In this section, we will now prove Theorem 3.5. The core of the proof of this theorem is a slight modification of the proof of Theorem 1 of [11] by Rajiv Gupta and M. Ram Murty. We will need the following lemma, which is reasonably standard.
Lemma 3.36. Let p be a prime. Let E be an elliptic curve over Q and
e
E its reduction modulo p. If Ee(Fp)[`] is not cyclic for some prime `, then
p≡1 (mod `).
Proof. If Ee(Fp)[`] is not cyclic for some prime `, the prime p must split
completely inQ(E[`]). By the existence of the Weil pairing, we haveQ(ζ`)⊂
Q(E[`]). Hence p splits completely inQ(ζ`). Now the theory of cyclotomic fields implies that p≡1 (mod `).
Theorem 3.37. For every elliptic curveE overQsuch that #E(Q)[2] = 2, the set of rational points of Km(E×E) lies dense in the space of its p-adic points for infinitely many primes p.
Proof. Take an elliptic curveE as in the statement of the theorem. When-
ever we writeEe, we will mean the reduction of E modulo the primepunder
consideration and Eet for its non-trivial quadratic twist.
In this proof, we will call a primep“good” for an elliptic curve E if the groups Ee(Fp) and Eet(Fp) are both cyclic, and “bad” otherwise. Applying
Lemma 3.31, Proposition 3.27 and Theorem 3.20 in turn, one sees that it suffices to prove that there exist infinitely many primes p that are good for
E. (Note that, since #E(Q)[2] = 2, the condition in Lemma 3.31 that the order of both groups be different frompis automatically satisfied if pis not equal to 2 and is a prime of good reduction.)
We will restrict to a set of primes among which the primes that are good for E are easier to count. Following Gupta and Murty, we define the following set of primes for each pair of positive real numbers and x:
S(x) =
p≤x prime :
E has good reduction at p, each odd prime divisor of p−1 is ≥x1/4+ and divides
p−1 only once, and p is non-split inQ(E[2])
In [11, Lemma 3], Gupta and Murty prove, using a result from sieve theory by Fouvry and Iwaniec [10], that there exists an >0 such that
#S(x)
x
3.7. Proof of Theorem 3.5 73
We choose an such that (3.7) holds, and we define S(x) = S(x). For every integera we letS(a, x)⊂S(x) be the subset of primespsuch that ap is equal to a, where ap =p+ 1−#Ee(Fp) is the trace of the Frobenius ofE
at p. By the Hasse–Weil bound, we have
S(x) = a
|a|≤2x1/2
S(a, x).
We claim that if x ∈ R is large enough, then for every integer a with |a| ≤2x1/2, there are primes`
aand`ta, both greater than or equal to x1/4+, such that, for all p∈ S(a, x), we have that Ee(Fp)[`] is cyclic for all primes
` 6= `a and Eet(Fp)[`0] is cyclic for all primes `0 6= `ta. Choose an integer a
such that |a| ≤ 2x1/2. First, assume that p ∈ S(a, x) and
e
E(Fp)[`] is not cyclic. Then ` must be odd, since p does not split in Q(E[2]). Then we must have
`2 |#Ee(Fp) =p+ 1−a. (3.8) We also have
` |p−1 (3.9)
by Lemma 3.37. This last fact implies, by the definition of S(x) and the fact that ` is odd, that we have
`≥x1/4+ (3.10)
Together, (3.8) and (3.9) imply ` | a−2. If x is large enough, then the integer a, whose absolute value is less than 2x1/2, has at most one prime divisor that is greater than or equal tox1/4+. Hence, if there is such a prime divisor `a, we have ` =`a. If there is no such prime divisor, we may set`a equal to any prime we want. For the other part, we assume thatp∈S(a, x) and Eet(Fp)[`0] is not cyclic. Now we use that `2 | #Eet(Fp) = p+ 1 +a.
Reasoning as before, we find that `0 must be an odd prime divisor ofa+ 2 that is greater than or equal to x1/4+. Again, there is at most one such a prime divisor for x large enough: if there exists one we will call it `t
a, and then we must have `0 = `ta; if not, we let `ta be arbitrary. This proves the claim made at the start of the paragraph.
Assuming that x is large enough as in the previous paragraph, we can now give a lower bound in terms of x on the number of primes pin S(a, x) such that p is good for E in the sense defined earlier. If p∈S(a, x) is bad for E, then we must have either `2a | p+ 1−a or (`ta)2 | p+ 1 +a. Since
74 Chapter 3. Density results for Kummer surfaces both `a and `ta are greater than or equal to x1/4+, and we have p ≤x, the number of p∈S(a, x) that are bad forE is bounded above by
x `2 a + x (`t a)2 +O(1)≤ x x1/2+2 + x x1/2+2 +O(1) = 2x 1/2−2+O(1). Summing the above over all integers a with |a| ≤ 2x1/2, we find that the total number of p inS(x) that is bad for E is at most
4x1/2· 2x1/2−2+O(1)
= 8x1−2+O(x1/2).
Comparing this with (3.7), we see that, for x large enough, the number of good primes in S(x) grows at least as fast asymptotically as logx2x times a
constant.
Chapter 4
Refinements and computations
4.1
Introduction
We recall the following definition from chapter 3.
Definition 4.1. Let S be a set of primes. (i) For (dp) ∈
Q
p∈SQ
∗
p and c ∈ Q∗, we call Ec a good twist of E with
respect to (dp) and S if for each p∈S we have c∈dpQ∗p2, and Ec(Q) is dense in Q
p∈SE c(
Qp).
(ii) We say E has good twists if, for all (dp) ∈ Qp∈SQ
∗
p, there is c ∈ Q
∗
such that Ec is a good twist of E with respect to (d
p) and S.
As before, if S = {p} for some prime p, and if E has good twists with respect to (dp) andS, we will also say that E has good twists with respect todp andp. If E has good twists with respect to S, we will also say thatE has good twists with respect to p.
4.1.1
Goal of this chapter
In this chapter we will establish criteria for an elliptic curve E over Q to have good twists with respect to a prime p. In view of Theorem 3.20, the existence of good twists of E with respect to p implies that the rational points on Km(E × E) lie p-adically dense. The crucial idea underlying all criteria established in this chapter is a construction of Jean-Fran¸cois Mestre [22], to be introduced in section 4.2.1. In section 4.7, we will use these criteria to perform a computer search for pairs (E, p) for which it is true that the rational points on Km(E×E) lie p-adically dense.
76 Chapter 4. Refinements and computations
4.1.2
Computer calculations
For an elliptic curveEoverQwhosej-invariant is different from 0 and 1728, we will introduce the notion of a lucky prime number p for E in Definition 4.34. Prime numbers that are not lucky for E are calledunluckyforE. The unlucky prime numbers include the prime numbers less than or equal to 7, and the primes for which E has bad reduction. It will be very easy to verify, using a Computer Algebra System, whether or not a prime numberp
is lucky forE. We will show in Proposition 4.35 that ifpis lucky forE, and if X = Km(E×E), then X(Q) lies dense in X(Qp). We have also created computer code (described in section 4.7) that computes the lucky prime numbers <2000 for all elliptic curves E overQ given by y2 =x3+ax+b, whereaand bare integers such that−5≤a≤5 witha6= 0, and 0< b≤5. Doing this, we have obtained the following result.
Theorem 4.2. Let S5,5 be the set of elliptic curves E over Q given by
y2 = x3 +ax+b, where a and b are integers such that −5 ≤ a ≤ 5 with
a 6= 0, and 0 < b ≤ 5. Then for all E ∈ S5,5 there are at most 8 prime
numbers pwith 7< p <2000which are unlucky for E. Furthermore, for all prime numbers p such that 109< p <2000 and all E ∈S5,5 we have that if
p is unlucky for E, then p is a prime of bad reduction for E. If E ∈ S5,5,
and X = Km(E×E), and p is a prime with 109 < p <2000 for which E
has good reduction, then X(Q) is dense in X(Qp).
The proof of Theorem 4.2 will be given at the end of section 4.7.