Bounds of the rank of the Mordell-Weil group of jacobians of hyperelliptic curves
Harris B. Daniels Álvaro Lozano-Robledo Erik Wallace


















Rank bounds for jacobians]Bounds of the rank of the Mordell–Weil group of Jacobians of Hyperelliptic Curves


Department of Mathematics and Statistics, Amherst College, Amherst, MA 01002, USA
hdaniels@amherst.edu
<http://www3.amherst.edu/ hdaniels/>


Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
alvaro.lozano-robledo@uconn.edu
<http://alozano.clas.uconn.edu/>


Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
erik.wallace@uconn.edu


Primary: 11G10, Secondary: 14K15.


In this article we extend work of Shanks and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves C: y^2=f(x) defined over , with f(x) of degree p, where p is a Sophie Germain prime, such that the rank of the Mordell–Weil group of the jacobian J/ℚ of C is bounded by the genus of C and the 2-rank of the class group of the (cyclic) field defined by f(x), and exhibit examples where this bound is sharp.

[
    Erik Wallace
    
================



§ INTRODUCTION


Let C/ be a hyperelliptic curve, given by a model y^2=f(x), with f(x)∈[x], and let J/ be the jacobian variety attached to C. The Mordell–Weil theorem shows that J() is a finitely generated abelian group and, therefore,

    J()≅ J()_tors⊕^R_J/,

where J()_tors is the (finite) subgroup of torsion elements, and R_J/=rank_(J())≥ 0 is the rank of J(). In this article we are interested in bounds of R_J/ in terms of invariants of C or f(x). 

In <cit.>, Washington showed the following bound for the rank of certain elliptic curves, building on work of Shanks on the so-called simplest cubic fields (see <cit.>).


Let m≥ 0 be an integer such that  m^2+3m+9 is square-free. Let E_m be the elliptic curve given by the Weierstrass equation

    E_m y^2 = f_m(x)= x^3+mx^2-(m+3)x+1.

Let L_m be the number field generated by a root of f_m(x), let Cl(L_m) be its class group, and let (L_m)[2] be the 2-torsion subgroup of (L_m). Then,

    rank_(E_m()) ≤ 1 + dim__2(Cl(L_m)[2]).



In this article, we extend Washington's result to curves of genus g≥ 2. In order to find other families of hyperelliptic curves of genus g≥ 2 where a similar bound applies, we use a method of 2-descent for jacobians described by Cassels, Poonen, Schaefer, and Stoll (see Section <ref>; in particular, we follow the implementation described in <cit.>). The examples come from hyperelliptic curves y^2=f(x) such that f(x) defines the maximal real subfield of a cyclotomic extension of , and the degree of f(x) is p, a Sophie Germain prime. We obtain the following result.


Let q≥ 7 be a prime such that p=(q-1)/2 is also prime, and let L=(ζ_q)^+ be the maximal totally real subfield of (ζ_q). Let f(x) be the minimal polynomial of ζ_q+ζ_q^-1 or -(ζ_q+ζ_q^-1), let C/ be the hyperelliptic curve y^2=f(x), of genus g=(p-1)/2, and let J/ be its jacobian. Then, there are constants ρ_∞ and j_∞, that depend on q, such that  

    _ (J())≤__2^(2)(,J)  ≤ρ_∞ + j_∞ + __2 (^+(L)[2]),

where ρ_∞+j_∞≤ p-1. Further, if one of the following conditions is satisfied, 


  * the Davis–Taussky conjecture holds (Conjecture <ref>), or

  * the prime 2 is inert in the extension (ζ_p)^+/, or

  * q≤ 92459,
 then ρ_∞=0 and j_∞=g=(p-1)/2, and __2^(2)(,J) ≤  g + __2((L)[2]).


In fact, if the Davis–Taussky conjecture holds (see Remark <ref>), then the bound of Theorem <ref> becomes __2^(2)(,J) ≤  g.

The organization of the paper is as follows. In Section <ref>, we review the method of 2-descent as implemented by Stoll in <cit.>. In Sections <ref>, <ref>, and <ref>, we specialize the 2-descent method for the situations we encounter in the rest of the paper, namely the case when f(x) defines a totally real extensions, or cyclic extension of , of prime degree. In Section <ref>, we give a new proof of Washington's theorem using the method of 2-descent. In Section <ref> we provide examples of hyperelliptic curves of genus g=(p-1)/2 where p is a Sophie Germain prime, and prove Theorem <ref>. Finally, in Section <ref>, we illustrate the previous sections with examples of curves, jacobians, and how their ranks compare to the bounds.


The authors would like to thank Keith Conrad, Gürkan Dogan, Franz Lemmermeyer, Paul Pollack, and Barry Smith, for several helpful comments and suggestions. We would also like to express our gratitude to David Dummit for very useful suggestions and noticing an error in an earlier version of the paper. Finally we would like to express our thanks to the referees who have given us helpful suggestions and pointed out some errors in previous versions of this paper.




§ STOLL'S IMPLEMENTATION OF 2-DESCENT

In this section we summarize the method of 2-descent as implemented by Stoll in <cit.>. The method was first described by Cassels <cit.>, and by Schaefer <cit.>, and Poonen-Schaefer <cit.> in more generality. Throughout the rest of this section we will focus on computing the dimension of the 2-Selmer group of the jacobian J of a hyperelliptic curve C, given by an affine equation of the form

    C:y^2=f(x),

where f∈[x] is square-free and (f) is odd (Stoll also treats the case when (f) is even, but we do not need it for our purposes). In this case, the curve C is of genus g=((f) - 1)/2 with a single point at infinity in the projective closure. Before we can compute the dimension of the 2-Selmer group, we must define a few objects of interest and examine some of their properties. We will follow the notation laid out in <cit.>.

Let ^(2)(,J) be the 2-Selmer group of J over , and let (,J)[2] be the 2-torsion of the Tate-Shafarevich group of J (as defined, for instance, in Section 1 of <cit.>). Selmer and Sha fit in the following fundamental short exact sequence:

    0[r]    J()/2J()[r]   ^(2)(,J)[r]   (,J)[2][r]    0.

With this sequence in hand we get a relationship between for the rank of J() and the _2-dimensions of the other groups that we defined.

    _ J() + __2 J()[2] + __2(,J)[2] = __2^(2)(,J).

Using equation (<ref>), we get our first upper bound on the rank

    _ J()≤__2^(2)(,J)-__2(,J)[2]≤__2^(2)(,J).

This upper bound is computable, in the sense that J()[2] and the Selmer group are computable, as we describe below.


For any field extension K of  and f∈[x], let L_K = K[T] / (f(T)) denote the algebra defined by f and let N_K denote the norm map from L_K down to K.


We denote L_K=K[θ], where θ is the image of T under the reduction map K[T]→ K[T]/(f(T)), and L_K is a product of finite extensions of K:

    L_K=L_K,1×⋯× L_K,m_K,

where m_K is the number of irreducible factors of f(x) in K[x]. Here, the fields L_K,j correspond to the irreducible factors of f(x) in K[x], and the map N_K:L_K→ K is just the product of the norms on each component of L_K. That is, if α=(α_1,α_2,…,α_m_K), then N_K(α)=∏_i=1^m_KN_L_K,i/K(α_i) where N_L_K,i/K:L_K,i→ K is the usual norm map for the extension of fields L_K,i/K.

In order to ease notation, we establish the following notational conventions: when K= we will drop the field from the subscripts altogether, and if K=_v, we will just use the subscript v. This convention will apply to anything that has a field as a subscript throughout the rest of the paper. As an example, L_v=_v[T]/(f(T)) and L=[T]/(f(T)).

Following standard notational conventions, we let _K, I(K), and (K) denote the ring of integers of K, the group of fractional ideals in K, and the ideal class group of K, respectively. We  define analogous objects for the algebra L_K as products of each component, as follows:

    _L_K   =_L_K,1×⋯×_L_K,m_K,
    
    I(L_K)   =I(L_K,1)×⋯× I(L_K,m_K),
    (L_K)   =(L_K,1)×⋯×(L_K,m_K).


 
Let K be a field extension of , and let L=[T]/(f(T)) be as before.
 

  * Let I_v(L) denote the subgroup of I(L) generated, in each component, by fractional ideals in L_,i with support above a prime v in . For a finite set S of places of , let

    I_S(L)=∏_v∈ S∖{∞} I_v(L).



  * For any field extension K of , let 

    H_K = ( N_K L_K^×/(L_K^×)^2→ K^×/(K^×)^2 ).

For any place v of , we let _v H→ H_v be the canonical restriction map induced by the natural inclusion of fields ↪_v.

  * Let (C) denote the group of degree-zero divisors on C with support disjoint from the principal divisor ÷(y).




In our case, the curve is given by C:y^2=f(x), and the support of ÷(y) is exactly the  points with coordinates (α,0), where α is a root of f, and the unique point at infinity. 

Now for each K, there is a homomorphism  

    F_K(C)(K)→ L^×_K,   ∑_P n_PP↦∏_P (x(P) - θ )^n_P,

and this homomorphism induces a homomorphism δ_K J(K) → H_K with kernel 2J(K) by <cit.>. By abusing notation, we also use δ_K to denote the induced map J(K)/2J(K) → H_K. 

All of these facts, together with some category theory, give us the following characterization of the 2-Selmer group of J over . 
 
The 2-Selmer group of J over  can be identified as follows:

    ^(2)(,J) = {ξ∈ H |_v(ξ) ∈δ_v(J(_v)) for all  v}.



In order to take advantage of this description of the Selmer group, we need some additional facts about the 2-torsion of J and the δ_K maps.

 Let K be a field extension of .


  (1) For a point P∈ C(K) not in the support of ÷(y), δ_K(P-∞) = x(P) - θ (L_K^×)^2.

  (2) Let f = f_1⋯ f_m_K be the factorization of f over K into monic irreducible factors. Then, to every factor f_j, we can associate an element P_j∈ J(K)[2] such that:


  (i) The points {P_j} generate J(K)[2] and satisfy ∑_j=1^m_KP_j = 0.

  (ii) Let h_j be the polynomial such that f = f_jh_j. Then 

    δ_K(P_j) = (-1)^(f_j)f_j(θ) + (-1)^(h_j)h_j(θ)  (L_K^×)^2.



  (3) J(K)[2] = m_K -1.





Let I_v = (N I_v(L)/I_v(L)^2 → I()/I()^2) and let _v H_v→ I_v be the map induced by the valuations on each component of L_v. Considering all primes at once, we get another map H → I(L)/I(L)^2. More specifically, the  map is the product of _v(_v) over all places v. 


Next, the following lemma helps us compute the dimensions of various groups when K is a local field or .


Let K be a v-adic local field, and let d_K=[K:_2] if v=2 and d_K = 0 if v is odd. Then


  (1) J(K)/2J(K) =  J(K)[2] + d_Kg =m_K-1+d_Kg.

  (2) H_K = 2 J(K)/2J(K) = 2(m_K-1+d_Kg).

  (3) I_K = m_K - 1.





With all of this machinery the description of ^(2)(,J) given in Proposition <ref> can be refined as follows.


Let S = {∞,2}∪{v : v^2  divides (f) }. Then 

    ^(2)(,J) = {ξ∈ H |(ξ)∈ I_S(L)/I_S(L)^2, _v(ξ)∈δ_v(J(_v)) for all  v∈ S}.


This new characterization suggests the following method to compute ^(2)(,J):



  S1: Find the set S.

  S2: For each v∈ S, determine J_v = δ_v(J(_v))⊆ H_v.

  S3: Find a basis for a suitable finite subgroup H⊆ L^×/(L^×)^2 such that ^(2)(,J)⊆H.

  S4: Compute ^(2)(,J) as the inverse image of ∏_v∈ S J_v under 
    ∏_v∈ S_vH→∏_v∈ S H_v.



Ignoring any complications that arise from computing and factoring the discriminant of f, we focus on steps 2 and 3. We omit the details of how to carry out step 4, since we are only interested in an upper bound for the _2-dimension of ^(2)(J,). Step 2 can be broken down into three substeps:


  S2.1: For all v∈ S∖{∞}, compute J_v = δ_v(J(_v)) and its image G_v = _v(J_v) in I_v.

  S2.2: If G_v = 0 for some v, with v odd, remove v from S. 

  S2.3: Compute J_∞. 


To complete step 2.3, we need the following lemma.

 With notation as above:


  (1) J()/2J() = m_∞-1 - g.

  (2) J_∞ is generated by {δ_∞(P-∞) : P∈ C()}.

  (3) The value of δ_∞(P-∞) only depends on the connected component of C() containing P. 



Next, for step 3, we see that if we let 

    G = ∏_v∈ S∖{∞} G_v ⊆ I(L)/I(L)^2,
 then the group 
{ξ∈ H : (ξ)∈ G}
contains ^(2)(,J). In fact, the larger group H = {ξ∈ L^×/(L^×)^2 :  (ξ)∈ G } also contains the 2-Selmer group and we can compute its basis using the following two steps.


  S3.1: Find a basis of V = ( L^×/(L^×)^2 → I(L)/I(L)^2).

  S3.2: Enlarge this basis to get a basis of H = ^-1(G). 

With notation as above, Stoll deduces an upper bound and a formula for the _2-dimension of the 2-Selmer group (see Lemma 4.10 and the discussion under Step 4), as follows.
 With notation as above,

    __2^(2)(,J)≤ (m_∞-1) + __2((L)[2]) + __2(G→(L)/2(L)).



In the next section, we modify the proof of the bound in Proposition <ref> to allow for extra conditions at infinity, before we specialize to totally real, and  cyclic extensions.



 §.§ About the proof of Proposition <ref>



The following commutative diagrams helps understand where the Selmer group fits:

    J()/2J()@^(->[rrr]^δ[d]           H[rrr]^[d]^∏_v res_v         I(L)/I(L)^2 [d] 
    ∏_v J(_v)/2J(_v)@^(->[rrr]^∏_v δ_v         ∏_v H_v[rrr]^∏_v _v         ∏_v I_v(L)/I_v(L)^2

The 2-Selmer group of J over  is then given, as in Prop. <ref>, by

    ^(2)(,J) = {ξ∈ H : (ξ)∈ I_S(L)/I_S(L)^2, _v(ξ)∈δ_v(J(_v)) for all  v∈ S}.

The Selmer group is thus contained in H⊆ L^×/(L^×)^2, and more precisely,

    ^(2)(,J)⊆{ξ∈ H : (ξ) ∈ G, res_∞(ξ) ∈ J_∞}

where J_∞=δ_∞(J()), the group G is the product ∏_v∈ S∖{∞} G_v ⊆ I(L)/I(L)^2, and recall that H is the kernel of the norm map from L^×/(L^×)^2 down to ^×/(^×)^2. Thus, ^(2)(,J) is contained in the larger group

    H= {ξ∈ L^×/(L^×)^2 : (ξ) ∈ G, res_∞(ξ) ∈ J_∞}.

We emphasize here that the definition of H in <cit.> does not impose a condition at ∞, but the definition of H does to improve the bounds accuracy (thus H⊆H). In an attempt to simplify notation, let L_J_∞ be the subspace of L^×/(L^×)^2 with a condition added at infinite primes by L_J_∞ = L^×/(L^×)^2 ∩_∞^-1(J_∞). Thus, H={ξ∈ L_J_∞ : val(ξ) ∈ G},
and ^(2)(,J)⊆H. 
 Note that H is the largest subgroup of L_J_∞ such that 
	
    (H)≅  G∩val(L_J_∞).

Let us show that indeed (H)≅  G∩val(L_J_∞). Indeed:


  * Suppose ξ∈H and consider val(ξ). By definition, since  ξ∈H, we have that val(ξ) is in G, and ξ∈ L_J_∞, thus val(ξ)∈val(L_J_∞). Hence, (ξ)∈ G∩val(L_J_∞).

  * Conversely, suppose g ∈ G∩val(L_J_∞). Then, there is some ξ∈ L_J_∞ such that val(ξ)=g. In particular, res_∞(ξ) ∈ J_∞ and since val(ξ)=g∈ G, it follows that ξ∈H. Hence, val(ξ)∈val(H).

Also, let us show that H is the largest subgroup of L_J_∞ such that 
(H)≅  G∩val(L_J_∞). Suppose that ξ∈ L_J_∞ and val(ξ) ∈ G∩val(L_J_∞). Then, ξ∈ L_J_∞ and val(ξ)∈ G, so ξ∈H by definition.



Next, we define subspaces V and W of L^×/(L^×)^2 as follows:


  * Let {ξ_i}_i=1^r be generators of G∩val(L_J_∞) with (G∩val(L_J_∞))=r, and for each 1≤ i≤ r pick one μ_i∈ L_J_∞ such that (μ_i)=ξ_i. Let W be the subspace generated by {μ_i}_i=1^r. Note that W⊆ L_J_∞⊆ L^×/(L^×)^2. In particular, _∞(w)∈ J_∞ for all w∈ W. Moreover, W and (W) are isomorphic by construction, so

    W≅(W) = G∩val(L_J_∞) ⊆ G∩val(L^×/(L^×)^2)=(G→(L)/2(L)).

Thus, r=(W)=(G∩val(L_J_∞))≤(G→(L)/2(L)).


  * Next, let us write V=( L_J_∞→ I(L)/I(L)^2). It follows that H = V ⊕ W (note that (V) is trivial, while (w) is non-trivial for every w≠ 0 in W).



 Let V=( L_J_∞→ I(L)/I(L)^2), let U=(_L^×/(_L^×)^2)∩ L_J_∞, and let (L_J_∞) be the class group defined as follows (L_J_∞)=I(L)/P(L_J_∞), where I(L) is the group of fractional ideals of L, and P(L_J_∞) is the group of principal fractional ideals 𝔄=(α) with a generator such that _∞(α)∈ J_∞.  Then, there is an exact sequence:
 
    0 ↦ U  → V →(L_J_∞)[2] → 0.

 

Consider 

    L^×∩_∞^-1(J_∞) [rr]^2[d]       L^×∩_∞^-1(J_∞)[rr][d]        L_J_∞[rr][d]^      0  
    
    0 [r]     I(L)[rr]^2        I(L)[rr]       I(L)/I(L)^2[rr]        0

and apply the snake lemma.



Let P(L) be the subgroup of all principal fractional ideals, let P^+(L) be the subgroup of principal ideals generated by a totally positive element, and let P(L_J_∞) be as above. Since the trivial signature (1,1,…,1)∈ J_∞, it implies that P^+(L) ⊆ P(L_J_∞) ⊆ P(L), and therefore there are surjections

    (L) ↞(L_J_∞) ↞^+(L),

where ^+(L)=I(L)/P^+(L) is the narrow class group of L.




Putting all this together (and writing  for __2), we obtain a bound

    ^(2)(,J)   ≤(H) =  U + (L_J_∞)[2] +  W 
        = (_L^×/(_L^×)^2∩_∞^-1(J_∞)) + (L_J_∞)[2] +  G∩val(L_J_∞)
       ≤(_L^×/(_L^×)^2∩_∞^-1(J_∞))  + ^+(L)[2] +  G∩val(L_J_∞).

We note here that (_L^×/(_L^×)^2∩_∞^-1(J_∞))≤ m_∞ -1, where we have used the fact that (_L^×/(_L^×)^2)=m_∞, and the fact that _∞(-1) is not in J_∞ (because J_∞=δ_∞(J())⊆ H_∞, which is the kernel of the norm map, so N(j)=1 for j∈ J_∞, but the norm N(-1)=-1 because the degree of L is odd). We will improve on the bound given by (<ref>) above by making certain assumptions about G and a more careful analysis of the dimension of the subgroup of totally positive units. Before we state our refinements, we review some of the results on totally positive units that we shall need.

It is worth pointing out that the third line of Eq. (<ref>) is not necessarily an improvement over the bound in Prop. <ref> if ^+(L)[2]> (L)[2]. In our setting, we will seek conditions where ^+(L)[2]= (L)[2] and then our bound in Eq. (<ref>) will be an improvement due to the more careful counting of units according to their infinite valuations.



 §.§ Totally Positive Units

Let L be a totally real Galois number field of prime degree p>2, with embeddings τ_i : L →, for i=1,…,p, and maximal order _L. Let (L) = (_L) be the ideal class group of L, and let ^+(L) be the narrow class group. Let V_∞ = {± 1}^p≅ (_2)^p and, by abuse of notation, we extend the map _∞ (as in Definition <ref>, where we note that H_∞≅ V_∞ and H⊆ L^×/(L^×)^2) to

    _∞: L^×/(L^×)^2 → V_∞

by _∞(α) = ((τ_1(α)),(τ_2(α)),…,(τ_p(α))). Let _L^× be the unit group of _L, and let _L^×, + be the subgroup of totally positive units. Thus,

    (_∞|__L^×/(_L^×)^2) = _L^×,+/(_L^×)^2.

We refer the reader to <cit.> for heuristics and conjectures about the dimension of the totally positive units (in particular, the conjecture on page 4). In the following theorem, we use the notation of <cit.>.
 
Let ρ, ρ^+, and ρ_∞ be defined by

    ρ = __2(L)/2(L), ρ^+ = __2^+(L)/2^+(L),  and ρ_∞ = __2_L^×,+/(_L^×)^2

Then,
 

  * ρ_∞= p-__2_∞(_L^×/(_L^×)^2)=__2{± 1 }^p/_∞(_L^×/(_L^×)^2).

  * We have

    0 →{± 1 }^p/_∞(_L^×/(_L^×)^2) →^+(L) →(L)→ 0.

In particular, max{ρ,ρ_∞}≤ρ^+≤ρ_∞+ρ, and ρ^+=ρ_∞+ρ if and only if the exact sequence splits.

  * (Armitage-Fröhlich) ρ^+-ρ≤ (p-1)/2.



For part (1), note that ρ_∞ = __2_L^×,+/(_L^×)^2 is the dimension of (_∞|__L^×/(_L^×)^2), and the dimension of _L^×/(_L^×)^2 is p. Thus, the dimension of the image of _∞|__L^×/(_L^×)^2 is p minus the dimension of the kernel. 

For part (2), see Section 2 of <cit.>, and in particular Equation (2.9). Part (3) is shown in <cit.>, where it is shown that ρ^+-ρ≤⌊ r_1/2 ⌋, where r_1 is the number of real embeddings of L. Here r_1=p is an odd prime, so the proof is concluded.


From the statement of the previous theorem, we see that ρ^+≥ρ_∞. However, ρ≥ρ_∞ is not necessarily true. In the following result, a condition is given that implies ρ≥ρ_∞ (see also <cit.>, Section 3).

 Let L/ be a finite abelian extension with Galois group of odd exponent n, and suppose that -1 is congruent to a power of 2 modulo n. Then, in the notation of Theorem <ref>, we have ρ=ρ^+. In particular, ρ≥ρ_∞.


We obtain the following corollary.


Let L/ be a cyclic extension of odd prime degree p, and suppose that the order of 2 in (/p)^× is even. Then, ρ=ρ^+. In particular, __2(L)[2] = __2^+(L)[2].


Suppose that (L/)≅/p for some prime p>2, such that the order of 2 in (/p)^× is even (since (L/) is cyclic of order p, this is equivalent to -1 being congruent to a power of 2 modulo p). Hence, Theorem <ref> applies, and ρ=ρ^+. 


The odd primes below 100 such that the order of 2 is odd modulo p are 7, 
23, 31, 47, 71, 73, 79, and 89, so the corollary applies to all other primes not in this list (i.e., 3, 5, 11, 13, 17, 19, etc.). 


Let K/ be an (imaginary) abelian extension of the rationals of degree n, let h_K be the class number of K, and let C_K⊆_K^× the groups of circular units of K (as defined in <cit.>, p. 376), and let C_K^+ ⊆_K^×, + be the subgroup of circular units that are totally positive. Let K^+ be the maximal real subfield of K, and let h_K^+ be its class number. Let h_K^- = h_K/h_K^+. Further, assume that each prime p which ramifies in K does not split in K^+. Then:


  * The index [_K^×,+:(_K^×)^2] is a divisor of the index [C_K^+:C_K^2].

  * If the discriminant of K is plus or minus a power of a prime, then h_K^- is odd if and only if [C_L^+:C_L^2]=1, where C_L is the subgroup of circular units of L=K^+.

  * Suppose the discriminant of K is plus or minus a power of a prime, [K^+:]=n is a power of an odd prime p, and the order of 2 p is even. Then, h_K^- is odd if and only if h_K^+ is odd.


 The results are, respectively, Lemma 5, Theorem 3, and Theorem 4 of <cit.>.


Next, we cite a results of Estes which extends work  of Davis (<cit.>), and Stevenhagen (<cit.>). See also <cit.>.

 Let q and p  be primes such that q=2p+1. If 2 is inert in (ζ_p)^+, where ζ_p is a primitive p-th root of unity, then the class number of (ζ_q) is odd.


The following result combines the results of Davis, Estes, Stevenhagen, and Garbanati, and gives a specific criterion to check that ρ_∞=0 for the maximal real subfield of a cyclotomic field.

 Let q and p>2  be primes such that q=2p+1, and let  L=(ζ_q+ζ_q^-1), where ζ_q is a primitive q-th root of unity. Further, assume that  the prime 2 is inert in the extension (ζ_p)^+/. Then, ρ_∞ = __2_L^×,+/(_L^×)^2 = 0.


Let K=(ζ_q) and let K^+ = L = (ζ_q+ζ_q^-1). We note that the discriminant of K is a power of a prime (namely q), and therefore the primes of K or L that are ramified (namely the primes above q), are totally ramified, so they do not split. Moreover, p=(q-1)/2 is prime and [L:]=p. Thus, the hypotheses of Theorem <ref> are satisfied for K and L.
	
    If 2 is inert in (ζ_p)^+/, then Theorem <ref> shows that h_K is odd, and therefore h_K^- is odd as well, since h_K^- = h_K/h_K^+ by definition. Since the discriminant of K is a power of q, Theorem <ref> part (2) shows that [C_L^+:C_L^2]=1 and therefore [_L^×,+:(_L^×)^2]=1 as well by part (1). We conclude that ρ_∞=0.
 

There is in fact a conjecture of Davis and Taussky that says that ρ_∞=0 in the case of L=(ζ_q+ζ_q^-1), where p=(q-1)/2 is a Sophie Germain prime. For more on the Davis–Taussky conjecture see <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.


Let q and p  be primes such that q=2p+1, and let  L=(ζ_q+ζ_q^-1), where ζ_q is a primitive q-th root of unity. Then, C^+_L=C_L^2. (Thus, it follows that ρ_∞ = __2_L^×,+/(_L^×)^2 = 0.)


In the next result we note that the Davis–Taussky conjecture is equivalent to the class number of (ζ_q) being odd. (We thank David Dummit for pointing out the following equivalence to us.)


	Let q and p  be primes such that q=2p+1, let  L=(ζ_q+ζ_q^-1), where ζ_q is a primitive q-th root of unity, and let K=(ζ_q). Then, C^+_L=C_L^2 if and only if the class number of K is odd.


	If h^+_K is even, then h^-_K is even (see, for instance, <cit.>, pp. 773-774, for a proof of this fact). Since h_K=h_K^+h_K^-, it follows that h_K^- is odd if and only if h_K is odd. Since the discriminant of K is a power of q (prime), Theorem <ref>, part (2), implies that h^-_K is odd if and only if C^+_L = (C_L)^2. Hence, the Davis–Taussky conjecture holds if and only if h_K^- is odd, if and only if h_K is odd.


We conclude this section with some remarks about how to compute an upper bound of ρ_∞ in the cyclic case, working in coordinates over _2. Let L be a cyclic extension of  of degree p>2, and let (L/)=⟨σ⟩. Then, L is totally real (since L is Galois over , then it is totally real or totally imaginary, so [L:]=p>2 is odd, and p=2r_2 is impossible). Let u≠± 1 be a fixed (known) unit in _L^× and let 

    res_∞(u) = (ε_1,…,ε_p)

where τ_1,…,τ_p are the real embeddings of L and ε_i is the sign of τ_i(u)∈. We order our embeddings in the following manner. Let g_u(x) be the minimal polynomial of u over , and let r_1,…,r_p be the real roots of g_u(x) ordered so that r_1<r_2<⋯ <r_p. Then, {r_i}={τ_j(u)}, and we choose τ_i so that τ_i(u)=r_i for all 1≤ i≤ p. With this notation, res_∞(u) = (-1,-1,-1,…,1,1,1), i.e., it consists of a non-negative number of -1 signs followed by a non-negative number of +1 signs. Recall that u is in the kernel of res_∞ if and only if u is a totally positive unit.
	
    Attached to  the generator σ∈𝒢=(L/), there is a permutation ϕ=ϕ_σ∈ S_p, where ϕ is considered here as a permutation of {1,2,…,n}, such that τ_i(σ(u))=r_ϕ(i), and therefore
    
    res_∞(σ(u)) = (ε_ϕ(1), …, ε_ϕ(p)).

    Since τ is an embedding (injective), if τ_i(α)=r_i for some α∈ L^×, then τ_i(σ(α)) = r_ϕ(i). It follows that τ_i(σ^n(u)) = r_ϕ^n(i), and 
       
    res_∞(σ^n(u)) = (ε_ϕ^n(1), …, ε_ϕ^n(p)),

       for all n≥ 1.
       
Now, in addition, suppose that u is a unit of norm 1, and let 𝒢· u be the subgroup of units generated by the conjugates of u, i.e., 

    𝒢· u = ⟨ u, σ(u), σ^2(u),…, σ^p-1(u)⟩⊆_L^×
 generate a subgroup of  _L^×. Note that the product ∏_n=0^p-1σ^n(u)=1, so  
    𝒢· u = ⟨ u, σ(u), σ^2(u),…, σ^p-2(u)⟩.
 Then, 

    res_∞(𝒢· u) = ⟨ 
    (ε_ϕ^n(1), …, ε_ϕ^n(p)) : 0≤ n≤ p-2⟩⊆ V_∞,

where we have defined V_∞ = {± 1}^p. If we fix an isomorphism ψ:{± 1}≅_2, and write f_i = ψ(ε_i), then the map res_∞: 𝒢· u → V_∞ can be written in _2-coordinates, and the corresponding p× (p-1) matrix over _2 is given by

    M_∞,u = ( f_ϕ^j(i))_1≤ i≤ p
     0≤ j ≤ p-2 = ([        f_1     f_ϕ(1)          ⋯ f_ϕ^p-2(1);        f_2     f_ϕ(2)          ⋯ f_ϕ^p-2(2);          ⋮          ⋮          ⋱          ⋮;        f_p     f_ϕ(p)          ⋯ f_ϕ^p-2(p);            ]).


Let u be a unit of norm 1, and let d_∞,u be the dimension of the column space of M_∞,u or, equivalently, the dimension of res_∞(𝒢· u). Then, ρ_∞≤ (p-1) - d_∞,u. In particular, if d_∞,u = p-1, then ρ_∞ =0.
 

If u is of norm 1, then -1∉𝒢· u, because the norm of -1 is -1, and the norm of every element in 𝒢· u is 1. In particular, ⟨_∞(-1), _∞(𝒢· u)⟩ is a space of dimension 1+d_∞,u. Hence, the kernel of _∞ is at most of dimension p-(1+d_∞,u). It follows that ρ_∞≤ (p-1)-d_∞,u as desired.




 §.§ Totally positive units in cyclic extensions of prime degree


Let L be a cyclic extension of prime degree p>2, and let _L^× be the unit group of _L. Let _L^×, 1 be the units of norm 1, so that _L^×≅{± 1 }×_L^×,1. In this section we show the following result:

 Let p≥ 3 be a prime, let L be a cyclic extension of degree p, and suppose that the polynomial ϕ_p(x)=(x^p-1)/(x-1) is irreducible over _2. Then, either ρ_∞=0 (i.e., _L^×,+ = (_L^×)^2), or ρ_∞ = p-1 in which case every unit in _L^×,1 is totally positive.


If every unit in _L^×,1 is totally positive, then ρ_∞=p-1 since we would have

    ρ_∞ = __2_L^×,+/(_L^×)^2 = __2_L^×,1/(_L^×)^2= p-1.

Otherwise, there must be a unit u∈_L^×,1 that is not totally positive (in particular, u is not in ± 1 ·  (_L^×)^2). Let G=(L/)=⟨σ⟩≅/p, and let u_i = σ^i(u) for i=0,…,p-1, be the conjugates of u. Let τ be a fixed embedding of L in , let τ(u_i)=ε_i ∈{± 1 } for i=0,…,p-1, and order the embeddings τ=τ_0,…,τ_p-1 of L such that _∞(u)=(ε_0,…,ε_p-1). In other words, τ_i = τ∘σ^i. Thus, 

    _∞(σ(u)) = _∞(u_1) = (ε_1,…,ε_p-1,ε_0).


Consider the class u∈_L^×/(_L^×)^2 and its non-trivial signature _∞(u). The group ring _2[G] acts on the module M=_2[G]·u. Since G is of prime order and by assumption ϕ_p(x) is irreducible, it follows that M is irreducible. Furthermore, since u≠± 1 it follows that  _∞(u)≠ (1,1,…,1) or (-1,-1,…,-1). Hence, _∞(σ(u))≠_∞(u) by our formula above, and therefore M is not 1-dimensional.

Moreover, 

    _2[G]≅_2[x]/(x^p-1)≅_2[x]/(x-1) ⊕_2[x]/(ϕ_p(x)),

Since we are assuming that ϕ_p(x) is irreducible over _2, the only irreducible representations of G over _2 are the trivial (1-dimensional) representation, and a representation of dimension p-1. Since the irreducible _2[G]-module M is not 1-dimensional, it must be (p-1)-dimensional. Finally, we note that M ⊆_L^×,1/(_L^×)^2, and every unit class in M has non-trivial signature (the norm is 1 and there are p>2 signs, so both 1 and -1 appear in the signature). Since the dimension of all possible signatures in _L^×,1 is p-1, and M is (p-1)-dimensional, we conclude that all signatures occur, and therefore ρ_∞=0, as desired. 
 



We conclude this section quoting a conjectural density of cubic fields and quintic fields with maximal ρ_∞, which is part of a broader conjecture of Dummit and Voight (see <cit.>). 


Let p=3 (resp. p=5). As L varies over all totally real fields of degree p ordered by absolute discriminant, the density of such fields with ρ_∞=2 (resp. ρ_∞=4) is approximately 1.9% (resp. 0.000019%). 


 By Prop. <ref>, if L is a cyclic field of degree p=5, and ρ_∞≠ 4, then it must be 0. By Conjecture <ref>, approximately 99.999981% of all totally real quintic fields conjecturally have ρ_∞≠ 4. Thus, we expect  that cyclic quintic fields with ρ_∞=0 must be quite abundant. Note, however, that cyclic quintic fields are a subset of density 0 among all totally real quintic fields, so the conjectures of Dummit and Voight do not apply directly here.




 §.§ Refinements of the bound on the rank


Now we are ready to improve the bound in Proposition <ref>. We will continue using the notation of Section <ref>.

 Let p be an odd prime, let C: y^2=f(x) with f(x) of degree p (and genus g=(p-1)/2), such that L, the number field defined by f(x), is totally real of degree p, and let J/ be the jacobian of C/. Let ρ_∞ = __2_L^×,+/(_L^×)^2, and  let j_∞ = __2 (_∞(_L^×/(_L^×)^2)∩ J_∞). Then:

    ^(2)(,J) ≤ j_∞ +ρ_∞ + ^+(L)[2] +  G∩(L_J_∞),

In particular, 


  * ρ_∞+j_∞≤ p-1.

  * j_∞≤ J_∞ = (p-1)/2 = genus(C).

  * ^+(L)[2] ≤ρ_∞ + (L)[2]. In particular,

    ^(2)(,J) ≤ j_∞ +  2ρ_∞ + (L)[2] +  G∩(L_J_∞),


  * If G∩(L_J_∞) is trivial, then 
    ^(2)(,J) ≤ j_∞ + ρ_∞ +  ^+(L)[2]≤ j_∞ +2ρ_∞ +  (L)[2].




Let p be an odd prime, let C: y^2=f(x) with f(x) of degree p, such that L, the number field defined by f(x), is totally real of degree p, and let J/ be the jacobian of C/. Recall that in Section <ref> we showed

    ^(2)(,J)   ≤(H) =  U +  V +  W 
        = (_L^×/(_L^×)^2∩_∞^-1(J_∞)) + (L_J_∞)[2] +  G∩val(L_J_∞).

Clearly,

    (_L^×/(_L^×)^2∩_∞^-1(J_∞))    = _∞(_L^×/(_L^×)^2)∩ J_∞ +  (_∞|__L^×/(_L^×)^2)
       = j_∞ + _L^×,+/(_L^×)^2 = j_∞ + ρ_∞.

Now, for part (1), notice that (_L^×/(_L^×)^2∩_∞^-1(J_∞))⊆_L^×/(_L^×)^2 so the dimension as a _2-vector space is at most p. Moreover, _∞(-1) is not in J_∞ (because J_∞=δ_∞(J())⊆ H_∞, which is the kernel of the norm map, so N_^L(j)=1 for j∈ L such that _∞(j)∈ J_∞, but N_^L(-1)=-1 since the degree of L is odd). Thus, j_∞+ρ_∞ = (_L^×/(_L^×)^2∩_∞^-1(J_∞))≤ p-1, as claimed.

For part (2), recall that we have defined J_∞ = δ_∞(J()/2J())⊆ H_∞. By Lemma <ref>, and the fact that δ_∞ is injective (Lemma 4.1 in <cit.>), we have 

    (J_∞) = m_∞-1-g = p-1 - p-1/2 = p-1/2,

where we have used the fact that L is totally real to claim that m_∞=p.

Part (3) follows from Theorem <ref>, which shows that ρ^+≤ρ_∞ + ρ. And part (4) is immediate from (3), so the proof is complete.
 

Now we can put together Corollary <ref> and Proposition <ref> to give a bound in the cases when the multiplicative order of 2 p is even.

 Suppose L is a cyclic, totally real number field of degree p>2, such that the order of 2 in (/p)^× is even. Then, in the notation of Proposition <ref>, we have

    ^(2)(,J)    ≤ j_∞ + ρ_∞ + (L)[2] + (G→(L)/2(L))
       ≤ p-1 + (L)[2] + (G→(L)/2(L)).

Moreover, if ρ_∞=0, then 

    ^(2)(,J)    ≤ g + (L)[2] + (G→(L)/2(L))
        = p-1/2 + (L)[2] + (G→(L)/2(L)).



Suppose L is a cyclic, totally real number field of degree p>2, such that the order of 2 in (/p)^× is even. Then, Corollary <ref> implies that ρ=ρ^+. Thus, the bound follows from Proposition <ref>. Note that the bound p-1+ (L)[2] + (G→(L)/2(L)) is the bound that appears in Proposition <ref>.




§ GENUS 1


The goal of this section is to show an alternative proof of Theorem <ref>, using Stoll's implementation of the 2-descent algorithm and the results we showed in the previous section. Let m be an integer such that D=m^2+3m+9 is square-free. Let C be the (hyper)elliptic curve given by the Weierstrass equation

    C: y^2 = f_m(x)= x^3+mx^2-(m+3)x+1.

Since C is elliptic, the jacobian J is isomorphic to C, so we will identify C with J. We will conclude the bound stated in the theorem as a consequence of Theorem <ref>. Since the order of 2≡ -1  3 is 2, Corollary <ref> shows that ρ^+=ρ. In Section <ref> we discussed that cyclic cubic fields with ρ_∞≠ 0 are rare. Let us show that in fact _L^×,+ = (_L^×)^2, and therefore ρ_∞=0, for the cyclic cubic fields L=L_m defined by f_m(x)=0.

 Let m be an integer such that m≡ 3  9, and let L be the number field defined by f_m(x)= x^3+mx^2-(m+3)x+1=0. Then, ρ_∞=0.


Let α be the negative root of f_m(x). Then α'=1/(1-α), and α”=1-1/α are the two other roots, and in fact they are units in _L^×. Moreover,

    -m-2<α<-m-1<0<α'<1<α”<2

and therefore all eight possible sign signatures may be obtained from α and its conjugates. Thus, every totally positive unit is a square, and ρ_∞=0, as claimed. 
 

Thus, in order to prove Theorem <ref>, it is enough to show that G is trivial.

 
Let m≥ 0 be an integer such that D=m^2+3m+9 is square-free. Let v=2, or let v be a prime divisor of D, and let f_m(x)=x^3+mx^2-(m+3)x+1. Then, f_m(x) is irreducible as a polynomial in  _v[x].


Let v=2. Then, if we consider f_m(x) as a polynomial in _2[x], we have

    f_m(x)=
    x^3+x^2+1   if m≡ 1  2,
    
    x^3+x+1   if m≡ 0  2.

In both cases, f_m is irreducible over _2, hence it is irreducible over _2.

Now, let v>2 be a prime divisor of D. Then, by assumption, v>3 and

    f_m(x-m/3)=x^3-D/3x+D· (2m+3)/27

is integral over _v. Since D=m^2+3m+9, it follows that 4D-(2m+3)^2=27 and therefore the greatest common divisor of D and 2m+3 divides 27. Since by assumption D is square-free, it cannot be divisible by 3, and so (D,2m+3)=1; this together with the fact that D is square-free implies that f_m(x-m/3) is Eisenstein over _v. Hence, f_m is irreducible over _v, as claimed.


We are now ready to prove Theorem <ref>.


Let m≥ 0 be an integer such that  m^2+3m+9 is square-free. Let E_m be the elliptic curve given by the Weierstrass equation

    E_m: y^2 = f_m(x)= x^3+mx^2-(m+3)x+1.

Let L_m be the number field generated by a root of f_m(x), and let Cl(L_m) be its class group. Then,

    rank_(E_m()) ≤ 1 + dim__2(Cl(L_m)[2]).




We shall use Theorem <ref>. The order of 2≡ -1  3 is 2, even, so ρ=ρ^+, and Lemma <ref> shows that ρ_∞=0, so it remains to compute G_m = ∏_v∈ S_m∖{∞} G_m,v. 

The discriminant of f_m is D^2=(m^2+3m+9)^2, so we have

    S_m={∞,2}∪{v| D}.

However, by Lemma <ref>, the polynomial f_m(x) is irreducible over _v for any finite prime v∈ S_m. It follows that the number of irreducible factors of f_m(x) over _v is 1, and  therefore I_m,v is zero-dimensional by Lemma <ref>. Since G_m,v⊆ I_m,v, we conclude that G_m,v is always trivial. Hence, G_m is trivial, and Theorem <ref> implies that 

    ^(2)(,J_m) ≤ g + (L_m)[2] + (G_m→(L_m)/2(L_m)) = g + (L_m)[2],

where J_m is the jacobian of the elliptic curve E_m. Since the genus of E_m is 1, then E_m ≅ J_m over . Hence,

    _(E_m())≤^(2)(,E_m) ≤ 1 + (L_m)[2],

as desired.
 







§ GENUS G=(P-1)/2, WHERE P IS A SOPHIE GERMAIN PRIME


The goal of this section is to find examples of hyperelliptic curves of genus g≥ 2 where the dimension of the Selmer group can be bounded in terms of a class group, as in Theorem <ref> (the genus 1 case). We begin by looking at polynomials f(x) that cut out extensions of degree p, contained inside a q-th cyclotomic extension, where q is another prime.


Let q>2 be a prime such that the multiplicative order of 2 q is either q-1 or (q-1)/2, and let p>2 be a prime dividing q-1. Let (ζ_q) be the q-th cyclotomic field, and let L be the unique extension of degree p contained in (ζ_q). Further, suppose that _L = [α] for some algebraic integer α∈ L, let f(x) be the minimal polynomial of α, and let J/ be the jacobian variety associated to the hyperelliptic curve C: y^2=f(x). Then,

    ^(2)(,J) ≤ρ_∞ + g + ^+(L)[2].

If in addition the multiplicative order of 2 p is even, then 
    ^(2)(,J) ≤ρ_∞ + g + (L)[2].



We apply Stoll's algorithm to y^2=f(x), in order to compute G. Note that f(x) is monic, integral, and irreducible over , since α generates _L as a ring. Moreover, since _L=[α], it also follows that

    (f(x))= (𝒪_L),

and since L⊆(ζ_q), the only prime dividing (_L) is q. Hence, the set S={∞,2,q}. We will show that G_2 and G_q are trivial, and therefore G is trivial as well. Indeed:


  * Let v=2. Since the order of 2 modulo q is q-1 or (q-1)/2 by hypothesis, it follows from <cit.> that 2 splits into 1 or 2 prime ideals in (ζ_q)/, and therefore 2 must be inert in the intermediary extension L/ of degree p. In particular, the polynomial f(x) is irreducible over _2, since it defines an unramified extension L_2/_2 of degree p. Hence, m_2=1, and the dimension of I_2 is m_2-1=0 by Lemma <ref>. Since G_2⊆ I_2, we conclude that G_2 is trivial as well.


  * Let v=q. Since L⊆(ζ_q) and q is totally ramified in the cyclotomic extension, it is also totally ramified in L/. Thus, f(x) is irreducible over _q because it defines a totally ramified extension L_q/_q of degree p. Thus, m_q=1 and arguing as above in the case of v=2, we conclude that G_q is trivial.

Since the only finite primes in S are 2 and q, it follows that G=G_2× G_q is trivial. Now, Proposition <ref> shows the bound ^(2)(,J) ≤ρ_∞ + g + ^+(L)[2]. If in addition the order of 2 p is even, and since L/ is cyclic of degree p, then Corollary <ref> shows that ρ=ρ^+. Hence ^(2)(,J) ≤ρ_∞ + g + (L)[2], as claimed.


The drawback, however, of the previous result is that there are very few subfields of cyclotomic extensions with a power basis, as the following result points out.


Let L be an extension of degree p≥ 5, and let _L be the maximal order of L. Then, _L has a power basis if and only if L=(ζ_q)^+ is the maximal real subfield of the q-th cyclotomic field, where q is a prime with q=2p+1. 


For instance, the unique cyclic number field of degree 5 with a power basis for the maximal order is (ζ_11)^+. Also, there is no cyclic number field of degree 7 with a power basis for its maximal order (since 15 is not a prime). Hence, we concentrate on those cyclic extensions of degree p, where p is a Sophie Germain prime, i.e., q=2p+1 is also prime. 


Let q≥ 7 be a prime such that p=(q-1)/2 is also prime, and let L=(ζ_q)^+ be the maximal real subfield of (ζ_q). Let f(x)∈[x] be any monic integral polynomial defining L, let C: y^2=f(x), and let J/ be its jacobian. Then,  

    ^(2)(,J) ≤ρ_∞ + j_∞ + ^+(L)[2] + (G→(L)/2(L)).

Moreover, if f(x) is the minimal polynomial of ζ_q+ζ_q^-1 or -(ζ_q+ζ_q^-1), then 

    ^(2)(,J) ≤ρ_∞ + j_∞ + ^+(L)[2].

Further, if one of the following conditions is satisfied, 


  * the Davis–Taussky conjecture holds, or

  * the prime 2 is inert in the extension (ζ_p)^+/,

  * q≤ 92459,

 then ρ_∞=0, and ^(2)(,J) ≤  g + (L)[2].

 The first bound follows from Proposition <ref>, so let us assume that f(x) is the minimal polynomial of ζ_q+ζ_q^-1 or -(ζ_q+ζ_q^-1).  The ring of integers _(ζ_q)^+ has a power basis, namely [ζ_q+ζ_q^-1]. Moreover, since p is a Sophie Germain prime (with q=2p+1 prime), it follows that the multiplicative order of 2 q is a divisor of 2p = 2· ((q-1)/2). Since q≥ 7, the order of 2 is bigger than 2, so it must be p=(q-1)/2 or q-1. Thus, Theorem <ref> applies and we obtain ^(2)(,J) ≤ρ_∞ + j_∞ + ^+(L)[2].

Further, if (1), (2), or (3) holds, then by Conjecture <ref>, or Theorem <ref>, respectively, we find that ρ_∞=0 and ρ=ρ^+.

If q≤ 92459, we shall use the computational approach at the end of Section <ref> to show that ρ_∞=0. Let ζ=ζ_q=e^2π i/q. Then, the ring of integers of (ζ_q)^+ has a power basis, namely _(ζ_q)^+ = [ζ_q+ζ_q^-1]. Let u=-(ζ+ζ^-1) if p≡ 1  4 and u=ζ+ζ^-1 if p≡ 3  4, thus chosen so that u is a unit in _L^× of norm 1. Moreover, we note that 
if p≡ 1  4, then

    -(ζ+ζ^-1) < -(ζ^2+ζ^-2) < ⋯ < -(ζ^p-1/2+ζ^-p-1/2) < 0 < -(ζ^p+1/2+ζ^-p+1/2) < ⋯ < -(ζ^p+ζ^-p) <1,

and if p≡ 3  4, then

    ζ^p+ζ^-p < ζ^p-1+ζ^-(p-1) < ⋯ < ζ^p+1/2+ζ^-p+1/2 < 0 < ζ^p-1/2+ζ^-p-1/2 < ⋯ < ζ+ζ^-1 <1.

Thus, according to our conventions described in this section, the embeddings τ_1,…, τ_p are numbered so that τ_i(u)=r_i∈ with 

    r_i = 
    -(ζ^i+ζ^-i)     if  p≡ 1  4,
    ζ^p+1-i+ζ^-(p+1-i)    if  p≡ 3 4,

for all 1≤ i ≤ p. Thus,

    res_∞(u) = 
     (-1,-1,-1,…,1,1,1) ∈ H_∞

 with (p-1)/2 minus ones when p≡ 1  4, and (p+1)/2 minus ones when p≡ 3  4.
 
 Now, the Galois group 𝒢=(L/) is cyclic of order p. Since q=2p+1 is prime, then the multiplicative order of 2 q is (q-1)/2 or q-1. Thus, either -2 or 2 is a primitive root mod q. Let γ(ζ)→(ζ) that sends γ(ζ)=ζ^2. It follows that σ = γ∈𝒢≅ (/q)^×/{± 1} is a generator. Thus, the automorphism σ(ζ+ζ^-1)=ζ^2+ζ^-2 generates 𝒢. Let ϕ=ϕ_σ∈ S_p be the permutation attached to σ as defined above. For instance, if p≡ 1  4, then r_1=-(ζ+ζ^-1), where ζ=e^2π i/q, so τ_1(σ(u))=r_2=-(ζ^2+ζ^-2)  and therefore ϕ(1)=2. However, if p≡ 3 4, then r_1=ζ^p+ζ^-p. We can find an integer 1≤ k≤ p, and k or -k≡ 2p  q, such that  σ(u)=ζ^k+ζ^-k. It follows that τ(σ(u))=r_k and so ϕ(1)=k in this case. In general, the permutation ϕ is defined by
 
    ϕ(i) = min{2· i  q, (-2· i)  q}

 when p≡ 1  4, and by
 
    ϕ(i) = p+1 - min{(2· (p+1-i))  q, q-(2· (p+1-i)  q)},

 when p≡ 3  4, where our representatives in /q are always chosen amongst {0,1,…,q-1}.
	With these explicit descriptions of res_∞(u) and ϕ_σ, we have computed  (using Magma) the matrix M_∞,u for all primes p and q, with q≤ 92459, as in the statement, and in all cases d_∞,u=p-1. Hence, ρ_∞=0 follows from Lemma <ref>.



	If the Davis–Taussky conjecture holds, then the class number h_K of (ζ_q) is odd (by Theorem <ref>), and therefore the class number h_K^+ of L=(ζ_q)^+ is odd as well (because h_K^+ is a divisor of h_K). Hence, if the Davis–Taussky conjecture holds, then (L)[2] is trivial, and the bound of Theorem <ref> becomes 
	
    ^(2)(,J) ≤  g.





§ EXAMPLES
 




 §.§ Curves of Genus 1



In this section we present some data that was collected on the curves described in Theorem <ref> (the data collected can be found at <cit.>). For m∈, let f_m(x), E_m, and L_m be as in Theorem <ref>. Using Magma, we attempted to compute (subject to GRH) the Mordell–Weil rank of E_m() and __2(L_m)[2] for every m∈ such that 1≤ m ≤ 20000 and m^2+3m +9 is square-free. There are 12462 such values of m in the given interval, and we were able to compute the rank of E_m/ for  12235 of them. For the other 227 curves, we were only able to get upper and lower bounds on their rank.

Of the 12462 curves that we tested, 10327 of them (about 82.87%) had rank equal to the upper bound given in Theorem <ref>. However, one might expect that the sharpness of this upper bound would decay as m gets larger and larger, and in fact that seems to be the case. Let us define a function to keep track of the sharpness of the bound in an interval. 

Let 
    M = {m∈ : 1≤ m ≤ 20000  and m^2+3m +9  is square-free}
 
and
S = {m∈ M : (E_m) = (L_m) +1}. Given I⊆ M, we define  (I) = #(S∩ I)/#(M∩ I).


The set S only includes curves whose rank was actually computable and, because of this, the (I) statistic only gives a lower bound for how sharp Washington's upper bound is over the set I. 

 
In order to see how the sharpness of the upper bound degrades as m grows, in Table <ref> we present (I) over disjoint intervals of length 1000. In the data, we clearly see that the number of curves for which Washington's bound is sharp in a given interval does start to decrease, but the bound is still sharp more often than not (notice, however, that the sharpness is inflated by the fact that the bound is sharp every time the bound is 1, since there is a point of infinite order P = (0,1) on E_m). 




To see how fixing the bound first affects its sharpness, we define the following sets

    T(r) = {m ∈ M:(E_m) =  r} and  B(b) = {m ∈ M: (L_m)[2]+1 = b}.
 In Table <ref>, for each bound b that occurs we give the number of curves of rank r whose bound is b for each r that occurs. We also give the percentage of the curves whose rank is exactly b and provide the totals of each column so that we can see how many curves of each rank we found (for similar statistics and conjectures in a broader context, see <cit.>).





From Table <ref> we can see that all of the curves that we computed, have odd rank less than or equal to 7. It also turns out that for all of the curves that we computed, Washington's bound is also odd and less than or equal to 7. In Table <ref>, we give the first m such that (E_m) = r and (L_m)+1= b for each pair (r,b) that occurred. 

It is also interesting to point out that the average rank among curves with b=1 is 1, the average rank among curves with b=3 is 2.23, among curves with b=5 is 3.38, and among curves with b=7 the average rank is 5.20 (see <cit.> for other examples of Selmer bias in genus 1). 




Lastly, for the sake of concreteness, we end this section with an explicit example. When m=143 we have that 

    E_143:y^2 = x^3 + 143x^2 - 146x + 1

with conductor 2^2· 20887^2. Using Magma, we can compute that 

    (L_143)≅/2 + /2 + /4 + /4,

and so Washington's bound for the Mordell–Weil rank is (L_143)[2] +1 = 4+1 = 5. Looking for points on E_143 we find 5 independent points of infinite order that generate the Mordel–Weil group. 

    E_143() ≅^5 = ⟨ (126/121 , -3023/1331), (90 , -1369), (65/64 , 577/512), (21/4 , 
    -461/8), (-1 , 17) ⟩.







 §.§ Examples in the Sophie Germain case


In this section we show examples of hyperelliptic curves that arise from Theorem <ref>. 


Let q=7 and p=3. Then, L=(ζ_7)^+ is the maximal real subfield of (ζ_7), which has degree 3, and class number 1 (see <cit.>). Note that the order of 2 in (/3)^× is 2=p-1, and therefore condition (2) of Theorem <ref> is met. Hence, if f(x) is the minimal polynomial of ±(ζ_7+ζ_7^-1), then the jacobian J() of y^2=f(x) satisfies

    _(J())≤__2^(2)(,J) ≤ g+(L)[2] = 1.

For instance, if f(x) is chosen to be the minimal polynomial of -(ζ_7+ζ_7^-1), then f(x)=x^3 - x^2 - 2x + 1 we in fact recover the elliptic curve E_-1 of Theorem <ref> for m=-1. The Mordell–Weil rank of the elliptic curve E_-1 y^2=x^3 - x^2 - 2x + 1 is 1, so the bound on the rank given by Theorem <ref> in this case is in fact sharp.


 
If q=11 and p=5, then L=(ζ_11)^+ is a field of degree 5 and trivial class group. Since 2 is a primitive root modulo 5, if f(x) be the minimal polynomial of ±(ζ_11+ζ_11^-1), then Theorem <ref> says 

    _(J())≤__2^(2)(,J) ≤ g+(L)[2] = p-1/2=2,

where J is the jacobian of y^2=f(x). If f(x) is the minimal polynomial of ζ_11+ζ_11^-1, then f(x)=x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1. Below we will describe a general method to find some rational points on the jacobian, and show that 2≤_(J)≤ 2. Thus, the rank is 2 and the bound is sharp. 


 
If q=23 and p=11, then L=(ζ_23)^+ is a field of degree 11 and trivial class group. Since 2 is a primitive root modulo 11, if f(x) be the minimal polynomial of ±(ζ_23+ζ_23^-1), then Theorem <ref> says 

    _(J())≤__2^(2)(,J) ≤ g+(L)[2] = p-1/2=5,

where J is the jacobian of y^2=f(x). If f(x) is the minimal polynomial of -(ζ_23+ζ_23^-1), then 

    f(x)=x^11 - x^10 - 10x^9 + 9x^8 + 36x^7 - 28x^6 - 56x^5 + 35x^4 + 35x^3 -    15x^2 - 6x + 1.
 Below we will show that 4≤_(J)≤ 5. A full 2-descent (via Magma) shows that the rank is, in fact, equal to 4.


We finish this section by describing a method to produce points in J (as in Theorem <ref>, and compute the rank of the subgroup generated by these points. Let p be a fixed Sophie Germain prime, let q=2p+1, and let L=(ζ_q)^+ be
the maximal real subfield of (ζ_q), where ζ_q 
is a primitive q-th root of unity. The minimal polynomial
for ζ_q+ζ_q^-1 has constant term 1 or -1 according to
whether p is congruent to 1 or 3  4. If f∈[x] is 
a polynomial with constant term 1, then the point (0,1) will 
be on the curve

    C:y^2=f(x)

and furthermore the factorization of f(x)-1 will provide points
on the jacobian J of C, allowing us to obtain a lower bound for
the Mordell–Weil rank of J().  For this reason, we define f to be the minimal polynomial
of  θ=(-1)^(p-1)/2(ζ_q+ζ_q^-1).

A lower bound for the rank can be computed by considering the
images of the factors of f-1 in L^×/(L^×)^2
under the map δ_:J()→ H_ (see Section <ref>). Let y_0∈, let g(x) be an
irreducible factor of f(x)-y_0^2, 
and let K be the splitting field of g(x).  Then, over K, we have
a factorization

    g(x)=∏_i=1^n (x-x_i),

and the points P_i=(x_i, y_0) are in C(K). Under the map δ_K J(K)→ H_K,
as a map from C(K) to L_K^×/(L_K^×)^2
we have

    P_i↦ (x_i-θ_K)(L_K^×)^2.

If f remains irreducible over K, then L_K is simply the composite
extension of K and L_q, and θ_K=θ. As a map from J(K) to L_K^×/(L_K^×)^2, δ_K is a homomorphism of groups, and hence the divisor P_1+P_2+⋯ +P_n maps to

    ∏_i=1^n (x_i-θ_K)(L_K^×)^2=(-1)^ng(θ_K)(L_K^×)^2.

On the other hand, the divisor P_1+P_2+⋯ +P_n can be regarded
as the base extension to K of a certain divisor D defined over , hence over  we have

    D ↦ (-1)^ng(θ)(L^×)^2,

via the map F_K of Section <ref>. Thus, for each irreducible factor g(x) we obtain a point on the jacobian J() that corresponds to the divisor D=D(g) defined above. Moreover, since the map J()/2J()→ H_ induced by δ_ is injective (<cit.>), in order to compute the rank of the subgroup generated by {D(g) }_g, it suffices to check the dimension of the (multiplicative) subgroup generated by {δ_(D(g)) } in H=H_.  


For example let q=11 and p=5, as in Example <ref>, so L=(ζ_11)^+ and f(x)=x^5+x^4-4x^3-3x^2+3x+1. Then, 
f(x)-1 factors as

    x(x^2 - 3)(x^2 + x - 1).

Their images in L^×/(L^×)^2 via δ_ are

    -θ(L^×)^2, (θ^2 - 3)(L^×)^2, and 
    (θ^2 + θ - 1)(L^×)^2

respectively. To obtain a lower bound for the rank of J it remains only to reduce {-θ, (θ^2 - 3),(θ^2 + θ - 1)} to a multiplicatively
independent subset modulo squares. Since the product of all three is

    -(f(θ)-1)(L^×)^2=(L^×)^2

we see that at most two are multiplicatively independent. On the
other hand -θ(θ^2 - 3) is not a square in L, so

    -θ(L^×)^2 and (θ^2 - 3)(L^×)^2

are multiplicatively independent give us a lower bound of 2 as the 
rank of J over . An upper bound of 2 for the rank was computed in Example <ref>, hence the rank is exactly 2.


In Table <ref>, we have collected the upper bound given by Theorem <ref>, together with some computational data of lower bounds for the rank of the jacobian J associated to the first few Sophie Germain primes, obtained using the method we have described here. It is worth pointing out that the bound given by Theorem <ref> in these examples is unconditional (i.e., not dependent on the Davis–Taussky conjecture) since q≤ 92459. We also note that 2 is inert in (ζ_p)^+ in some cases such as p=5,11,23,29,53,83,131,173 but not in others such as p=41,89,113.





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