A note on the order of the Tate--Shafarevich group modulo squares
Alexandros Konstantinou


On the order of the Tate–Shafarevich group modulo squares]A note on the order of the Tate–Shafarevich group modulo squares

University College London, London WC1H 0AY, UK
alexandros.konstantinou.16@ucl.ac.uk
[2010]11G10 (11G40)

We investigate the order of the Tate–Shafarevich group of abelian varieties modulo rational squares. Our main result shows that every square-free natural number appears as the non square-free part of the Tate–Shafarevich group of some abelian variety, thereby validating a conjecture of W. Stein. 

[
    Alexandros Konstantinou
    April 29, 2024
===========================




§ INTRODUCTION

In the following note, we study the order of the Tate–Shafarevich group (A/K) of an abelian variety A defined over a number field K. Its non-zero elements correspond to principal homogeneous spaces for A/K with a rational point over all completions of K though lacking a point in K itself. Despite its central role in the study of rational points, several foundational questions regarding (A/K) remain open. These include the Tate–Shafarevich conjecture which stipulates its finiteness. In addition, a conjecture of Stein focuses on the possible values that the order of the Tate–Shafarevich group can take (if finite) modulo rational squares. In the following note, we validate Stein's conjecture, which can be phrased as follows.

For every positive square-free integer n, there exists an abelian variety A/ℚ with finite Tate–Shafarevich group of order nm^2 for some integer m ≥ 1.



 §.§ Previous work
 In the case of principally polarised abelian varieties, the Cassels–Tate pairing, 
    ⟨,⟩: (A/K) ×(A/K) →ℚ/ℤ,
 lays down significant constraints on the structure of the Tate–Shafarevich group. In particular, for elliptic curves, this is known to be alternating, and therefore the order of the Tate–Shafarevich group of an elliptic curve must be a square, if finite. 

The work of Flach <cit.> establishes the (slightly weaker) skew symmetric condition ⟨ x, y ⟩ = - ⟨ y , x⟩ for principally polarised abelian varieties. From this, one can only deduce that the Tate–Shafarevich group is either a square or twice a square. Building upon this, Poonen and Stoll present a criterion for deducing if the Tate–Shafarevich group (provided it's finite) is of square order purely based on the vanishing of ⟨ c, c ⟩ for an element c ∈(A/K)[2] canonically associated to A. They then use this criterion to present the first example of an abelian variety for which the order of its Tate–Shafarevich group is twice a square.



 Subsequently, Stein <cit.> presents the first examples of abelian varieties defined over ℚ in which the order of their Tate–Shafarevich groups satisfy |(A/ℚ)  | = p m^2 for p < 25000 an odd prime. His construction relies on the existence of an abelian variety defined over ℚ which becomes isomorphic to a product E^p-1 of elliptic curves over a cyclic extension. 
Following this, Keil's work <cit.> presents examples of abelian surfaces, given as quotients of a product of two elliptic curves, for which |  | = p m^2 for p ≤ 13 an odd prime. 

 


 §.§ Acknowledgements
 I would like to extend my thanks to Vladimir Dokchitser for his guidance and many constructive discussions during the course of this work. In addition, I would like to thank Adam Morgan for many helpful suggestions on a draft version of this paper and for many useful discussions that came before that. 



§ EXPRESSION FOR  MOD SQUARES
 
Our proof relies on breaking down the Weil restriction of scalars of an abelian variety up to isogeny followed by an application of a formula of Cassels–Tate. This allows us to express the size of the Tate–Shafarevich group modulo squares in terms of Birch–Swinnerton-Dyer constants.



 §.§ Cassels–Tate formula
 In the following subsection, we briefly recall a result of Cassels and Tate which proves invariance of the Birch–Swinnerton-Dyer conjecture under isogeny. We first fix some notation. 
 We let K be a number field. Given a K-rational isogeny ϕ: A_1 → A_2, we write 

    Q(ϕ) =   |(ϕ: A_1(K)/A_1(K)_→ A_2(K)/A_2(K)_)| 
    ×   |(ϕ: (A_1/K)_→(A_2/K)_)|,

where _ denotes the divisible part of . 

For an abelian variety A/K with a fixed non-zero global exterior form ω, we denote the Birch–Swinnerton-Dyer periods by 
    Ω_ℂ(A,ω)=2^A∫_A(ℂ) |ω∧ω |,      Ω_ℝ(A,ω)=∫_A(ℝ) | ω |.

We write Ω(A/K,ω) = ∏_v |∞Ω_K_v(A,ω) where K_v = ℝ (resp. ℂ) if v is real (resp. complex). In addition, for a non-archimedean place v, we write w_A,v^ for a Néron minimal exterior form on A, c_v(A/K) for its Tamagawa number and C(A/K,ω)=∏_v ∤∞c_v(A/K)|ω/ω_A,v^ |_v. We write _0 = /_ and □ to denote the square of a rational number.


The following theorem is a version of the Cassels–Tate formula as found in <cit.>.
 
Let ϕ:A_1 → A_2 be a K-rational isogeny. Then, for any non-zero global exterior forms ω_1 and ω_2 on A_1 and A_2,
9.5pt9pt

    Q(ϕ^∨)/Q(ϕ)=|A_2(K)_ |/|A_1(K)_ ||A_2^∨(K)_ |/|A_1^∨(K)_ |C(A_1/K,ω_1)/C(A_2/K,ω_2)Ω(A_1/K,ω_1)/Ω(A_2/K,ω_2)∏_p |(ϕ) | _0(A_1/K)[p^∞] |/ | _0(A_2/K)[p^∞] |.



Given a field extension L/K, we write ϕ_L for the base change of ϕ to L. The following proposition shows that for carefully chosen L and working modulo squares, the Cassels–Tate formula simplifies.
 
Let L/K be a quadratic extension such that all bad primes of A_1/K and A_2/K and all places v |(ϕ) split in L. In addition, suppose that (ϕ) is a square. Then,

    Q(ϕ_L^∨)/Q(ϕ_L)  = |A_2(L)_ |/|A_1(L)_ ||A_2^∨(L)_ |/|A_1^∨(L)_ |∏_p |(ϕ) | _0(A_1/K)[p^∞] |/ | _0(A_2/K)[p^∞] |□.



We note that C(A_1/L,ω_1) Ω(A_1/L,ω_1) is independent of ω_1. By our assumptions on L/K, the only terms in this product we have to consider are complex places of L which lie above real places of K and non-archimedean places w ∤(ϕ). As in the proof of <cit.>, when w is complex, we have  Ω_ℂ(A_1,ϕ^*ω_2)/Ω_ℂ(A_2,ω_2) = (ϕ) = □. For a non-archimedean place w ∤(ϕ), |(ϕ^*ω_2/ω_A_1,w^)/(ω_2/ω_A_2,w^) |_w =1 (see <cit.>). To conclude, C(A_1/L,ω_1)/C(A_2/L,ω_2)Ω(A_1/L,ω_1)/Ω(A_2/L,ω_2)=□ giving the required result. 



 §.§ A decomposition for the Weil-restriction of scalars
  In what follows, we let A/K be a principally polarised abelian variety defined over a number field K. We let p be an odd prime, F/K a cyclic extension of degree p and  _K^FA the restriction of scalars from F to K. For brevity, we write B=_K^FA. This is an F/K-twist of A^p, and the product polarisation on A^p descends to a principal polarisation on B/K. Then, there are K-morphisms 

    i: A → B,      :B → A.

These are the inclusion and the trace map respectively. 

After identifying the base change B_F=A^p as principally polarised abelian varieties, the base changed morphisms i_F and _F coincide with the diagonal inclusion and the summation map respectively. In addition, writing i^∨: B → A for the dual of i (having suppressed the canonical principal polarisations on both sides), we have i^∨= Finally, we write R to denote the kernel of the trace map. This is an abelian variety defined over K and is an F/K-twist of A^p-1 (see <cit.>).


  Let ϕ: R × A → B be given by (x,y) ↦ x + i(y). Then, ϕ is a K-rational isogeny with  (ϕ) ≅ A[p].


It suffices to show (ϕ) ≅ A[p]. After base change to K, we identify R_K = A^p-1 and B_K=A^p. Then, on K-points ϕ is given by (x_1,…,x_p-1,y) ↦ (y+x_1,…,y+x_p-1,y-∑_i=1^p-1x_i) . Thus, restricting the projection R × A → A to (ϕ) allows us to deduce that (ϕ) ≅ A[p].




 §.§ An expression for (R/L) modulo squares

In this subsection, we relate  | (R/L)[p^∞]| □ to the torsion subgroup of R and the rank of A/L.

  Let L/K be a number field extension such that L⊗_KF is a field. Then, the following hold.


  * For all odd primes q ≠ p, | _0(R/L)[q^∞] | = □.

  * If, in addition, A is an elliptic curve, | _0(R/L)[2^∞] | = □.



By Lemma <ref>,  (ϕ)=p^2(A). When q ≠ p, this isogeny induces an isomorphism _0(A/L)[q^∞] ×_0(R/L)[q^∞] ∼→_0(B/L)[q^∞]. Since A and B are principally polarised, <cit.> asserts that _0(A/L)[q^∞] and _0(B/L)[q^∞] are both of square order for all odd q. This proves (1). Further, by <cit.>, B_L ≅_L^L⊗_KFA, and by combining <cit.> with Shapiro's Lemma, _0(B/L)[2^∞] ≅_0(A/L⊗_KF)[2^∞]. Therefore, if A is an elliptic curve, both _0(A/L)[2^∞] and _0(B/L)[2^∞] are of square order by <cit.>. Claim (2) follows.

In what follows, we write _p(A/K) for the ℤ_p-corank of the p^∞-Selmer group _p^∞(A/K):= _n ≥ 1_p^n(A/K). This coincides with the Mordell–Weil rank of A plus the multiplicity of ℚ_p/ℤ_p in its Tate–Shafarevich group.
  Let L/K be a quadratic extension such that all bad primes of A/K and all primes dividing either (ϕ) or the relative discriminant Δ(F/K) split in L. Then, the following hold.


  * Up to rational squares, 

    | _0(R/L)[p^∞] | ≡Q(ϕ_L)/Q(ϕ^∨_L) | R(L)[p^∞]|/ | R^∨(L)[p^∞]|□.


  * If, in addition, _p(A/L)= _p(A/L⊗_KF), then

    | _0(R/L)[p^∞] | ≡ p^_p(A/L) | R(L)[p^∞]|/ | R^∨(L)[p^∞]|□.




Let v be a place of bad reduction for B/K. By <cit.>, this can only happen when A/K has bad reduction at v or when v |Δ(F/K). This shows that all bad primes of B/K split in L.
Since A/K is principally polarised, |A(L)[p^∞]|= | A^∨(L)[p^∞]  | and |_0(A/L)[p^∞]|= | _0(A^∨/L)[p^∞]  |. The same holds for B. Part (1) follows by combining  Lemma <ref> and Lemma <ref>. For the second claim, it suffices to show that if _p(A/L)= _p(A/L⊗_KF), then Q(ϕ_L)/Q(ϕ^∨_L) =p^_p(A/L). After suppressing the principal polarisations A=A^∨, B=B^∨, we view ϕ^∨ as a morphism B → A × R^∨. Therefore, 
Q(ϕ_L)/Q(ϕ^∨_L)≡Q(ϕ^∨_L∘ϕ_L: A × R → A × R^∨) □. Since the degree of ϕ^∨∘ϕ is a power of p, it induces an isomorphism (A× R/L)_[q^∞] ∼→(A×R^∨/L)_[q^∞] for q ≠ p. From this, we deduce (ϕ_L^∨∘ϕ_L  |  _) = (ϕ_L^∨∘ϕ_L  |  _[p^∞]). Since B_L ≅_L^L⊗_KFA (see <cit.>), then by combining <cit.> with Shapiro's Lemma gives _p^∞(B/L) ≅_p^∞(A/L⊗_KF). By assumption, _p(A/L)=_p(A/L⊗_K F), from which we deduce _p(R/L)=_p(R^∨/L)=0. As a result, Q(ϕ^∨_L∘ϕ_L) coincides with Q(i_L^∨∘ i_L), where we view i_L^∨∘ i_L as an endomorphism of A_L. Using the description of i and i^∨ from <ref>, i^∨∘ i=[p]_A. This gives the required result. 





§  OF NON-SQUARE ORDER


We retain all the notation from <ref>. In this section,  we restrict to the case when A=E is an elliptic curve, and so R is an abelian variety of dimension p-1. In addition, we fix the following notation.

 We write K_∞ to denote the cyclotomic ℤ_p-extension of ℚ, and K_n=ℚ(ζ_p^n + ζ_p^n^-1) for the n^ layer of this extension. For d a positive, square-free integer, we let L_n= K_n ·ℚ(√(-d)). 

We first show that for n sufficiently large, (R/L_n) is finite for any choice of d. Further, having carefully selected d, we show that |(R/L_n)| ≡ p □. We begin by proving a lemma involving torsion on R and R^∨.
 
Let E be an elliptic curve defined over ℚ. Then, for any n ≥ 0 and any prime p ≥ 11 both R(L_n)[p^∞] and R^∨(L_n)[p^∞] are trivial.


As  abelian varieties over L_n+1, both R and R^∨ are isomorphic to  E^p-1. It therefore suffices to show that E(L_n+1)[p^∞] is trivial for p ≥ 11. Let E_-d be the quadratic twist of E by -d. Then, there exists a map E(L_n+1) → E(K_n+1) × E_-d(K_n+1) whose kernel and cokernel are 2-groups. Then, for any odd prime p, this induces an isomorphism  
    E(L_n+1)[p^∞] ∼→ E(K_n+1)[p^∞] × E_-d(K_n+1)[p^∞].
 By <cit.>,  E(K_∞)[p^∞]=E(ℚ)[p^∞] (similarly for E_-d) for any p≥ 5. If p≥ 11, Mazur's theorem <cit.> asserts that E(ℚ)[p^∞] and E_-d(ℚ)[p^∞] are both trivial. Therefore, E(L_n+1)[p^∞] is trivial.

 
The following result is a formal consequence of a theorem by Kato and Rohrlich. 


Let E be an elliptic curve defined over ℚ. Then, there exists a positive integer n such that


  * _s=1L(E/L_n,s) = _s=1L(E/L_n+1,s),

  * (R/L_n) is finite, 

  * _p(E/L_n+1) = _p(E/L_n).



As E is defined over ℚ, _K_n^L_nE is isogenous to E × E_-d giving an equality of L-functions

    L(E/L_n,s) = L(E/K_n,s)L(E_-d/K_n,s),

By Rohrlich <cit.>, the sequence {_s=1L(E/K_n,s)}_n ≥ 1 stabilises (similarly for E_-d), and so _s=1L(E/L_n,s) =_s=1L(E/L_n+1,s) for n sufficiently large. This proves (1). 
Part (2) is a consequence of a result by Kato <cit.>. Following the notation introduced there, for G an abelian group, τ∈G, and a G-module M, we let M^(τ) :={ x ∈ M : I_τ· x =0} where I_τ⊆ℤ[G] denotes the kernel of the map ℤ[G] →ℂ^× induced by τ. We apply this to G=(L_n+1/ℚ) ≅ℤ/2p^n+1ℤ. For brevity, we write H= (L_n+1/L_n)≅ℤ/pℤ. Further, the composition i ∘ from <ref> is equal to N_H:= ∑_h ∈ H h viewed as a K-endomorphism of B.  It follows that the composition R ↪ B N_H→ B is the zero map, which in turn gives a homomorphism (R/L_n) →(N_H  |  (E/L_n+1)). We claim this has finite kernel. To see this, we look at the short exact sequence of abelian varieties 

    0 → R → B   E → 0.
 Taking Galois cohomology gives an exact sequence

    0 → E(L_n)/_L_n^L_n+1E(L_n+1) → H^1(L_n,R) →  H^1(L_n,B).

By the Mordell–Weil theorem E(L_n)/pE(L_n) is finite, from which we deduce that E(L_n)/_L_n^L_n+1E(L_n+1) must be finite. Therefore, the induced map (R/L_n) →(N_H  |  (E/L_n+1)) has finite kernel. For a primitive character τ∈G, we have N_H ∈ I_τ. Therefore, for such τ, (N_H  |  (E/L_n+1))⊆(E/L_n+1)^(τ). By Artin formalism,
  
    L(E/L_n+1,s) = L(E/L_n,s)∏_χ∈H
    χ≠L(E/L_n, χ,s).
  
We let n be as in part (1) and χ a non-trivial character of H. Then, L(E/L_n, χ,s)=L(E/ℚ, _H^Gχ,1) ≠ 0. Therefore, L(E/ℚ, τ,1) ≠ 0 for all primitive characters τ∈G. Then, by <cit.>, the τ-part (_n+1)^(τ) is finite for all primitive τ∈G. This completes the proof of (2). For (3), it suffices to show that R(L_n) is finite. This follows in the same way as (2).  
 
 
 
Let p ≥ 11 be a prime. Then, there exists an abelian extension L/ℚ and an abelian variety R/L such that | (R/L)  | = p ·□. 


We let E/ℚ be an elliptic curve defined over ℚ and d a positive integer such that p and all bad primes of E/ℚ split in ℚ(√(-d)).
For the remainder of this proof, we fix n as in Theorem <ref> so that _p(E/L_n+1)=_p(E/L_n) and (R/L_n) is finite. In view of Lemma <ref>(1)-(2), we deduce that | (R/L_n)  | ≡ | _0(R/L_n)[p^∞]  |. Combining this with Proposition <ref>(2) (applied with K=K_n, F=K_n+1,L=L_n) and Lemma <ref>, we get

    | (R/L_n) | ≡ p^_p(E/L_n)□. †

Since E is defined over ℚ, its global root number ω(E/ℚ(√(-d)))=-1. This is because the only contribution in the root number computation comes from the unique archimedean place of ℚ(√(-d)). Since _ℚ^ℚ(√(-d)) E is isogenous to E × E_-d, we deduce that
10.5pt9pt

    (-1)^_p(E/ℚ(√(-d))) = (-1)^_p(E/ℚ) (-1)^_p(E_-d/ℚ)= w(E/ℚ)w(E_-d/ℚ).

In view of the isogeny, w(E/ℚ(√(-d)))= w(E/ℚ)w(E_-d/ℚ), from which we deduce that _p(E/ℚ(√(-d)))  ≡ 1  2. Since the parity of p^∞-Selmer ranks remains unchanged in odd degree Galois extensions <cit.>, then _p(E/L_n) ≡_p(E/ℚ(√(-d))) ≡ 1  2. Combining this with (<ref>) gives the required result. 


 Let p be a rational prime. Then, there exists an abelian variety R'/ℚ such that | (R'/ℚ) |=p·□.


For p≥ 11, we consider R' = ^L_ℚR where R is the abelian variety from Theorem <ref>. Since (R'/ℚ) ≅(R/L) (combine Shapiro's lemma with <cit.>), the result follows by Theorem <ref>. For p≤ 7, see <cit.> and <cit.>.


 
For every positive square-free integer n, there exists an abelian variety defined over ℚ with finite Tate–Shafarevich group of order n ·□.
 

Given abelian varieties A_1, A_2, (A_1× A_2) ≅(A_1) ×(A_2). The result follows from Theorem <ref>.


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