cid:88)(cid:96)f (cid:90) x (cid:88) 1 (cid:88) d (cid:18) 1 (cid:19) (cid:35)
− L(1,χ )logp √ dS +E(x,t),
d (p)
B(r)f √ k k dS SlogS
i=1 k≤2 S B(r)<p≤S1/f
(k,2r)=1 p≡af,i (modBK)
k2|r2−4pf
where

loglogx+ x3/2logx if f = 1,

 √ t
E(x,t) (cid:28) 1+ xlogx if f = 2,
t

1 if f ≥ 3.
The next lemma consists of a result of James [11, Proposition 2.1] and a straightforward
generalization of a result of David and Pappalardi [5, Lemma 2.2]. Denote by [C,D] the
least common multiple of C and D, and let
(cid:88) 1 (cid:88) κr,A,B(n)
K := k ,
r,A,B k nφ([B,nk2])
k∈N n∈N
where
(cid:88) (cid:16)a(cid:17)
κr,A,B(n) := .
k n
a∈(Z/4nZ)
a≡0,1 (mod4)
(r2−ak2,4nk2)=4
4A≡r2−ak2 (mod(4B,4nk2))
Also, let
(cid:88) 1 (cid:88) 1 (cid:88) (cid:16)a(cid:17)
(2) C := C (a,n,k),
r,A,B k nφ(4Bnk2) n r
k∈N n∈N a∈(Z/4nZ)∗
(k,2r)=1
where
C (a,n,k) := #{b ∈ (Z/4Bnk2Z)∗ : b ≡ A (mod B) and 4b2 ≡ r2 −ak2 (mod 4nk2)}.
r
ThenwehavethefollowingestimatesfortheweightedsumsofL-seriesappearinginLemma2.3.
Lemma 2.4. For every c > 0,
 (cid:16) (cid:17)
(cid:88) 1 (cid:88) Kr,A,B ·x+O logxcx if f = 1,
L(1,χd (p))logp = √ (cid:16) √ (cid:17)
√ k k C · x+O x if f = 2.
k≤2 x B(r)<p≤x1/f r,A,B logcx
(k,2r)=1 p≡A (modB)
k2|r2−4pf
Our third lemma gives an Euler product expansion for the constant appearing in the f = 2
case of Lemma 2.4 (for the f = 1 case see [11, Theorem 1.1]).
5
Lemma 2.5. We have
   (cid:16) (cid:16) (cid:17)(cid:17)
q q2 −q −1− −1
2 (cid:89) q (cid:89) q
Cr,A,B = 3φ(B) · q −(cid:16)−1(cid:17)  (q −1)(q2 −1) cr,A,B.
q(cid:45)B q q(cid:45)2rB
q|r
Proof of Theorem 2.2. First suppose f = 1. We combine Lemmas 2.3 and 2.4 to obtain
(cid:34) (cid:32) (cid:33)
1 (cid:88) m 1 (cid:88)(cid:96)1 (cid:18) x (cid:19)
πr,1 (x) = √ K ·x+O
|C | E(cid:126)v,w(cid:126) π xlogx r,a1,i,BK logcx
t
((cid:126)v,w(cid:126))∈Ct i=1
(cid:32) (cid:33) (cid:35)
(cid:88)(cid:96)1 (cid:90) x (cid:18) S (cid:19) d (cid:18) 1 (cid:19)
− K ·S +O √ dS
r,a1,i,BK logcS dS SlogS
B(r)
i=1
(cid:18) x3/2logx(cid:19)
+O loglogx+ .
t
In [11] James proved that
2 (cid:89)(cid:18) q2 (cid:19) (cid:89) q(q2 −q −1)
K = · k ,
r,A,B 3φ(B) q2 −1 (q +1)(q −1)2 r,A,B
q(cid:45)B q(cid:45)2rB
q|r
and thus (cid:80)(cid:96)1 K = πDr,1,K. Moreover, one can show that
i=1 r,a1,i,BK 2m
√
(cid:90) x S d (cid:18) 1 (cid:19) (cid:18) x (cid:19)
· √ dS = O ,
logc(S) dS SlogS logc+1(x)
B(r)
which yields
(cid:34) √ (cid:35)
1 (cid:88) D x (cid:90) x d (cid:18) 1 (cid:19)
πr,1 (x) = r,1,K − S √ dS
|C | E(cid:126)v,w(cid:126) 2 logx dS SlogS
t 2
((cid:126)v,w(cid:126))∈Ct
√
(cid:18) x x3/2logx(cid:19)
+O + .
logc+1x t
Integrating by parts gives
√ √
(cid:90) x d (cid:18) 1 (cid:19) x 2
S √ dS = − −2π (x),
1/2
dS SlogS logx log2
2
and our result now follows.
Now suppose f = 2. Here Lemmas 2.3 and 2.4 yield
(cid:34) (cid:32) √ (cid:33)
1 (cid:88) m 1 (cid:88)(cid:96)2 √ (cid:18) x (cid:19)
πr,2 (x) = √ C · x+O
|C | E(cid:126)v,w(cid:126) π xlogx r,a2,i,BK logcx
t
((cid:126)v,w(cid:126))∈Ct i=1
(cid:32) √ (cid:33) (cid:35)
(cid:88)(cid:96)2 (cid:90) x √ (cid:18) S (cid:19) d (cid:18) 1 (cid:19)
− C · S +O √ dS
r,a2,i,BK logcS dS SlogS
B(r)2
i=1
√
(cid:18) (cid:19)
xlogx
+O 1+ .
t
6
It is easy to see that the first term in the brackets is O(1). Moreover, integrating by parts
gives
(cid:90) x √ d (cid:18) 1 (cid:19) 1
S √ dS = − loglogx+O(1)
dS SlogS 2
B(r)2
and
√
(cid:90) x S d (cid:18) 1 (cid:19)
· √ dS = O(1).
logcS dS SlogS
B(r)2
Therefore
√
1 (cid:88) m(cid:18)(cid:88)(cid:96)2 (cid:19) (cid:18) xlogx(cid:19)
πr,2 (x) = C loglogx+O 1+ ,
|C | E(cid:126)v,w(cid:126) 2π r,a2,i,BK t
t
((cid:126)v,w(cid:126))∈Ct i=1
and since Lemma 2.5 implies that
(cid:88)(cid:96)2 2πD
r,2,K
C = ,
r,a2,i,BK m
i=1
our result follows.
Finally, suppose f ≥ 3. By (9) and (10) it suffices to show that the sum on the right hand
side of (9) is O(1), and this follows easily from (7). (cid:3)
3. Counting Curves
In this section we gather results that will aid us in estimating the number of ((cid:126)v,w(cid:126)) ∈ C
t
such that E reduces to a given elliptic curve.
(cid:126)v,w(cid:126)
Note first that #{n ∈ Z : |n| ≤ t} = 2t+O(1). Moreover, given (cid:126)v ∈ Zm there are at most
two values of w(cid:126) ∈ Zm such that ∆ = 0. It follows that
(cid:126)v,w(cid:126)
(3) |C | = 4mt2m +O(t2m−1).
t
Let p ⊂ O be a prime ideal of degree f which lies above an unramified rational prime p > 3.
K
Recall (see [16]) that in order to reduce an elliptic curve E/K modulo p one first obtains a
minimal model for the curve at p, and then reduces the coefficients of this model modulo p.
We denote the resulting curve by Ep, and for γ ∈ O we denote by γp its image in O /p. In
K K
order to obtain our estimate we will use the fact that if ord (R((cid:126)v)) < 4 or ord (R(w(cid:126))) < 6,
p p
then the model (1) of E is minimal at p.
(cid:126)v,w(cid:126)
Next note that since
pO ⊆ p ⊆ O ,
K K
we have that
(O /pO )
K K ∼
= O /p,
K
(p/pO )
K
and therefore
|p/pO | = pm−f.
K
Set s = pm−f, and suppose {ρ ,...ρ } is a complete set of distinct coset representatives for
1 s
p/pO . Fix γ ∈ O . Then for (cid:126)v ∈ Zm we have that
K K
R((cid:126)v) ≡ γ (mod p) ⇐⇒ R((cid:126)v)−γ ≡ ρ (mod pO ) for some 1 ≤ i ≤ s.
i K
7
If we define, for each 1 ≤ i ≤ s, integers c (1 ≤ j ≤ m) by γ +ρ = (cid:80)m c α , then
i,j i j=1 i,j j
#{(cid:126)v ∈ Zm : (cid:107)(cid:126)v(cid:107) ≤ t and R((cid:126)v)p = γp}
s
(cid:88)
= #{(k ,k ,...,k ) ∈ Zm : |c +pk | ≤ t for all 1 ≤ j ≤ m}
1 2 m i,j j
(4)
i=1
2mtm (cid:18)tm−1(cid:19)
= +O
pf pf−1
when t ≥ p.
4. The Average in Terms of L-Series
In this section we prove Lemma 2.3. We begin by recalling the Hurwitz class number (see,
for example, [13]), which is a weighted sum over the equivalence classes of binary quadratic
formsf(x,y) = ax2+bxy+cy2 ofagivendiscriminant. Moreprecisely, ifwelet∆ = b2−4ac
f
denote the discriminant of f, then for ∆ > 0,

1 if some g ∈ [f] is proportional to x2 +y2,

(cid:88) 2
H(∆) := c , where c = 1 if some g ∈ [f] is proportional to x2 +xy +y2,
f f 3

[f] 1 otherwise.
∆ =−∆
f
The Kronecker class number K(−∆), meanwhile, is simply the number of equivalence classes
of binary quadratic forms of discriminant −∆. For our purposes it will be more convenient
to work with H(∆) (note that H(∆) = K(−∆)+O(1)). We recall (see [5]) that
(cid:88) h(−∆/k2)
(5) H(∆) = 2 ,
w(−∆/k2)
k2|∆
−∆/k2≡0,1 (mod4)
where h(d) and w(d) denote respectively the Dirichlet class number of, and the number of
units in, the imaginary quadratic order of discriminant d. Moreover, Dirichlet’s class number
formula states that
w(d)|d|1/2
(6) h(d) = L(1,χ ).
d
2π
Let p > B(r) be prime. Then 4pf −r2 > 0, and for a positive integer k with k2 | r2−4pf
we have that L(1,χ ) (cid:28) logp (see [13, p. 656]). Noting that d (p) ≡ 1 (mod 4) since r
d (p) k
k
is odd, we therefore obtain the following useful estimate:
(cid:112)
(cid:88) 4pf −r2
(7) H(4pf −r2) = L(1,χ ) (cid:28) pf/2log2p.
d (p)
πk k
k2|r2−4pf
Since p > 3, any elliptic curve over F may be written in the form
pf
E : y2 = x3 +ax+b (a,b ∈ F ).
a,b pf
8
Recalling that E ∼= E over F if and only if there exists u ∈ F∗ such that a(cid:48) =
a(cid:48),b(cid:48) a,b pf pf
u4a and b(cid:48) = u6b, it follows that

pf−1 if a = 0 and pf ≡ 1 (mod 3),

 6
#{(a(cid:48),b(cid:48)) ∈ F2 : E ∼= E } = pf−1 if b = 0 and pf ≡ 1 (mod 4),
pf a(cid:48),b(cid:48) a,b 4
pf−1 otherwise.
2
Following Schoof [15] we define N(r) to be the number of F -isomorphism classes of elliptic
pf
curves with pf + 1−r points defined over F . Since p (cid:45) r, by Deuring’s Theorem (see [6]
pf
or [15, Theorem 4.6])
N(r) = K(r2 −4pf) = H(4pf −r2)+O(1).
Letting
T (r) := #{(a,b) ∈ F2 : #E (F ) = pf +1−r},
pf pf a,b pf
we have the following result.
Theorem 4.1 (Deuring). T (r) = pf ·H(4pf −r2)+O(pf).
pf 2
Proof. Let E˜ denote an F -isomorphism class of elliptic curves. Since there are at most ten
pf
isomorphism classes containing other than pf−1 curves E , we have that
2 a,b
(cid:88) (cid:88) (cid:18)pf −1(cid:19) (cid:18)pf −1(cid:19)
T (r) = 1 = N(r)+O(pf) = H(4pf −r2)+O(pf).
pf 2 2
ap(E˜)=r (a,b)
E ∈E˜
a,b
Our result now follows from (7). (cid:3)
Proof of Lemma 2.3. Note first that
1 (cid:88) 1 (cid:88) (cid:88) 1 (cid:88) (cid:88)
πr,f (x) = 1 = 1.
|C | E(cid:126)v,w(cid:126) |C | |C |
t t t
((cid:126)v,w(cid:126))∈Ct ((cid:126)v,w(cid:126))∈Ct N(p)≤x N(p)≤x ((cid:126)v,w(cid:126))∈Ct
degK(p)=f degK(p)=fap(E(cid:126)v,w(cid:126))=r
ap(E(cid:126)v,w(cid:126))=r
For a prime ideal p of degree f lying above a rational prime p > B(r) we have that
(cid:88) (cid:88)
1 = |C (E )|,
t a,b
ap((cid:126)(vE,w(cid:126)(cid:126)v),w∈(cid:126))C=tr #E ((aF,b)∈)F=2ppff+1−r
a,b pf
where
C (E ) := {((cid:126)v,w(cid:126)) ∈ C : Ep = E }.
t a,b t (cid:126)v,w(cid:126) a,b
If Ep = E , then either R((cid:126)v)p = a and R(w(cid:126))p = b, or the model (1) is not minimal at p.
(cid:126)v,w(cid:126) a,b
Since in the latter case we have R((cid:126)v)p = R(w(cid:126))p = 0, by (4) and Theorem 4.1 it follows that
(cid:88) (cid:18)4mt2m (cid:18)t2m−1(cid:19)(cid:19)(cid:18)pf (cid:19)
1 = +O H(4pf −r2)+O(pf)
p2f p2f−1 2
((cid:126)v,w(cid:126))∈Ct
ap(E(cid:126)v,w(cid:126))=r
for t ≥ p.
9
Next note that the conditions N(p) ≤ x and deg (p) = f together imply that p ≤ x1/f.
K
Then our assumption that t ≥ x1/f, along with (3) and the fact that the prime ideals p lying
above primes p ≤ B(r) do not affect our average, allows us to conclude that
1 (cid:88)
πr,f (x)
|C | E(cid:126)v,w(cid:126)
t
((cid:126)v,w(cid:126))∈Ct
(cid:34)
(cid:88) (cid:18) 1 (cid:18) 1 (cid:19)(cid:19)(cid:18)4mt2m (cid:18)t2m−1(cid:19)(cid:19)
= +O +O
4mt2m t2m+1 p2f p2f−1
B(r)f<N(p)≤x
deg (p)=f
K
(cid:35)
(cid:18)pf (cid:19)
· H(4pf −r2)+O(pf) .
2
If p is unramified in K, then deg (p) = f if and only if there are g(p) = m/f primes in O
K K
which lie above p. It follows that
1 (cid:88)
(8) πr,f (x)
|C | E(cid:126)v,w(cid:126)
t
((cid:126)v,w(cid:126))∈Ct
(cid:34)
m (cid:88) (cid:18) 1 (cid:18) 1 (cid:19)(cid:19)(cid:18)4mt2m (cid:18)t2m−1(cid:19)(cid:19)
= +O +O
f 4mt2m t2m+1 p2f p2f−1
B(r)<p≤x1/f
g(p)=m/f
(cid:35)
(cid:18)pf (cid:19)
· H(4pf −r2)+O(pf) .
2
Using the bound in (7), we find that the summand on the right hand side of (8) is
(cid:32) (cid:33)
H(4pf −r2) 1 log2p
+O + .
2pf pf tpf/2−1
Since t ≥ x1/f, we may therefore write
1 (cid:88) m (cid:88) H(4pf −r2)
(9) π (x) = +E(x,t)
|C | Er,f 2f pf
t (cid:126)v,w(cid:126)
((cid:126)v,w(cid:126))∈Ct B(r)<p≤x1/f
g(p)=m/f
where, by standard estimates,

loglogx+ x3/2logx if f = 1,
(cid:88) (cid:18) 1 log2p (cid:19)  √ t
(10) E(x,t) (cid:28) + (cid:28) 1+ xlogx if f = 2,
pf tpf/2−1 t

B(r)<p≤x1/f 1 if f ≥ 3.
g(p)=m/f
Using the equality in (7) we may rewrite the main term on the right hand side of (9) as
(cid:112)
m (cid:88) (cid:88) 4pf −r2
(11) L(1,χ ).
2πf kpf dk(p)
B(r)<p≤x1/f k2|r2−4pf
g(p)=m/f
10