Open Access

The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point

Research in the Mathematical Sciences20141:15

DOI: 10.1186/s40687-014-0015-4

Received: 31 July 2014

Accepted: 9 October 2014

Published: 2 December 2014

Abstract

In recent work, Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over is exactly 3, and Bhargava and Ho have shown that the average size of the 2-Selmer group in the family of elliptic curves with a marked point is exactly 6. In contrast to these results, we show that the average size of the 2-Selmer group in the family of elliptic curves with a two-torsion point is unbounded. In particular, the existence of a two-torsion point implies the existence of rational isogeny. A fundamental quantity attached to a pair of isogenous curves is the Tamagawa ratio, which measures the relative sizes of the Selmer groups associated to the isogeny and its dual. Building on previous work in which we considered the Tamagawa ratio in quadratic twist families, we show that, in the family of all elliptic curves with a two-torsion point, the Tamagawa ratio is essentially governed by a normal distribution with mean zero and growing variance.

Background

In recent work [1], Bhargava and Shankar showed that when all elliptic curves over are ordered by height, the average size of the 2-Selmer group is equal to 3. Similar work in a preprint of Bhargava and Ho shows that the average size is 6 when the average is taken over all elliptic curves with a marked point. This result has the same flavor as that of Bhargava and Shankar, in that, after discounting for the known contribution of the marked point, the average size is 3. Here, we consider the related case where the marked point is of order two. Unlike the case of the generic marked point (which is almost always of infinite order) considered by Bhargava and Ho, the existence of this point affects the average size of the 2-Selmer group in an essential way - in particular, we show that the average size is no longer bounded.

Results

Given an elliptic curve E / with a rational isogeny ϕ:EE of degree p, one can associate to E a finite p-group called the ϕ-Selmer group, which we denote by Sel ϕ ( E / ) . Similarly, one can also associate to the dual isogeny ϕ ̂ : E E the p-group Sel ϕ ̂ ( E / ) . The Tamagawa ratio is defined to be
T ( E / E ) : = | Sel ϕ ( E / K ) | | Sel ϕ ̂ ( E / K ) | .

In this work, we consider the distribution of T ( E / E ) as E ranges over the set of elliptic curves with a rational two-torsion point.

Let EA,B:y2=x3+A x2+B x denote a generic such curve, and let ϕ : E A , B E A , B be the degree two isogeny corresponding to the rational subgroup generated by the point (0,0). We are interested in the distribution of the (logarithmic) Tamagawa ratio
t ( A , B ) : = ord 2 T ( E A , B / E A , B ) .

Let E ( X ) : = { ( A , B ) 2 : | A | , B 2 X , A 2 4 B 0 , and, for all p , if p 4 B , then , p 2 A } be the set of A and B in a box for which the model EA,B is minimal. Our main theorem is that, as we vary over elements of E ( X ) , t(A,B) becomes normally distributed.

Theorem 1.

As X, the set { t ( A , B ) : ( A , B ) E ( X ) } becomes normally distributed with mean 0 and variance 2 log logX. That is, for any z , we have that
lim X 1 # E ( X ) # { ( A , B ) E ( X ) : t ( A , B ) z 2 log log X } = 1 2 π z e t 2 / 2 dt.

Remark.

Lemma 10 below shows that # E ( X ) 4 X 3 / 2 / ζ ( 6 ) .

This theorem has a nice consequence for the distribution of 2-Selmer ranks of the elliptic curves EA,B, owing to the fact that | Sel ϕ ( E A , B / ) | is essentially a lower bound for | Sel 2 ( E A , B / ) | . As remarked above, for the family of all elliptic curves over , Bhargava and Shankar [1] have shown that average size of the 2-Selmer group is exactly 3, and for the family of curves with a marked point, but where that point is not required to be torsion, Bhargava and Ho have shown that the average size of the 2-Selmer group is exactly 6. In contrast to these results, Theorem 1 implies the following corollary.

Corollary 2.

For any integer r≥0, we have that
liminf X 1 # E ( X ) # { ( A , B ) E ( X ) : dim 𝔽 2 ( Sel 2 ( E A , B / ) ) r } 1 2 .

In particular, the average size of Sel 2 ( E A , B / ) is unbounded.

Remark.

Of course, Corollary 2 contradicts neither Bhargava and Shankar’s result nor Bhargava and Ho’s, as the set of elliptic curves with a two-torsion point is of density zero in either family.

Remark.

In the forthcoming work, Kane and the first author, using different techniques, show that the average size of Sel ϕ ( E A , B / ) for E A , B E ( X ) is log X , from which it follows that the average size of Sel 2 ( E A , B / ) is log X .

In recent work [2], the authors considered the analogous problem in the family of quadratic twists and proved the analog of Theorem 1. The key insight in that case is that the Tamagawa ratio is essentially an additive function, which could be studied by proving a variant of the classical Erdős-Kac theorem. For the family under consideration in this paper, the Tamagawa ratio is no longer an additive function. However, it can be decomposed into two pieces which are individually additive. We adapt the proof of the Erdős-Kac theorem due to Billingsley [3] to show that these two pieces are independently and normally distributed, from which Theorem 1 follows. In the forthcoming work, we consider in greater generality these joint Erdős-Kac style theorems and we apply them to the study of simultaneous twists of elliptic curves.

Selmer groups

We begin by briefly recalling the definition of the ϕ-Selmer group of E.

If E ( ) has a point P of order two, then there is a two-isogeny ϕ:EE between E and E with kernel C=〈P〉. We have a short exact sequence of G modules
0 C E ( ¯ ) ϕ E ( ¯ ) 0
(1)
which gives rise to a long exact sequence of cohomology groups
0 C E ( ) ϕ E ( ) δ H 1 ( , C ) H 1 ( , E ) H 1 ( , E )

The map δ is given by δ(Q)(σ)=σ(R)−R where R is any point on E ( ¯ ) with ϕ(R)=Q.

This sequence remains exact when we replace by its completion v at any place v, which gives rise to the following commutative diagram.

We define a distinguished local subgroup H ϕ 1 ( v , C ) H 1 ( v , C ) as the image
δ v E ( v ) / ϕ ( E ( v ) ) H 1 ( v , C )
for each place v of and we define the ϕ -Selmer group of E / , denoted Sel ϕ ( E / ) , by
Sel ϕ ( E / ) = ker H 1 ( , C ) Res v v of H 1 ( v , E [ 2 ] ) / H ϕ 1 ( v , C ) .

The isogeny ϕ on E gives rise to a dual isogeny ϕ ̂ on E with kernel C=ϕ(E[ 2]). Exchanging the roles of (E,C,ϕ) and ( E , C , ϕ ̂ ) in the above defines the ϕ ̂ -Selmer group, Sel ϕ ̂ ( E / ) , as a subgroup of H 1 ( , C ) . The groups Sel ϕ ( E / ) and Sel ϕ ̂ ( E / ) are finite dimensional 𝔽 2 -vector spaces and their ranks are related to that of the 2-Selmer group Sel 2 ( E / ) via the following theorem.

Theorem 3.

The ϕ-Selmer group, the ϕ ̂ -Selmer group, and the 2-Selmer group sit inside the exact sequence
0 E ( ) [ 2 ] / ϕ ( E ( ) [ 2 ] ) Sel ϕ ( E / ) Sel 2 ( E / ) ϕ Sel ϕ ̂ ( E / ) .
(2)

Proof.

This is a well-known diagram chase. See Lemma 2 in [4], for example.

Tamagawa ratios

Our methods take advantage of a natural duality which exists between the groups Sel ϕ ( E / ) and Sel ϕ ̂ ( E / ) . This global duality is a consequence of a local duality between the distinguished local conditions H ϕ 1 ( , C ) and H ϕ ̂ 1 ( , C ) which is established in the following two lemmas.

Lemma 4.

The sequence
0 C / ϕ E ( v ) [ 2 ] δ v H ϕ 1 ( v , C ) H [ 2 ] 1 ( v , E [ 2 ] ) ϕ H ϕ ̂ 1 ( v , C ) 0
(3)

is exact.

Proof.

This is a well-known result. See Remark X.4.7 in [5], for example.

Lemma 5(Local Duality).

For each place v of , there is a local Tate pairing H 1 ( v , C ) × H 1 ( v , C ) { ± 1 } induced by a pairing [,]:C×C→{±1} given by [ Q , R ~ ] = Q , R , where 〈Q,R〉 is the Weil pairing and R is any pre-image of R ~ under ϕ. The subgroups defining the local conditions H ϕ 1 ( v , C ) and H ϕ ̂ 1 ( v , C ) are orthogonal complements under this pairing.

Proof.

Orthogonality is Equation 7.15 and the immediately preceding comment in [6]. Counting dimensions of the terms in (3) shows that H ϕ 1 ( v , C ) and H ϕ ̂ 1 ( v , C ) are not only orthogonal, but are in fact orthogonal complements.

Global duality motivates the following definition.

Definition 1.

The ratio
T ( E / E ) = Sel ϕ ( E / ) Sel ϕ ̂ ( E / )

is called the Tamagawa ratio of E.

What is important for our application is that the Tamagawa ratio can be computed using a local product formula.

Theorem 6(Cassels).

The Tamagawa ratio T ( E / E ) is given by
T ( E / E ) = v of H ϕ 1 ( v , C ) 2 .

Proof.

This is a combination of Theorem 1.1 and Equations 1.22 and 3.4 in [6]. Alternatively, this follows from combining Lemma 5 with Theorem 2 in [7].

Remark.

The product in Theorem 6 converges because H ϕ 1 ( p , C ) 2 = 1 for primes p different from 2 where E has good reduction. More generally, because H 1 ( v , C ) v × / ( v × ) 2 , H ϕ 1 ( v , C ) 8 for all places v of .

This next Lemma gives an easy formula for computing H ϕ 1 ( p , C ) for p≠2.

Lemma 7.

For p different from 2, H ϕ 1 ( p , C ) = c p c p , where c p and c p are the Tamagawa fudge factors at p for E and E, respectively.

Proof.

This is a combination of Lemmas 4.2.(2) and 4.3 in [8].

Local conditions

If E is an elliptic curve with a single point of order two, then E is given by a model of the form y2=x3+A x2+B x, where the point (0,0) has order two. If we insist that we do not have both p2A and p4B for any prime p, then E has a unique model of this form, and this model will be minimal except possibly at 2.

Given such a model, we can easily read off the reduction type of E at any prime p≠2.

Proposition 8.

Let p be a prime different from 2.
  1. (i)

    E has good reduction at E if pB(A 2−4B).

     
  2. (ii)

    E has additive reduction at E if pB and pA 2−4B.

     
  3. (iii)

    E has multiplicative reduction at p if p divides exactly one of A 2−4B and B. If pA 2−4B, then this reduction is split if and only if 2 AB p = 1 ; if pB, then this reduction is split if and only if B p=1.

     

Proof.

This follows easily from Tate’s algorithm. See Section IV.9 in [9], for example.

Proposition 8 tells us that for a given prime p, the probability that a curve E has multiplicative reduction at p is 2 p + O ( 1 p 2 ) and the probability E has additive reduction at p is O ( 1 p 2 ) . This leads us to expect that the dominant contribution towards T ( E / E ) will come from primes of multiplicative reduction and we therefore compute the contribution at such places.

Proposition 9.

Suppose that E has multiplicative reduction at p different from 2. Then
| H ϕ 1 ( p , C ) | = c p c p = 4 if ord p ( A 2 4 B ) is odd or ( 2 AB p ) = 1 1 if ord p B is odd or ( B p ) = 1 2 otherwise

Proof.

It is easy to check that E and E have Kodaira types I n and I n respectively, where n=ord p (A2−4B)+2ord p B and n=2ord p (A2−4B)+ord p B. The equality on the right is then immediate from Tate’s algorithm combined with Proposition 8 (iii). The equality on the left is Lemma 7.

The distribution of the Tamagawa ratio

Recall from Theorem 6 that the Tamagawa ratio T ( E / E ) can be expressed as a product of local factors,
T ( E / E ) = v 2 Δ∞ T v ( E / E ) ,
one for each place of bad reduction. For the elliptic curve EA,B:y2=x3+A x2+B x with a two-torsion point, we can therefore express t ( A , B ) = ord 2 T ( E / E ) as a sum over such places,
t ( A , B ) = v 2 Δ A , B t v ( A , B ) ,
which we can further split as
t ( A , B ) = t mult ( A , B ) + t add ( A , B ) + O ( 1 ) ,

where tmult(A,B) is the contribution from the primes of multiplicative reduction, tadd(A,B) is the contribution from the primes of additive reduction, and the O(1) term comes from the places 2 and . As observed earlier, Proposition 8 shows that the probability that a given prime p is of multiplicative reduction is 2/p+O(1/p2) and the probability it is of additive reduction is O(1/p2) (Though it is likely clear that these are roughly the correct probabilities, Lemma 10 below makes this precise.). We therefore expect that the primes of additive reduction will have a finite contribution to the distribution of the Tamagawa ratio, owing to the convergence of 1 / p 2 , whereas the primes of multiplicative reduction will not. Before establishing this, we make our intuition on probabilities precise.

Lemma 10.

For each prime p and for any integers a and b, let
δ ( p ; ( a , b ) ) : = p 4 p 6 1 if p a or p b , and p 4 1 p 6 1 if p a and p b.
Let q be a square-free integer, and let δ ( q ; ( a , b ) ) = p q δ ( p ; ( a , b ) ) . We then have that
# { ( A , B ) E ( X ) : ( A , B ) ( a , b ) ( mod q ) } = δ ( q ; ( a , b ) ) · 4 X 3 / 2 ζ ( 6 ) + O ( q 2 X + q 6 X 3 / 8 ) ,

where ζ(s) is the Riemann zeta function.

Proof.

For each prime p, consider the class (a,b) (mod p). If (a,b)(0,0) (mod p), then it lifts to p6 classes (mod p4), each of which is occupied by elements of E ( X ) . On the other hand, if (a,b)≡(0,0) (mod p), there will be p2 lifts (mod p4) which are not occupied. Thus, a class (a,b) (mod q), with q square-free, can be lifted (mod q4) in exactly
p q p a or p b p 6 p q p a and p b ( p 6 p 2 )
ways that will occur in E ( X ) . Let (a,b) be such a lift. We then have that
( A , B ) E ( X ) : ( A , B ) ( a , b ) ( mod q 4 ) 1 = B 2 X B b ( mod q 4 ) | A | X A a ( mod q 4 ) p 2 A if p 4 B 1 = B 2 X B b ( mod q 4 ) 2 X q 4 p 4 B , p q 1 1 p 2 + O p 4 B , p q p 2 = : 2 X q 4 B 2 X B b ( mod q 4 ) f q ( B ) + O B 2 X B b ( mod q 4 ) p 4 B , p q p 2 ,
say, where f q (B) is multiplicative. Let g q :=f q μ, so that f q ( B ) = d B g q ( d ) ; note that g q (d)=0 if (d,q)>1. The summation in the main term is thus
B 2 X B b ( mod q 4 ) f q ( B ) = d X 1 / 2 ( d , q ) = 1 g q ( d ) | B | X 1 / 2 / d B b d 1 ( mod q 4 ) 1 = 2 X 1 / 2 q 4 d X 1 / 2 ( d , q ) = 1 g q ( d ) d + O d X 1 / 2 | g q ( d ) | .
We note that the Dirichlet series L(s,g q ) satisfies
L ( s , g q ) = p q 1 p 2 4 s and L ( 1 , g q ) = ζ ( 6 ) 1 p q 1 p 6 1 ,
so that
B 2 X B b ( mod q 4 ) f q ( B ) = 2 X 1 / 2 q 4 ζ ( 6 ) p q 1 p 6 1 + O ( 1 ) .
Similarly, we also find that
B 2 X B b ( mod q 4 ) p 4 B , p q p 2 X 1 / 2 q 4 + X 3 / 8 ,
whence
( A , B ) E ( X ) : ( A , B ) ( a , b ) ( mod q 4 ) 1 = 4 X 3 / 2 q 8 ζ ( 6 ) p q 1 p 6 1 + O X q 4 + X 3 / 8 .

Summing over lifts (a,b), the result follows.

We are now ready to prove Theorem 1.

Proof of Theorem 1.

We proceed via the method of moments, adapting an approach due to Billingsley [3] to prove the classical Erdős-Kac theorem.

We first note that the set of ( A , B ) E ( X ) for which A2−4B is a square is O(X), and so, in view of the fact that # E ( X ) 4 X 3 / 2 / ζ ( 6 ) , such (A,B) will have no contribution to the limiting distribution. We therefore assume in the sequel that A2−4B is not a square, which amounts to assuming that (0,0) is the only non-trivial two-torsion point on E A , B / .

Consider the functions
g 1 ( A , B ) : = p A 2 4 B 1 and g 2 ( A , B ) : = p B 1 ,
and note that
t ( A , B ) = g 1 ( A , B ) g 2 ( A , B ) + t add ( A , B ) + O p 2 A 2 4 B or p 2 B 1 ,

where the implied constant may to be taken to be 1. Let T = ε log log X and consider the error term. There are O(X3/2/p2) pairs ( A , B ) E ( X ) with either p2A2−4B or p2B, whence there are O(X3/2/T) pairs satisfying these divisibility conditions for some prime p>T. For the remaining full-density subset, the contribution from the sum is manifestly ≤T. Similarly, there are O(X3/2/T) pairs (A,B) for which tadd(A,B)>T. We will now show that g1(A,B) and g2(A,B) are asymptotically independent and normally distributed, each with mean and variance log logX, from which Theorem 1 therefore follows.

For any prime p, let ρ(p)=(p5−1)/(p6−1). A simple calculation with Lemma 10 reveals that ρ(p) is both the probability that ( A , B ) E ( X ) satisfies pA2−4B and the probability that pB. We therefore expect that g1(A,B) and g2(A,B) should be normal with mean μ(X) and variance σ2(X) both given by
μ ( X ) , σ 2 ( X ) = p < X ρ ( p ) = log log X + O ( 1 ) .
Let z=X δ for some δ>0. For each odd prime p<z, denote by D p and D p random variables which are 1 with probability ρ(p) and 0 with probability 1−ρ(p), and are such that
Prob ( D p = 1 and D p = 1 ) = p 4 1 p 6 1 .
In view of Lemma 10, we think of D p and D p as modeling the events pB and pA2−4B. If we set
D ( z ) : = p < z D p and D ( z ) : = p < z D p ,

the multidimensional central limit theorem (with Lindeberg’s criterion, say) implies that, as z, D(z) and D(z) become independent and normally distributed with mean and variance each log logz. We will show that the (k1,k2)-mixed moment of g1(A,B) and g2(A,B) agrees with the (k1,k2)-mixed moment of D(z) and D(z), and since mixed moments determine the multinormal distribution, the result will follow.

First, let g1(A,B;z) and g2(A,B;z) be defined by
g 1 ( A , B ; z ) : = p A 2 4 B p < z 1 and g 2 ( A , B ; z ) : = p B p < z 1 .
For any integers k1,k2≥0, set z = X 1 / 7 ( k 1 + k 2 ) . Using Lemma 10, we compute that
1 # E ( X ) ( A , B ) E ( X ) g 1 ( A , B ; z ) k 1 g 2 ( A , B ; z ) k 2 = p 1 , , p k 1 < z q 1 , , q k 2 < z prime 1 # E ( X ) ( A , B ) E ( X ) : p i A 2 4 B i q j B j 1 = p 1 , , p k 1 < z q 1 , , q k 2 < z prime P ( p ; q ) + O ( X 1 / 14 )
where P(p;q) is the density of ( A , B ) E ( X ) for which each p i A2−4B and each q j B. We also observe that
𝔼 D ( z ) k 1 D ( z ) k 2 = p 1 , , p k 1 < z q 1 , , q k 2 < z prime P ( p ; q )
by the construction of D p , D p . We therefore have, letting μ(z)= log logz, that
1 # E ( X ) ( A , B ) E ( X ) g 1 ( A , B ; z ) μ ( z ) k 1 g 2 ( A , B ; z ) μ ( z ) k 2 = j 1 = 0 k 1 j 2 = 0 k 2 ( μ ( z ) ) j 1 + j 2 k 1 j 1 k 2 j 2 · · 1 # E ( X ) ( A , B ) E ( X ) g 1 ( A , B ; z ) k 1 j 1 g 2 ( A , B ; z ) k 2 j 2 = j 1 = 0 k 1 j 2 = 0 k 2 ( μ ( z ) ) j 1 + j 2 k 1 j 1 k 2 j 2 𝔼 D ( z ) k 1 j 1 D ( z ) k 2 j 2 + O ( X 1 / 14 ) = 𝔼 ( D ( z ) μ ( z ) ) k 1 ( D ( z ) μ ( z ) ) k 2 + O ( X 1 / 14 ) .
Thus, g1(A,B;z) and g2(A,B;z) have the same moments as D(z) and D(z). Finally, for i=1,2, we see that
g i ( A , B ) μ ( X ) = g i ( A , B ; z ) μ ( z ) + O ( 1 ) ,
so that, by the binomial theorem and the Cauchy-Schwarz inequality,
( A , B ) E ( X ) g 1 ( A , B ) μ ( X ) k 1 g 2 ( A , B ) μ ( X ) k 2 = ( A , B ) E ( X ) g 1 ( A , B ; z ) μ ( z ) k 1 g 2 ( A , B ; z ) μ ( z ) k 2 + O μ ( X ) k 1 + k 2 1 2 ,

Thus, the mixed moments of g1(A,B) and g2(A,B) converge to those of D(z) and D(z), and the result is proved.

Declarations

Acknowledgements

The second author was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. Both authors read and approved the final manuscript.

Authors’ Affiliations

(1)
Center for Communications Research
(2)
Department of Mathematics, Stanford University

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Copyright

© Klagsbrun and Lemke Oliver; licensee Springer. 2014

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