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

In recent work, Bhargava and Shankar have shown that the average size of the $2$-Selmer group of an elliptic curve over $\mathbb{Q}$ 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.


Introduction and statement of results
In recent work [BS10], Bhargava and Shankar showed that when all elliptic curves over Q are ordered by height, the average size of the 2-Selmer group is equal to 3. Similar work by Bhargava and Ho [BH12] shows that the average size is six 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 three. 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, the average size is no longer bounded.
Given an elliptic curve E/Q with a rational isogeny φ : E → E ′ of degree p, one can associate to E a finite p-group called the φ-Selmer group, which we denote by Sel φ (E/Q) (see Section 2 for the definition). Similarly, one can also associate to the dual isogenŷ φ : E ′ → E the p-group Selφ(E ′ /Q). The Tamagawa ratio is defined to be 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 E A,B : y 2 = x 3 + Ax 2 + Bx 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 The second author is supported by an NSF Mathematical Sciences Postdoctoral Fellowship.
1 Let E(X) := {(A, B) ∈ Z 2 : |A|, B 2 ≤ X, A 2 − 4B = 0, and, if p 4 | B, then p 2 ∤ A} be the set of A and B in a box for which the model E A,B is minimal. Our main theorem is that, as we vary over elements of E(X), t(A, B) becomes normally distributed.
This theorem has a nice consequence for the distribution of 2-Selmer ranks of the elliptic curves E A,B , owing to the fact that |Sel φ (E A,B /Q)| is essentially a lower bound for |Sel 2 (E A,B /Q)|. As remarked above, for the family of all elliptic curves over Q, Bhargava and Shankar [BS10] 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 [BH12] have shown that the average size of the 2-Selmer group is exactly 6. In contrast to these results, Theorem 1.1 implies the following corollary.
r:2-selmer Corollary 1.2. For any integer r ≥ 0, we have that In particular, the average size of Sel 2 (E A,B /Q) is unbounded.
Remark. Of course, Corollary 1.2 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 forthcoming work, Kane and the first author, using different techniques, show that the average size of Sel φ (E A,B /Q) for E A,B ∈ E(X) is ≍ √ log X, from which it follows that the average size of Sel 2 (E A,B /Q) is ≫ √ log X.
In recent work [KLO13], the authors considered the analogous problem in the family of quadratic twists and proved the analogue of Theorem 1.1. 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 [Bil74] to show that these two pieces are independently and normally distributed, from which Theorem 1.1 follows. In forthcoming work [KLO14], 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 sec:selmer
We begin by briefly recalling the definition of the φ-Selmer group of E. If E(Q) has a point P of order two, then there is a two-isogeny φ : E → E ′ between E and E ′ with kernel C = P . We have a short exact sequence of G Q modules which gives rise to a long exact sequence of cohomology groups

The map δ is given by
This sequence remains exact when we replace Q by its completion Q v at any place v, which gives rise to the following commutative diagram.
The isogeny φ on E gives gives rise to a dual isogenyφ on . Exchanging the roles of (E, C, φ) and (E ′ , C ′ ,φ) in the above defines theφ-Selmer group, Selφ(E ′ /Q), as a subgroup of H 1 (Q, C ′ ). The groups Sel φ (E/Q) and Selφ(E ′ /Q) are finite dimensional F 2 -vector spaces and their ranks are related to that of the 2-Selmer group Sel 2 (E/Q) via the following theorem.
gss Theorem 2.1. The φ-Selmer group, theφ-Selmer group, and the 2-Selmer group sit inside the exact sequence Proof. This is a well known diagram chase. See Lemma 2 in [FG08] for example.

Tamagawa Ratios awa ratios
Our methods take advantage of a natural duality which exists between the groups Sel φ (E/Q) and Selφ(E/Q). This global duality is a consequence of a local duality between the distinguished local conditions H 1 φ (Q, C) and H 1 φ (Q, C ′ ) which is established in the following two lemmas.
Proof. This is a well-known result. See Remark X.4.7 in [Sil09] for example. Proof. Orthogonality is equation (7.15) and the immediately preceding comment in [Cas65]. Counting dimensions of the terms in (3.1) shows that H 1 φ (Q v , C) and H 1 φ (Q v , C ′ ) are not only orthogonal, but are in fact orthogonal complements.
Global duality motivates the following definition.
Definition 3.1. The ratio What is important for our application is that the Tamagawa ratio can be computed using a local product formula. prodform2 Theorem 3.3 (Cassels). The Tamagawa ratio T (E/E ′ ) is given by Proof. This is a combination of Theorem This next Lemma gives an easy formula for computing H 1 φ (Q p , C) for p = 2.

Local Conditions
conditions If E is an elliptic curve with a single point of order two, then E is given by a model of the form y 2 = x 3 + Ax 2 + Bx, where the point (0, 0) has order two. If we insist that we don't have both p 2 | A and p 4 | B 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. Proof. This follows easily from Tate's algorithm. See Section IV.9 in [Sil94], for example.
Proposition 4.1 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 4.2. Suppose that E has multiplicative reduction at p different from 2. Then Proof. It is easy to check that E and E ′ have Kodaira types I n and I n ′ respectively, where n = ord p (A 2 − 4B) + 2ord p B and n ′ = 2ord p (A 2 − 4B) + ord p B. The equality on the right is then immidiate from Tate's algorithm combined with Proposition 4.1.(iii). The equality on the left is Lemma 3.4.

The distribution of the Tamagawa ratio stribution
Recall from Theorem 3.3 that the Tamagawa ratio T (E/E ′ ) can be expressed as a product of local factors, one for each place of bad reduction. For the elliptic curve E A,B : y 2 = x 3 + Ax 2 + Bx with a two-torsion point, we can therefore express t(A, B) = ord 2 T (E/E ′ ) as a sum over such places, which we can further split as where t mult (A, B) is the contribution from the primes of multiplicative reduction, t add (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 4.1 shows that the probability that a given prime p is of multiplicative reduction is 2/p + O(1/p 2 ) and the probability it is of additive reduction is O(1/p 2 ). (Though it is likely clear that these are roughly the correct probabilities, Lemma 5.1 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. Let q be a squarefree integer, and let δ(q; (a, b)) = p|q δ(p; (a, b)). We then have that 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 p 6 classes (mod p 4 ), each of which is occupied by elements of E(X). On the other hand, if (a, b) ≡ (0, 0) (mod p), there will be p 2 lifts (mod p 4 ) which are not occupied. Thus, a class (a, b) (mod q), with q squarefree, can be lifted (mod q 4 ) 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 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 We note that the Dirichlet series L(s, g q ) satisfies L(s, g q ) = p∤q 1 − p −2−4s and L(1, g q ) = ζ(6) −1 p|q 1 − p −6 −1 , so that Similarly, we also find that Summing over lifts (a ′ , b ′ ), the result follows.
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1. We proceed via the method of moments, adapting an approach due to Billingsley [Bil74] to prove the classical Erdős-Kac theorem. We first note that the set of (A, B) ∈ E(X) for which A 2 − 4B is a square is O(X), and so, in view of the fact that #E(X) ∼ 4X 3/2 /ζ(6), such (A, B) will have no contribution to the limiting distribution. We therefore assume in the sequel that A 2 − 4B is not a square, which amounts to assuming that (0, 0) is the only non-trivial two-torsion point on E A,B /Q.
where the implied constant may to be taken to be 1. Let T = ǫ √ log log X, and consider the error term. There are O(X 3/2 /p 2 ) pairs (A, B) ∈ E(X) with either p 2 | A 2 − 4B or p 2 | B, whence there are O(X 3/2 /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(X 3/2 /T ) pairs (A, B) for which t add (A, B) > T . We will now show that g 1 (A, B) and g 2 (A, B) are asymptotically independent and normally distributed, each with mean and variance log log X, from which Theorem 1.1 therefore follows.
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 5.1, we think of D p and D ′ p as modeling the events p | B and p | A 2 − 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 log z. We will show that the (k 1 , k 2 )-mixed moment of g 1 (A, B) and g 2 (A, B) agrees with the (k 1 , k 2 )-mixed moment of D(z) and D ′ (z), and since mixed moments determine the multinormal distribution, the result will follow. First, let g 1 (A, B; z) and g 2 (A, B; z) be defined by For any integers k 1 , k 2 ≥ 0, set z = X 1/7(k 1 +k 2 ) . Using Lemma 5.1, we compute that 1 #E(X) where P (p; q) is the density of (A, B) ∈ E(X) for which each p i | A 2 − 4B and each q j | B.
Thus, g 1 (A, B; z) and g 2 (A, B; z) have the same moments as D(z) and D ′ (z). Finally, for i = 1, 2, we see that (g 1 (A, B; z) − µ(z)) k 1 (g 2 (A, B; z) − µ(z)) k 2 , Thus, the mixed moments of g 1 (A, B) and g 2 (A, B) converge to those of D(z) and D ′ (z), and the result is proved.