A LGA NT Master Thesis in Mathematics
C JACOBIANS Gürkan Dogan, advised by Richard Griffon 14 June 2018 R EG ENSBURG L EIDEN 2
4
Let C/Q be a hyperelliptic curve, given affinely by a model y 2 = f ( x ), with f(x) ∈ Q[x] monic, squarefree and of odd ddegree. We consider the Jacobian variety J/Q attached to C . Since J is an Abelian variety, the Mordell-Weil Theorem shows that its Q-rational points J(Q) form a finitely generated Abelian group. Therefore, we have a decomposition J(Q) ≥ Zrank J(Q) ⊕ tors(J(Q)), where tors(J(Q)) is a finite group containing elements in J(Q) of f inite order, and rank J(Q) is the rank of J/Q. In this thesis, we are interested in bounding rank J(Q) in terms of the invariants of C To this end, we define the Selmer group Sel( 2 (Q, J) of J over Q, and find a more practical description of it. As the inequality (see Theorem 2.3.1) rank J(Q) ≤ dim Sel (2) (Q, J) – dim J(Q)[2] suggests, we study rank J(Q) by means of the Selmer group Sel( 2) (Q, J). Here, the dimensions are taken over F 2 and J(Q)[2] denotes the 2-torsion subgroup of J(Q). Moreover, we will see that the computation of dim J(Q)[2] depends only on the shape of the factorisation of f ( x ) in Q[ x ] and is therefore "easy to compute". Our main approach to studying Sel( 2 )(Q, J ) is to use Stoll's Implementation of 2-descent. Fol- lowing [23], we firstly give a more concrete description of the 2-Selmer group, referring to [16] for general and complete proofs. In [16], Edward F. Schaefer replaces the cohomology groups appear- ing in the original definition of the Selmer group by the kernels of certain norm maps of the algebra L = Q[T]/(ƒ(T)) defined by ƒ(x), making Sel(2)(Q,J) more tangible to work on.  See Theorem 3.2.3 for the precise statement. We will see that this concrete description of the 2-Selmer group can further be simplified so as to yield an upper bound on dim Sel (2) (Q, J) as follows: dim Sel( 2 )(Q, J ) ≤ m ∞ − 1 + dim Cl( L )[2] + dim ker( G → Cl( L )/ 2 Cl( L )), where G is a group defined by finitely many local conditions at finite places of Q and m ∞ is the number of irreducible factors of f(x) in R[x]. The main step in deriving that bound is to find a suitable finite subgroup Sel ⊂ L× ((L×)² containing Sel(²)(Q, J), whose dimension can be studied relatively easily. The details can be found in Chapter 3. In the next chapter, we will concentrate on a more current consideration.  Following [4], we will introduce a subgroup Sel(2)(Q, J) ⊆ Sel ⊆ L×/(L×)² with Sel ⊆ Sel and obtain the following bound: dim Sel (2) (Q, J) ≤ dim μ ∞ + dim Cl + (L)[2] + dim(G ∩ V ∞ ), (*) where μ∞ ⊆ Ož/(Ož)² and G ∩ V∞ ⊆ 1(L)/I(L)² are defined by certain local conditions at infinity, I ( L ) is the group of fractional ideals of L and Cl + ( L ) is the narrow class group of L . Furthermore, we will study the dimension of the totally positive units Ox,+ more carefully and will be able to make the bound (*) stronger for some families of hyperelliptic curves. For example, when L is a cyclic number field of odd prime degree p, the order of 2 in ( Z/pZ× ) is even and O X + = (O X ) 2 , we will see that the bound (*) improves to the one dim Sel (2) (Q, J) ≤ g + dim Cl(L)[2] + dim ker(G → Cl(L)/2 Cl(L))
where g = (p − 1)/2 is the genus of C. See Theorem 4.5.2 for details. In the last chapter, we will present an infinite family E n of elliptic curves for which we are able to use the bound (**) and estimate dim ker(G → Cl(L n )/2Cl(L n )). This will yield the following optimal bound on rank E n (Q): rank E n (Q) ≤ 3 + dim Cl(L n )[2] I would like to sincerely thank my advisor, Richard Griffon, for he has constantly supported me in all of my academic pursuits during the period he has been advising
This chapter is intended to contain preliminary definitions and results we will need in the next sections for our main exposition. Some proofs are omitted due to their being long or technical, but we then provide references for them. Every title has a rich literature on its own, so one probably cannot expect to get a full account of them through this chapter.
1.1
Hyperelliptic curves
We give a definition of hyperelliptic curves in this section. Let k be a field of characteristic 0 and let f(x) ∈ k[x] be a monic, squarefree polynomial of odd degree d = 2g + 1 with g ≥ 1 (so that d ≥ 3). Write f =
The polynomial f(x) being squarefree, we have that ho(x,y) is irreducible in k[x,y] and hence, the ideal ( h 0 ) ⊆ k [ x, y ] is prime. Let C 0 ⊆ A 2 be the affine set corresponding to the ideal ( h 0 ) ∈ k [ x, y ]. One can see that C 0 is s mooth because f(x) is squarefree. Therefore, we conclude that C 0 ⊆ A 2 is a smooth affine algebraic variety of dimension 1, defined over k . Let C o ς P 2 be the projective closure of C 0 . Then one can check that there is a unique point at infinity on C 0; namely, [0 : 1 : 0], which is Q-rational. One can also check that the point [0 : 1 : 0] is singular if d > 3. Consider the projective algebraic set C ⊂ P9+2, whose ideal is generated by the following 2g homogeneous polynomials of degree 2: Let φ : A² → P9+² be the map given by (x,y) → [1 : x : x ² : · · : x 9 + 1 : y] We claim that φ is an isomorphism of algebraic varieties between C 0 and the dehomogenisation C ∩ { x 0 = 1}. Indeed, an induction argument shows that for all [ x 0 : x 1 : · · : x g+2] ∈ C , one has
77 for all j = 1,...,g + 1. By definition, C ∩ { x 0 = 1} is defined by dehomogenisations of the above equations Q 1 , . . Q 9 , Q 9 +1 , . . Q 2 9 –1 , H with respect to the 0 th variable x 0; i.e., by the following polynomials: Moreover, for all j = 1,...,g + 1, (*) reads x j = x j for the points on the "dehomogenised curve" CN{x 0 = 1}. It follows that all points on CN{x 0 = 1} are of the form [1 : x 1 : x 2 : . .  .x 9 +1 : x 9 +2], meaning φ−1 ( ∩ { x 0 = 1}) ⊆ C 0. Moreover, the inverse of φ is given by φ − 1 | Cn { x 0 = 1} : [1 : x 1 : x 2 : · · : x 9 + 1 , x 9 + 2] ↦ (x 1 , x 9 + 2) ∈ C 0 , for [1 : x 1 : x 1 : x 1 : x 1 +1 : x 9+2] ∈ C ∩ {x 0 = 1}. Hence φ|c o and φ −1 cn {x =1} are mutually inverse, and moreover, they are both given by polynomial functions. Thus, it follows that φ is an isomorphism of algebraic varieties, as claimed. Notice that CN{x 0 = 0} consists of a unique point P ∞ = [0 :۰۰۰: 0 : 1 : 0], the point at infinity, and this point is k-rational. A straightforward computation shows that the rank of the Jacobian matrix of C at t the point P ∞ is g − 1; that is, C is smooth at P ∞ . Since C 0 is smooth and we have established an isomorphism C 0 ≤ C ∩{ x 0 = 1} above, we conclude that C = C 0 ∪ { P ∞} ⊆ P +2 is a smooth projective variety of dimension 1, defined over k , containing C 0 as a dense open subset.
squarefree polynomial of odd  degree d = 2g + 1 ≥ 3.  Let C  be the smooth projective variety of dimension 1 constructed above.  We then call C/k the hyperelliptic curve associated to f In what follows, we show that if C/k is the hyperelliptic curve over a field k of characteristic 0, associated to f with f monic, squarefree and of odd degree d = 2g + 1 ≥ 3, then the genus of the curve C is g To this end, let C 0 be as above and consider the morphism π 0 : C 0 → A 1 , ( x, y ) → x, defined over k. There exists a unique finite morphism π : C → P1 extending π 0 via P ∞ ↔ [1 : 0]. Let p ∈ P ¹ be a point such that p̸= [1 : 0]; i.e., p ∈ A ¹ . Then, whenever f(p) ≠ 0 (which holds for almost all p ∈ P 1 ), the fibre f- i (p) consists of two points (p,±√ƒ(p)) where √ƒ(p) ∈ k. This implies that the degree of the morphism π is 2 (see [21, Proposition 2.6|). On the other hand, whenever p ∈ P ¹ is s such that f(p) = 0, the fibre π ¹ (p) consists of a single point and the ramification index ep is 2. Note also that π-1[1 : 0] = P ∞ and π ramifies at t the point p = [1 : 0], meaning that e p = 2. Hence, the set of ramification points is Ω = {p ∈ k : f(p) = 0} ∪ {[1 : 0]} Now, let gc denote the genus of the hyperelliptic curve C , then the Riemann-Hurwitz formula (see [21, Chapter II, Theorem 5.9]) reads
Since the genus g =1 of P 1 is 0 and deg(f) = 2g + 1, we get that gc = g, as claimed. 1.2
Jacobians of hyperelliptic curves
Let k be a field of characteristic 0 and let C : y 2 = f ( x ) be a a hyperelliptic curve over k with f monic, squarefree and of odd degree 2g +1. Let P ∞ be the point at infinity of C. In this section, we want to sketch the construction of the Jacobian variety J(C) attached to the curve C. Let Div( C ) be the group of divisors on C (over the algebraic closure k ) and let Pic( C ) be the quotient of Div(C) by the principal divisors div(f) where f ∈ k (C) ×  Then there is a group homomorphism where [ D ] ∈ Pic( C ) denotes the linear equivalence class of a divisor D ∈ Div( C ). We define Definition 1.2.1. The Jacobian variety of C is given by J(C) = ker(deg : Pic(C) → Z) The kernel ker(deg) is sometimes denoted by Pic (C) and called the class group of C. Note that we have to "justify" this definition; that is, we must endow the Abelian group J(C) with the structure of an algebraic variety. This is explicitly done in [10] by D. Mumford over an algebraically closed field. In the following, we present a sketch of this construction, closely following [10]. For two divisors D 1 , D 2 ∈ Div( C ), we write D 1 ≡ D 2 to mean [ D 1 ] = [ D 2 ] in Pic( C ). ● Let C 9 = C × · · × C be the product of g copies of C . Then C 9 is an algebraic variety of dimension g. Let S q be the symmetric group on g letters. There is an action of S q on C 9 by permutation of the factors. We can therefore form the quotient Symm 9 C = C 9 / S q of C 9 by the action of S g . It can be shown that Symm g C is also an algebraic variety of dimension g Now if [ D ] is any element of J ( C ), then one can show that there are points P 1,..., P g such - g.P ∞ (see [10, p. 3.29]). This yields a surjection
Define Θ as the subset of J(C) of divisor classes of the form explicitly define a subvariety Z ⊆ Symm 9 C of dimension g such that the restriction
is a bijection. Hence one can transfer the structure of algebraic variety from Z to J(C) \ Θ. ● The aim in this step is to cover J(C) by affine pieces which are in bijection with to Z which we know is an algebraic variety of dimension g from the previous step. The following type of elements of J(C) is particularly important: Definition 1.2.2. Let Ω = {Pa = (α,0) ∈ C(k) : f(α) = 0} ∪ {P∞}. Let T ⊂ Ω be a subset of even cardinality.  We define The following lemma summarises some important properties of eT. We will make a heavy use of this lemmma in the next section while computing the dimension of the 2-torsion of the Jacobian . Lemma 1.2.3. Let T,T i ,T 2 C Ω be some subsets of even cardinality (so that the symmetric difference T 1 △T 2 = (T 1 ∪ T 2 ) \ (T 1 ∩ T 2 ) also has even cardinality).  Then (i) 2er = 0, (ii) eT, + eT, = eT, ∆T, (iii) eT, = eT, if and only if T, = T, or T, = Ω \ T,2. Thus, the set of the eT 's forms a group isomorphic to (Z/2Z)29 Proof. See [10, Lemma 2.4].
●
We achieve our aim by the following lemma: Lemma 1.2.4. We have where T  varies over the subsets T C Ω of even cardinality and eτ is defined as above. Proof. See [10, Lemma 2.5].
□
We know from the previous step that Z and J(C) \ Θ are in bijection. Since (J(C) \ Θ) + eT is a translation of J(C) \ Θ by an element e r using the group structure of J(C), we have an isomorphism (J(C) \ Θ) + cτ ≈ J(C) \ Θ.  Hence, Z and (J(C) \ Θ) + cτ are in bijection, which allows us to define the structure of an algebraic variety on (J(C) \ Θ) + e r. Thus, we have covered J(C) with the "affine pieces" isomorphic Z, as desired. ● Take one copy of Z for each T ⊂ Ω. To finalise the construction of J(C), one has to show in this last step that these copies can be glued together to give J ( C ) as a variety according to their identification as subsets of the Jacobian. This amounts to checking the compatibility of algebraic variety structure on the intersection ((J(C)) \ Θ) + e τ) ∩ ((J(C)) \ Θ) + e τ) with T , T ′ ∈ Ω arbitrary subsets of even cardinality. For the proofs, we refer to [10, Lemma 2.6] and [10, Proposition 2.7]. Since there exists a surjection C 9 → J(C ) and C is complete, it follows that J(C) is a complete variety. Moreover, J(C) is an Abelian variety. By an Abelian variety, we mean the following: Definition 1.2.5. An Abelian variety is a complete variety A with an Abelian group law such that the addition A x A → A and inverse A → A are morphisms. Indeed, we just observed that J ( C ) is a complete variety and by definition, it has an Abelian group structure. To show that the addition and inverse are morphisms, one relies on [10, Lemma 2.9]. Therefore, we conclude that given a hyperelliptic curve C over k, there exists an Abelian variety J(C)/k naturally attached to C of dimension the genus g of C ; namely, the Jacobian J(C). One can actually show that J(C) is indeed defined over k (see [9] for more details).
1.3
2-torsion of the Jacobian
Let k/Q be a field extension and C : y 2 = f ( x ) be a hyperelliptic curve over k with f ( x ) ∈ k [ x ] monic, squarefree and of odd degree. Let f = f1 · ·· fmk be the factorisation of f over k into monic, irreducible factors f 1 , . . , f mk. Let G k = Gal(k/k) be the absolute Galois group of k. Then the action of G k on the points P ∈ C ( k ) linearly extends to an action on J(C)[2], the 2-torsion subgroup of J[2]. One can then consider the G k -invariant points J(C)[2]Gk := J(k)[2] of J(C)[2] under the action of Gk. One of our aims in this section is to prove the following theorem: Theorem 1.3.1. In the above set-up, to every factor fi of f in k[x], we associate an element Di E J(k)[2] such that the Di 's for i = 1, . . ,mk, generate J(k)[2] and satisfy Moreover, we have dim J(k)[2] = mk − 1; i.e., J(k)[2] = (Z/2Z)mk-1 Since J(C)/k is an Abelian variety of dimension g over a field k of characteristic 0, it is known (see [9]) as a general fact that
Note that this fact is a special instance of Theorem 1.3.1 when f factorises completely in k[x] and in particular, this happens in k[x]. By Lemma 1.2.3, we know that {eτ}τςΩ, where T ⊆ Ω varies over the subsets of even cardinality, form a subgroup of J(C) isomorphic to ( Z/2Z )²9 . Therefore, we conclude that the same collection_{eτ}τco is the whole J(C)[2]. Now, let Ω = { P α = (α,0) ∈ C ( k ) : f (α) = 0}. For any α ∈ k such that f (α) = 0, we define e'a := e{p ,∞} = [P α + ∞ − 2∞] = [P α − ∞] We have the following lemma: Lemma 1.3.2. Let e'a where f(a) = 0 be defined as above. We have (i) The classes {e'a}f(a)=o generate J(C)[2] as a group, (ii) The only relation between the e'a, where α varies over the roots of f(x), is given by
Proof. (i) Note that if Pa1, Pa2 € Ω0 are distinct, then the part (ii) of Lemma 1.2.3 gives that e'a, + e'a, = e { P a,, P a,}  This implies that if T = { P a,, . . , P a2k }  ⊆ Ω , where k ≤ g, is a subset of even cardinality, then er = e'a, + · · + e'a2k Likewise, if T ⊆ Ω is a subset of even cardinality of the form T = { Pα1, . . , Pα2k-1, P∞}, where k ≤ g, then er = e'α1 + · · + e'α2k-1 Since {eτ}τςς with T ς Ω varying over subsets of even cardinality generates J(C)[2], it thus follows that {e'a}f(a)=o also generate J(C)[2], proving part (i) (ii) Note that we can write
where T 1 = {P 1,...,P 2g} and T 2 = {P 2g+1,00}. Since T 1 = Ω \ T 2, part (ii) and (iii) of Lemma 1.2.3 gives the relation (*). On the other hand, the "if" part of part (iii) of Lemma 1.2.3 also implies that this is the only relation between e 's. Indeed, this follows because as J(k)[2] is in particular a ( Z /2Z )-vector space, any relation between { e a } is given by La ∈ A ∈ a = 0 with A ⊂ Ω. Thus, the proof is complete.口Recall that we put G k = Gal( k/k ). Clearly there is an action of G k on Ω given by σ.(α,0) = (σ(α),0) for σ ∈ G k and α ∈ k a root of f(x). This yields an action of G k on { e a}f(α)=0 via σ.e'a = e'σ(α)  Denote by O i the orbit of an element e'a, ∈ {e'a}f(α)=0 so that where r is the number of orbits. For all i = 1,...,r, we define Note that part (i) of Lemma 1.2.3 implies that =0; 2e' = 0; which means that D i ∈ J(k)[2]. Moreover, since { c } f(α)=0 generates J(C)[2], it follows that {Di}i=1,...,r generates J(k)[2] as a group. Therefore, the following lemma concludes the proof of Theorem 1.3.1: Lemma 1.3.3. In the above set-up, we have (i) r = mk = the number of irreducible factors of f in k[x], (ii) J(k)[2] ≒ (Z/2Z)mk 1 Proof. (i) Note that by definition, the number r of orbits of the action of Gk on {&a}f(a)=0 is the same as the number of orbits of the action of G k on Ω . But this number is clearly m k Indeed, write f = f 1 . . f mk in k[x] with f s monic and irreducible for i = 1,. . ,m k . If α is a root of f ( x ), then there is a unique i ∈ {1, . . ,m k } such that f i ( α ) = 0. Moreover since each f i is irreducible, the orbit G k .α contains all the roots of the irreducible factor f i . Thus the result follows. (ii) The previous part now implies that J(k)[2] is generated by the elements { D i } i =1,...,m k Moreover, part (ii) of Lemma 1.3.2implies that the only relation between the D i for i = 1, . . ,m k is given by As J(k)[2] ⊆ J(C)[2] ≒ (Z/2Z) ²g, it t thus follows that J(k)[2] is a (Z/2Z)-module of rank mk − 1, finishing the proof. ● We now present some results about the dimensions of the F 2 -vector spaces J(R)/2J(R) and J(Q p )/2.J(Q p ), where Q p denotes the p adic field for a prime number p . Theorem 1.3.4. Let p be a prime and consider the p-adic extension Q , of Q. Let
Let g be the genus of C; i.e., g = (deg(f) - 1)/2, and let mp denote the number of irreducible factors of f (x) in Qp[x]. Then we have dim J(Q p )/2.J(Q p ) = dim J(Q p )[2] + d p g = m p − 1 + d p g Proof. See [8], where the author proves the fact that J(Q p ) contains a subgroup of finite index isomorphic to g copies of the ring of integers in Q p . This leads to the desired conclusion via [16, ● prop. 2.4]. Theorem 1.3.5. Let m ∞ denote the number of irreducible factors of f ( x ) in R [ x ] and g be the genus of C . Then we have dim J(R)/2J(R) = m ∞ − 1 − g Proof  See [18, Proposition 5.4].
Galois cohomology
1.4
We review the basics of cohomology theory of groups, avoiding technicalities and tedious constructions, and aiming at t the results and constructions we will need in the subsequent sections. Our main reference for this section is [13]. Definition 1.4.1. Let G be a group. Then an Abelian group A is a (continuous) G-module if there e xists a (continuous) composition ρ : G × A → A, (σ,α) → σα, which satisfies, for all a,b ∈ A, σ,σ1,σ2 E G, and the unit 1 E G, the following conditions: (i) σ(a + b) = σα + σb, (ii) σ1(σ2α) = (σ1σ2)α, (111)  1a = a. Example 1.4.2. Let K be a field of characteristic 0. Let G = Gal(K/K) be the absolute Galois group of K. For a smooth curve C of genus g ≥ 1 over K, let J denote its Jacobian variety J(C). Then we know that J is an Abelian variety over K. Moreover J(K) is a G-module. An instance of this example when K = Q will subsequently be of our particular interest. Let G be a group. For a G-module A, consider the Abelian group A c of G-invariant elements of A. This association defines a functor F from the category of G-modules to the category of Abelian groups. Theorem 1.4.3. The functor F from the above discussion is a left-exact covariant functor. Hence one can construct right derived functors H i (G, ●), i = 1,2,. . . , of F, from the category of G-modules to the category of Abelian groups.  They are characterised by the property that, given a short exact sequece 0 → A → B → C → 0 of G-modules, there exists an associated long exact sequence, called long exact cohomolgy sequence, 0 → AG → BG → CG → H 1 (G, A) → H 1 (G,B) → H 1 (G,C) → H 2 (G, A) → …., where the map δ is a connecting morphism. Proof. See [5, p. 204, Theorem 1.1A]. ● Definition 1.4.4. For an i ≥ 1, the Abelian group H i ( G , A ) is called a cohomology group.  In case where G = Gal(L/K) is a Galois group (of a field extension K ⊆ L), the cohomolgy group H i (G, A) is called a Galois cohomology group.  When G = Gal(K/K) is the absolute Galois group of a field K , we often abreviate H i ( G , A ) to H i ( K , A ). For a subgroup H of G , there are canonical morphisms H i ( G , A ) → H i ( H , A ) for i = 1,2,..., called restriction maps. Since we will be frequently using the restriction map on the first cohomol- ogy groups, we want to give a description of H 1 ( G, A ) and see how this restriction map is induced. We will also define the inflation map under the assumption that H is a normal subgroup of G. We will finally present a fundamental exact sequence relating these two maps. Following [13], we define the group Z 1 ( G,A ) of 1 -cocyles as the maps x : G → A with the property that, for all σ,τ ∈ G,
x(στ) = σx(τ) + x(σ) The subgroup B 1 ( G , A ) ⊆ Z 1 ( G , A ) of 1-coboundaries are the maps x : G → A of the form x(σ) = σα − α for a fixed a ∈ A , where σ ∈ G . It can be shown that
Remark 1.4.5. Observe that when the action of G on A is trivial, one gets H 1 (G, A) = Hom(G, A), where Hom(G, A) is the group of homomorphisms from G to A Let H ⊆ G be a subgroup. If x : G → A ∈ Z 1 (G, A) is a 1-cocycle, then its restriction x H to H is clearly a 1-cocyle in Z 1 (H, A). Therefore, we get a map res : H 1 (G, A) → H 1 (H, A), sending the class of a 1-cocycle to the class of its restriction to H. Now, assume that H is a normal subgroup of G and consider the quotient G/H. Let A H be the group of H -invariant elements of A. Then G/H acts on AH via σH.α = σ(α) for σH ∈ G/H and a ∈ A H . Indeed, for any τ ∈ H , T(σ(α)) = σ(τ(α)) = σ(α) So σ(α) ∈ A H and A H has a G/H -module structure. Moreover, if x : G/H → A H ∈ Z 1 (G/H, A H ) is a 1-cocylcle, then it induces a 1-cocycle in Z 1 (G, A) via the composition G → G/H → AH → A Hence we have obtained a map inf : H 1 (G/H, A 1 ) → H 1 (G, A), called inflation. We have an inflation-restriction exact sequence: Theorem 1.4.6. Let A be a G-module and let H be a normal subgroup of G. Then the sequence
is e va c t. Proof. See [13, p. 34, Theorem 4.6]. ● Example 1.4.7. In line with Example 1.4.2 above, we want to write the inflation-restriction exact sequence explicitely when G = Gal(K/K) and A = J(K).  To this end, let L/K be a Galois e xtension and let H = Gal(K/L).  Then H is a normal subgroup of G.  Hence, we have an isomorphism G/H ~ Gal(L/K).  Also, since the action of H is continuous on J(K), we have J(K)H = J(L). Therefore, the exact sequence reads 0 → H 1 (Gal(L/K), J(L))  ≡ > H 1 (G, J(K))  ≡ > H 1 (Gal(K/L), J(K)).
In this chapter, we will define the 2-Selmer group Sel ( 2 ) (Q, J ), properly introduce the notion of rank, and put a bound on the rank in terms of Sel(2)(Q,J). For this chapter and the rest of the thesis, we restrict attention to the hyperelliptic curves over Q.
2.1
The 2-Selmer and Shafarevich-Tate groups
Let C be a hyperelliptic curve over Q. Let J be the Jacobian of C . Let K be an extension of Q and let K denote the algebraic closure of K . Let G = Gal( K / K ) be the absolute Galois group of K . Consider the K -rational points J(K) of J . We give a first definition of the 2-Selmer group Sel(2)(Q, J) of J over Q in terms of Galois cohomology groups. We closely follow the write up [15] of B. Poonen Since multiplication by 2 map [2] is surjective on the K -rational points J(K) of J, we have a short exact sequence
where J(K)[2] is the group of 2-torsion points of J(K). Recall from Example 1.4.2 that there is an action of G on J ( K ), which turns it into a continuous G -module. By definition, we can define this action by defining it on a divisor. Indeed, if σ ∈ G and D = Σ P ∈ n p (P) ∈ Div (C) is a divisor on C, then we define the action as follows: It is easy to see that if D is of degree 0, then so is σ(D).  Moreover, if D is a 2-torsion, then 2σ(D) = σ(2D) = σ(0) = 0. This implies that G also induces an action on the 2-torsion subgroup J(K)[2] of J(K) Associated to the above short exact sequence, by Theorem 1.4.3 we have a long exact cohomolgy sequence 0 → J(K)[2] → J(K) [2], J(K) → H¹(G,J(K)[2]) → H¹(G,J(K))  [2] , H¹(G,J(K)) → …., where the map [2]* is functorially induced from the multiplication by 2 map.  Thus, simplifying the notation, we obtain the following short exact sequence:
where δ : J(K)/2J(K) → H¹(K, J[2])
15
CHAPTER 2. BOUNDING THE RANK IN TERMS OF THE SELMER GROUP
is called the coboundary morphism Let v be a place of Q. Then v is either a finite place or is infinity. In the first case, v is a prime number p and Q v = Q p , the p-adic (local) field; and in the second, Q v = R , the real numbers. For a discussion of this, see [12, §1]. Qv) be the absolute Galois group of Qv. We have an For any place v of Q, let Gv = Gal(0 injection (see [24, Section 1]) giving J(Q) a natural G v -module structure and yielding a restriction map on Galois cohomology groups as follows:
Therefore, writing the exact sequence (*) with K = Q and K = Q v for all places v, the construction of the res, maps defined above yields the following commutative diagram with exact rows: where the products are over all the places v of Q Definition 2.1.1. The 2-Selmer group Sel (2) (Q, J) of J over Q is defined as The Shafare vich-Tate group Ш(Q, J) of J over Q is de fine where Res is the product Π v res , of restriction maps over all the places v. Remark 2.1.2. The Shafarevich-Tate group Ш(Q,J) has a useful interpretation with regard to the Hasse's (Local-Global) Principle. Also it is not known in general if the Shafarevich-Tate group III(Q, J) is finite although il is conjectured to be so. See [6, §C.4] for a detailed discussion.
The rank of a Jacobian
2.2
In this section, we will properly define rank J(Q).  We obtain the following commutative diagram with exact rows from the diagram (**) above: to obtain the short exact sequence 0 → J(Q)/2.J(Q) → Sel(²)(Q, J) → Ш(Q, J)[2] → 0,
where Ш(Q, J)[2] is the 2-torsion of the Shafarevich-Tate group Ш(Q, J). In what follows, we give a sketch of the proof to the important fact that the 2-Selmer group Sel( 2 )(Q, J ) is finite group. For a detailed and general proof, we refer to [6, p. 282, Proposition C .4.21. Definition 2.2.1. Let J be the Jacobian of a hyperelliptic curve C/Q. Let v be a place of Q and let Iv ⊂ G = Gal(Q/Q) be an inertia for v. Consider the 2-torsion subgroup J [2] of J (Q). We say that a cohomology class φ ∈ H 1 (Q, J[2]) is unramified at ν if the restriction map res : H¹(Q, J[2]) → H¹(I v, J[2]) For a finite set S of places of Q, we denote by H 2 (Q,J[2]) the subgroup of is trivial on φ. H 1 (Q, J[2]) consisting of cohomology classes that are unramified at all places not in S. Theorem 2.2.2. Let C : y 2 = f ( x ) be a hyperelliptic curve with f ( x ) ∈ Q [ x ] squarefree, and let J be its Jacobian. Then the 2-Selmer group Sel(2)(Q, J) is a finite group of exponent 2. Proof. Let S be the set
It can be shown that S contains the primes of bad reduction of J together with ∞ and 2 (see the statement of [6, C.4.2]). Then the strategy is to prove that the subgroup H 2 (Q, J[2]) of H 1 (Q, J[2]) containing cohomology classes which are unramified outside S is finite, and then to see that Sel(2)(Q, J) is contained in H ξ(Q, J[2]) Since J[2] is the kernel of the isogeny J(Q) [2] , J(Q), it is known to be finite. Hence we have a continuous action of G on the finite set J[2]. This implies that G contains an open subgroup that acts trivially on J[2]. Since the open subgroups of G are of the form Gal(K/Q) where K is a finite extension of Q, we conclude that there exists a finite extension K/Q such that Gal(K,Q) acts trivially on J[2]. Let G K be the absolute Galois group of K . An instance of Example 1.4.7 yields an exact sequence
Since res( H ξ (Q, J[2])) ⊆ res( H ξ (G K , J[2])), this shows that proving H ξ (G K , J[2]) is finite would imply H ξ (Q, J[2]) is finite. Hence, we can replace Q by K in the previous paragraph and assume that G acts trivially on J[2]. By Remark 1.4.5, this implies that
H 1 (Q, J[2]) = Hom(G, J[2])
Note that by definition, any element of J[2] is killed by 2. This implies that the elements of Hom(G,J[2]) correspond to finite Abelian extensions of Q whose Galois group has exponent 2 by means of Kummer Theory (see [13]). Hence, the elements of H ξ (Q, J[2]) correspond to finite Abelian extensions of Q of exponent 2 that are unramified outside of S. But it is known that there are only finitely many maximal Abelian extensions of exponent 2 that are unramified outside S, see [6, p. 265, Corollary C.1.8]. Thus, it follows that H 3 (Q, J[2]) is a finite set. Now, let φ ∈ Sel ( 2 ) (Q, J ) and let v be a place not in S. Then v is a prime p̸= 2, at which J has good reduction J. Furthermore, the reduction map
J(Q)[2] → J(Fp)
is injective; see [6, p. 263, Theorem C.1.4]. One can show that this implies that φ is unramified at v and hence, φ ∈ H ξ(Q, J). Finally, it follows by definition of Sel ( 2 ) (Q, J ) that it is a group of exponent 2. □ Having the exact sequence (***), this important theorem has two immediate finiteness conse- quences. The first one is that the 2-torsion III(Q, J)[2] of the Shafarevich-Tate group is finite (cf. Remark 2.1.2). And the second one is Corollary 2.2.3 (Weak Mordell-Weil). The group J(Q)/2.J(Q) is a finite group. The Weak Mordell-Weil Theorem together with the theory of Néron-Tate heights (see [6, B.4]) implies the Mordell-Weil Theorem: Theorem 2.2.4 (Mordell-Weil). J(Q) is a finilely generaled Abelian group and we have a decom- position J(Q) ⇒ Trank J(Q) ⊕ tors(J(Q)), where tors(J(Q)) is a finite group containing elements in J(Q) of finite order, and rank J(Q) is called the rank of J/Q Proof  See [3, §5, pp. 158-161]. ● Remark 2.2.5. The proof of Mordell-Weil nevertheless does not give a recipe to compute the rank So it still remains a mysterious invariant of the Jacobian.
2.3
Bound on the rank
Going back to the exact sequence (***), it is now clear that the members of the sequence all have a structure of an F 2 -vector space. Therefore, dim F 2 J(Q)/2J(Q) + dim F 2 II(Q, J)[2] = dim F 2 Sel(2) Moreover, if r = rank J(Q), then Mordell-Weil implies that J(Q)/2J(Q) = (Z/2Z)r ⊕ tors(J(Q))/2 tors(J(Q)) On the other hand, tors( J(Q))/2 tors( J(Q)) ≒ J(Q)[2] Therefore, dim F 2 J(Q)/2J(Q) = rank J(Q) + dim F 2 J(Q)[2] Thus, we have proved Theorem 2.3.1. Let C/Q be a hyperelliptic curve and let J/Q denote its Jacobian. Dropping the subscripts F2 from the notation, we obtain rank J(Q) ≤ dim Sel (2) (Q, J) − dim J(Q) [2]
In this chapter, our aim is to find a practical description of the 2-Selmer group, which allows us to have an upper bound on dim Sel( 2 ) (Q, J). Our main reference throughout this chapter is M. Stoll's article, [23].
3.1
Not at ion
We fix some notation and give definitions that will be used throughout the thesis. We want to restrict attention to the hyperelliptic curves C/Q, given by y 2 = f(x) such that f(x) E Z[x] is monic, squarefree and of odd degree. However, we remark that M. Stoll also treats the case where f ( x ) is of even degree. See [ 23 , §5 ]. Observe that since f ( x ) is of odd degree, C has a unique point at infinity that is Q-rational. We denote this point by ∞. For any field extension K of Q , we consider the algebra LK = K [ T ]/( f ( T )) defined by f ( x ) and write it as a product of finite field extensions of K as follows: L K = L K,1 × · · × L K,m K ! where m K is the number of irreducible factors of f ( x ) in K [ x ] and each L K,m, corresponds to an irreducible factor of f ( x ) in K [ x ] via the Chinese Remainder Theorem. The notions like the ring of integers, the group of fractional ideals and the class group are available for the algebra LK ; namely, we define
Moreover, for an element α = (α 1 , . . ,α mk ) ∈ L K , we define the norm N K (α) of α by where, for each i = 1,...,mK, NLK,i : LK,i → K is the usual field norm.
F    Finally, we remark that all the dimensions in this thesis are taken over the finite field F2.
3.2
A more concrete description of the 2-Selmer group
The following group will replace the rather abstract cohomology group H 1 (Q, J[2]): Definition 3.2.1. We put H K = ker( NK : L K /( L K )² → K × /( K × )²), where the map NK is induced by the norm map we have defined above. At this point, we want to make some simplifications on the notation. When K = Q, we will omit the field from subscripts, and if K = Q v for a place v of Q, we will just use the subscript v. With this convention, when we write L , for example, we actually mean L Q = Q[T]/(f(T)).  Or, will be writing H v = ker(N v : L x /(L x ) 2 → Q e x /(Q e x ) 2 ) . This convention is to apply to any object in this thesis that has a field as subscript. Let Div 1 denote the degree 0 divisors on C with support disjoint from the support of the principal divisor div( y ). Note that the support of div( y ) contains the points with coordinates (α,0), where α is a root of f ( x ), and the point at infinity. Then for any extension K of Q, we have a homomorphism
where θ is the image of T under the reduction K[T] → K[T]/(f(T)). More precisely, θ is an mk-tuple (θ 1 , . . . ,θ mk ) with each θ i being a root of the i th irreducible factor of f ( x ). Hence the map FK is given by Theorem 3.2.2. The map FK above induces a homomorphism δκ : J(K) → HK, with kernel 2J(K). The induced injective map J(K)/2J(K) → Hk will be denoted by δκ as well. Proof. The proof essentially relies on two lemmas. The first one shows that the function FK is a well-defined map which maps to L × /( L × ) ² . This part makes use of Weil's Reciprocity and is not very technical. And the second one shows that any point in J(K) can be represented by a divisor from Div 1 ( C )( K ). For details, see [16, Proposition 3.3] or [17, Lemma 2.1 and Lemma 2.2]. E □ Schaefer's treatment in [16] is much more general, though. The following theorem finally establishes the important isomorphism H ≥ H 1 (Q, J[2]). Theorem 3.2.3. For all extensions K of Q, there exists a natural isomorphism H K ≒ H 1 (K, J[2]) Moreover, δκ from Theorem 3.2.2 above composed with this isomorphism is the coboundary mor- phism
from §2.1. Proof. Let K be an extension of Q. We want to relate the two seemingly different groups H K and H 1 ( K, J [2]). To this end, assume that deg( f ) = d and remember that we assume d to be odd. Let {α 1 , . . . ,α d} be roots of f ( x ) in K . Then Theorem 1.3.1 implies that the classes of the divisors D 1 = (α1,0) − generate J[2]. Define L to be the algebra K[T]/(f(T)) and Z to be the algebra R[T]/(f(T)). So we have
Let µ²( Σ ) denote the 2 nd roots of unity in Σ. Clearly, (L) ≈ {±1} d   Let e2 : J[2] × J[2] → {±1} denote the Weil pairing on J[2]. We define a map via
for all D ∈ J[2] ≈ Pic 0 (C)[2]. Then, by functoriality of Galois cohomology, w induces a map w* : H 1 (K, J[2]) → H 1 (K, μ 2 (L)] On the other hand, we have an isomorphism k : H 1 (K,μ2 (L)) → L×/(L×) 2 Indeed, since Z is a finitely generated K -algebra, a generalised version of Hilbert's 90 implies that H 1 (K, T × ) = 0 (see [19, p. 152]). Therefore, k is a Kummer isomorphism which is obtained from the long exact cohomology sequence associated to the following short exact sequence:
Now, we want to prove that the composition maps to HK and gives rise to a group isomorphism between H 1 (K,J[2]) and its image H K . To this end, we first want to see that the sequence
is exact, where N denotes the norm map. The map w is injective by non-degeneracy of the Weil pairing and the norm map N is clearly surjective. Note, moreover, that dim F2 J[2] = d − 1, dim F2 μ2( L ) = d, and dim F2 μ2( K ) = 1, where the first equality follows from Theorem 1.3.1. Hence, to prove the exactness at the middle, it is enough to show that N ০ w = 1. So let D ∈ J [2]: by Theorem 1.3.1 and the additivity of the Weil pairing, we get
Therefore, and the sequence (*) is exact. Hence Theorem 1.4.3 yields a long exact cohomology sequence:
H 1 (K, μ 2 (K)) is induced functorially by μ 2 (L)  → μ 2 (K) where H 1 (K, μ 2 (L)) But we have the following Kummer isomorphisms: H 1 (K, μ 2 ( L )) ≥ L × /( L × )² and H 1 (K, μ 2 ( K )) ≥ K × /( K × )² . −1) ∈ μ 2 ( L ) is mapped to −1 as d is odd. So the norm map N is surjective Moreover, (−1,..., and hence H 1 (K, J[2]) ≈ ker( N ∗ ), as desired. Therefore, we have proved the first assertion of the theorem. For the second assertion, one can prove that the maps FK and k o w o δ coincide as injections from J(K)/2J(K) to L×/(L×)², where δ is the coboundary morphism. Since by Theorem 3.2.2, F K induces δ K , this can be shown to imply the desired result. For details and general case, refer to [16, Theorem 2.3] or [17, Theorem 1.2]. ● Suppose for simplicity that L is a number field; i.e. f ( x ) is irreducible over Q. Suppose α ∈ L with Nξ (α) = 1. Then, if the minimal polynomial of α over Q has degree d and constant term a d , we have Nξ(α) = (−1) d ad = 1. Now, write
where, as usual, m v is the number of irreducible factors of f ( x ) in Q v [ x ] and L v,i /Q v are finite field extensions for i = 1, . . . , m v . We then have a natural inclusion map
which is induced by the inclusion Q → Q v . Assume that the minimal polynomial of α ∈ L v , i over Q v has constant term C i ∈ Q v for all i = 1,...,m v so that N Q v ,i (α) = (−1)[Lv,i:Qv]C i . Then
Therefore, we have obtained a map of groups H = ker( N : L × /( L × )² → Q × /( Q × )²) → H v = ker( N v : L x /( L x )² → Q x /( Q x )²). This reasoning carries over to the case where L is an algebra; i.e., to the case where f ( x ) is only assumed to be squarefree. We conclude that for any place v of Q, the inclusion Q → Qv induces a map
Remember that we have seen in Theorem 3.2.2 that there are induced maps δν : J(Q v ) → H v We obtain Theorem 3.2.4. The 2-Selmer group of J over Q can be identified to
where res v : H → H v is the restriction map defined above. Proof. By Theorem 3.2.3 we can form the following commutative diagram with exact rows: where the products are taken over all the places v. Assume ξ ∈ ker(φ) = Sel 2 (Q, J). By commu- tativity of the right square, this is equivalent to requiring that By exactness of the second row, this is the same as requiring that Since 2J(Qv) is in the kernel of δv, this holds if and only if res v (ξ) ∈ δv (J(Qv)) for all places v, as needed
3.3
A finite description of the 2-Selmer group
Although the new description of the 2-Selmer group we gave in Theorem 3.2.4 seems fairly less abstract than the original definition, it still takes us to check infinitely many conditions to determine if an element ξ ∈ H belongs to Sel(Q, J ), simply because there are infinitely many places v of Q. In this section, however, we will be able to reduce the set of places to consider to a finite set S . It is not a coincidence that S will turn out to be the same S that appeared in the statement of Theorem 2.2.2. Write L = Q[T]/( f (T)) = L 1 × . . × L m , where L i /Q are number fields for i = 1, . . , m . Recall that we have defined the group of fractional ideals of L as follows:
where I(Li) is the usual group of ideals of Li for each i = 1,...,m. Let v be a finite place of Q; i.e., v = p is a prime, and let Ip(L) denote the subgroup of I(L) consisting of ideals with support above p . In other words, I p ( L ) is the subgroup of I ( L ) generated by ideals of the form p1 × ··· × pi × ··· × pm where, for each i = 1,..., m, p i is a prime ideal of L i above p. Therefore, a typical element of Ip(L) can be written as where, for each j = 1, . . . , m, p j,i is a prime ideal of Li above p for a . , g j and n j j i E Z For any prime p, the norm of an element of Ip(L) is given by This motivates the following definition: Definition 3.3.1. Let L be as above and p be a prime. We de I p = ker(N : I p (L)/I p (L)² → I p (Q Observe that for any prime p, we have an isomorphism (see [22, Theorem 2.14]) Ip(L) = I(Lp) To see this, suppose for simplicity that L is a number field. Let p be a prime number and assume p 1 , . . . , p n are primes of L over p . Then we have I(Lp) = I(L &Q Qp) ≥ I(Lp,) × ··· × I(Lp,) Therefore, this isomorphism assigns to an ideal J ∈ I p ( L ) the n -tuple ( J p , , , , , , , , , , , ) of localisations of I at the various prime ideals of L above p . One then generalises this reasoning to an algebra L . Definition 3.3.2. For any prime p, denote by the map induced by L ∞ → I(L p ) that sends α ∈ L ∞ to the ideal (α) ∈ I(L p ) it generates. Via the identification Ip(L) ~ I(Lp), it is clear that the map val is well-defined. The following is the last definition of this section: Definition 3.3.3. LetL = Q[T]/(f(T)) be as above. (i) For a finite set S of places of Q, we put that is, IS(L) is the group generated by ideals in I(L) with support above S \ {00}. (11) We define val : H → I(L)/I(L)² The map val is induced by the map associating to an element α ∈ H the principal ideal (α) ∈ I(L) it generates. In the following lemma, we present some useful dimension results for various spaces we have defined so far. Lemma 3.3.4. Let p be a prime and consider the p-adic field Q p . Let d p be as in Theorem 1.3.4. We have (i) dim H p = 2 dim J(Q p )/2J(Q p ) = 2(m p − 1 + d p g), (ii) dim I p = m p − 1. Proof. (i) If M is a p-adic local field; that is, if it is a finite extension of Q p for a prime p, then we know that (see [20, Chapter II, Corollary to Theorem 3.3]) dim M×/(M×)² = 2 + d p [M : 0 where [ M : Q p ] is the degree of the extension M/Q p . Recall that d p = 0 if p̸= 2 and d p = 1 if p = 2. Since L p is the product of m p -many p -adic local fields, we obtain dim L ∞ /( L ∞ )² = 2m p + deg( f )d p Moreover, as deg( f ) = 2g + 1 is odd by assumption, the norm map N p : L p /(L p ) 2 → Q x /(Q x ) 2 is surjective. Thus, dim H p = dim L p /(L p )² − dim Q p /(Q p )² = 2m p + (2g + 1)d p − 2 − d p = 2(m p − 1 + d pg), as required. Finally, this is equal to 2dim J(Qp)/2J(Qp) by Theorem 1.3.4. (ii) Remember we noted that I p (L) ≥ I(L p ) ≥ I(L p 1 ) × · · × I(L p n ), where p s 's are the prime ideals of L lying above p . But the number of such primes is precisely m p Therefore dim I p (L)/I p (L)² = dim I (L p )/I (L p )² = m p On the other hand, the norm map N : Ip(L)/Ip(L)2 → Ip(Q)/Ip(Q)2 is clearly surjective. There- fore, dim I p = dim I p (L)/I p (L)² − dim I p (Q)/I p (Q)² = m p − 1 as needed. □ Let p be a prime. Recall that we have defined the maps δρ : J(Q p )/2J(Q p ) → H p (see §3.2) and val p : H p → I p . Therefore, one can form the sequence
and study if it is exact or not. The following technical lemma, the proof of which is omitted due to its being so, will immediately imply that this sequence is actually exact for almost all primes p Lemma 3.3.5. Suppose that p is an odd prime such that p2 does not divide disc(f). Then the composition
is the trivial map Proof  See [23, Lemma 4.5]. □ Corollary 3.3.6. If p is an odd prime such that p2 does not divide the discriminant disc(f) of f, then the sequence is exact. Proof. We know that the map δ p is injective and val p is surjective. We also know from the previous lemma that val p δ p = 0 Moreover, since p is odd, Lemma 3.3.4 together with Theorem 1.3.4 implies dim H p = 2(m p − 1) = dim J(Q p )/2J(Q p ) + dim I p This proves the exactness at H p and concludes the proof.
The above corollary will be crucial in simplifying the description of the 2-Selmer group to a finite set S. Before we state this simplification as a theorem, we want to study a commutative diagram which will be helpful in proving it. Suppose that L is a number field. Note that there exists an isomorphism where the product runs over all primes p . This isomorphims induces an isomorphism
Moreover, via the natural identification we will not distinguish between the elements of I(L)/I(L)² and Π n Ip(L)/Ip(L)². Therefore, we can form the following commutative diagram:
(*) where the products are over all the primes p. The commutativity of the left square is clear by the isomorphisms H ≥ H 1 (Q, J[2]) and H p ≥ H 1 (Q p, J[2]) we obtained in Theorem 3.2.3. And the commutativity of the right square follows from the isomorphism we have defined above and the identification I n ( L ) ≥ I ( L p ). Note that whenever the products are taken over odd primes p such that p 2 does not divide disc( f ), the sequence at the bottom is exact by Corollary 3.3.6 We have also proved Lemma 3.3.7. We have where p runs over all the primes. We finally obtain the following practical description of the 2-Selmer group: Theorem 3.3.8. Let S = {∞, 2} ∪ {p | p² divides disc(f)}. Then Sel(²)(Q, J) = {ξ ∈ H | val(ξ) ∈ I S (L)/I S (L)² and res v (ξ) ∈ δ v (J(Q v )) for all v ∈ S}. Proof. Recall that from Theorem 3.2.4, we have Sel 2 (Q, J) = {ξ ∈ H | res v (ξ) ∈ δ v ( J(Q v )) for all places v }. Call the set given in the theorem A. We must show that A = Sel(2), First, assume that ξ ∈ Sel ( 2 ) ( Q , J ) . This means that
for all places v.  This implies in particular that res,(ξ) ∈ δν(J(Qv)) for all v ∈ S.  So, to show that ξ ∈ A, we only need to see that val(ξ) ∈ Is(L)/Is(L) 2 . By Lemma 3.3.7, this is equivalent to showing that
has support above S ; i.e. that if p f S , then val p res p = 0. But we know that whenever p f S , the sequence at the bottom of the diagram ( * ) is exact. Hence, since res v ( ξ ) ∈ δ v ( J ( Q v )) for all places v, we get val p res p (ξ) = 0 if p ξ S. Thus, ξ ∈ A and Sel (2) (Q, J) ⊆ A. Conversely, assume that ξ ∈ A. Then to show that ξ ∈ Sel ( 2 ) (Q, J ), we only need to show that whenever p ξ S, res p (ξ) ∈ δ p (J(Q p )) That ξ ∈ A implies that val(ξ) ∈ I S (L)/I S (L) 2 , which implies that val p res ρ (ξ) = 0 as p ξ S . Hence by Corollary 3.3.6, it follows that res p (ξ) ∈ δρ (J(Qρ)). Thus, ξ ∈ Sel (2) (Q, J) and A ⊆
3.4
A first bound on the dimension of the 2-Selmer group
Based on the new description of the 2-Selmer group we gave in Theorem 3.3.8, we derive a first bound on dim Sel( Q, J ), hence on rank J ( Q ). We continue to consider L as before. Moreover, throughout this section,
as above. Definition 3.4.1. For any prime p E S, we put Jp = 6p Hp, Gp = valp(Jp) ⊆ Ip, and de fine Notice that when p & S, we have val p &p(J(Qp)) = 0 by Lemma 3.3.5. We approximate Sel(2)(Q, J) by
Lemma 3.4.2. We have
Moreover, Sel is a finite group of exponent 2 and hence, has a structure of a finite-dimensional F2-vector space. Proof. Let ξ ∈ Sel ( 2 )(Q,J).  Then in particular, res p (ξ) ∈ δ p (J(Q p )) for all p ∈ S \ {∞}. val p res p (ξ) ∈ val p δ p (J(Q p )) for all p ∈ S \ {∞}. Since val p res p (ξ) = 0 for all p ∈ S , this means that val(ξ) ∈ G, as needed. Moreover, since it is the image of a group under a homomorphism, G p is a group for each p ∈ S \ {∞}. Also, since G p ⊆ I p , it is a finite group by Lemma 3.3.4. Hence, since G is a finite product of G p 's, it is a finite group.  But we have Sel = val − 1 ( G ) and val is a group homomorphism. Thus, Sel is a finite group. Finally observe that Sel ⊂ H, which implies that Sel is of exponent 2 ● as H is. In the proof of the following important lemma, we show that the containment above is a proper one: Lemma 3.4.3. We have dim Sel ( 2 ) < dim Sel Proof.  Consider the norm map
By Theorem 3.4.2 above, Sel( 2 )(Q,J) ⊆ Sel and by definition, we have Sel( 2 )(Q,J) ⊆ ker( N ). Therefore, Sel(²)(Q, J) ⊆ Sel ∩ ker(N) ⊆ Sel. It is clear that −1 ∈ Sel as val(−1) ∈ G . On the other hand, we have N(−1) = (−1)des(f) = −1 as deg( f ) is odd. This means that −1 f ker( N ). Hence we have found an element, namely −1, in Sel which is not in the intersection Sel ∩ ker( N ). Thus, the second containment in ( * ) is indeed proper. Since Sel is finite over F 2 by the above lemma, this implies that dim Sel(²) (Q, J) ≤ dim (Sel ∩ ker( N )) < dim Sel yielding in particular
as needed □ We derive an upper bound on dim Sel ( 2 ) (Q, J ) by finding the dimension of the larger group Sel To this end, consider the subgroup W 0 = ker(val : L×/(L×)² → I(L)/I(L)²) ⊆ L×/(L×)² where the val map is induced by the maps on each "component" Li of the algebra L. Since 1 E G, it is obvious that Wo E Sel. Moreover, with the following lemma, the dimension of W 0 turns out to be easy to compute. Lemma 3.4.4. There is an exact sequence Proof. Apply the Snake Lemma to the diagram to get an exact sequence from which we extract the desired short exact sequence. Therefore, we get dim W 0 = dim O ξ /( O ξ )² + dim Cl( L )[2] Secondly, we let W = G ∩ val(L×/(L×) ⊆ I(L)/I(L)². We have Lemma 3.4.5. In the above set-up, we have (i) dim Sel = dim W 0 + dim W , (ii) W = ker(G → Cl(L)/2Cl(L)) Proof. (i) Consider the following group homomorphism Sel ⇒ W = G ∩ val( L × /( L × ) ² ), ξ → val(ξ) The map ϕ is surjective by definition of Sel and by surjectivity of val onto val( L × /( L × )²). We claim that W 0 = ker(ϕ). Notice first that the inclusion ker(ϕ) ⊆ W 0 is clear. To show that W 0 ⊆ ker(ϕ), suppose ξ ∈ W 0 . This means that val(ξ) = 0. Looking at its image under the right-most vertical map in the commutative diagram (*) from the previous section, we see that at each p ∈ S \ {∞}, val(ξ) maps to 0 in I p (L)/I p (L) 2 . This means, by commutativity of the diagram (*), that val p res p (ξ) = 0 for all p ∈ S \ {∞}. But obviously for each p ∈ S \ {∞}, 1 ∈ δρ(J(Qρ)) as δρ(J(Qρ)) is a group. Thus, val(ξ) ∈ G. This implies that ξ ∈ ker(ϕ) and hence Wo ⊆ ker( ϕ ). Therefore, it follows that dim Sel = dim ker(ϕ) + dim im(ϕ) = dim W 0 + dim W 1 as needed. (ii) First we claim that W ⊂ ker( G → Cl( L )/2Cl( L )). So let val( ξ ) ∈ W be arbitrary where ξ ∈  Sel.  Since val(ξ) is the class of the principal (ξ) ideal generated by ξ, we see that val(ξ) ∈  ker(G → Cl(L)/2 Cl(L)), as claimed. Conversely, if I ∈ ker(G → Cl(L)/2Cl(L)), then I is the square of a principal ideal, hence is a itself principal ideal, say, (ς). Then ζ ∈ Sel with val(ς) = ∑ and hence ∑ ∈ G ∩ val( L × /( L × ) 2 ). Thus, ker(G → Cl(L)/2Cl(L)) ⊆ W The following number-theoretical lemma is the last result we prove in this section before the main theorem. Lemma 3.4.6. We have dim O ξ /(O ξ )² = m ∞ Proof. Let L = L 1 × · · × L m as before. Let r be the sum of the numbers of real embeddings of all the number fields L i in L and s the sum of the numbers of pairs of complex embeddings of L i in L . We begin by observing that m ∞ = r + s . To see this, factorise f into its irreducible factors in R[x]: f(x) = f1 . . . fr+1 . . . fm∞ . where f i are of degree 1 for i = 1,...,r and are of degree 2 for i = r + 1,...,m ∞  Hence, deg f = r + 2(m ∞ − r). Since deg(f) = r + 2s, this gives that m ∞ − r = s. Note that Dirichlet's Unit Theorem (see [12, Chapter 1, §7]) is applicable to the algebra L as Thus, we obtain where r + s − 1 is called the rank of O ξ . Since O ξ [2] = μ 2 (L) = {±1}, we obtain
from which the result follows. □ The following is the main theorem of this section: Theorem 3.4.7 (Stoll). One has dim Sel 2 (Q, J) ≤ m ∞ − 1 + dim Cl(L)[2] + dim ker(G → Cl(L)/2 Cl(L)). Proof.  By Lemma 3.4.4 and Lemma 3.4.6, we get dim W 0 = dim O ξ /( O ξ )² + dim Cl( L )[2] = m ∞ + dim Cl( L )[2]. Hence, using Lemma 3.4.3 and Lemma 3.4.5, we get
= m ∞ − 1 + dim Cl(L)[2] + dim ker(G → Cl(L)/2 Cl(L)), as desired. □ Corollary 3.4.8. Let C : y 2 = f(x) be a hyperelliptic curve with f(x) E Q[x] monic, squarefree and of odd degree.  One has the following bound on rank J(Q): rank J(Q) ≤ m ∞ − m + dim Cl(L)[2] + dim ker(G → Cl(L)/2 Cl(L)), where m is the number of irreducible factors of f(x) in Q[x]. Proof  This directly follows from Theorem 2.3.1 and Theorem 3.4.7.口29
We describe the idea in the article [4] of Harris B. Daniels, Àlvaro Lozano-Robledo and Erik Wallace. They obtain stronger bounds on dim Sel( 2 ) (Q, J ) for certain special classes of hyperelliptic curves.
4.1
Adding a condition at infinity
Similarly to the previous chapter, we approximate Sel( 2 ) (Q, J ) by a larger subgroup of L × /( L × ) 2 whose dimension is easier to estimate. Recall that we have defined for all p ∈ S \ {∞}, Jp = δρ(J(Qp)). For the place at infinity, we define in a similar way:
Recall also the maps res ∞ : H → H ∞ and val : H → I(L)/I(L)² we have defined in the previous chapter. Finally, for S = {∞, 2} ∪ {p | p² divides  disc( f )}, recall that we have defined G p = val p (J p ) and put
With the notations as above, we now put Sel = {ξ ∈ L×/(L×)² | val(ξ) ∈ G,  res∞(ξ) ∈ J∞} Clearly, one has Sel = Sel ∩ res − 1 ( J∞ ) Being the intersection of the finite subgroup Sel of exponent 2 with the subgroup res 2 (J ∞ ) of L×/( L×) 2 , the above equality shows that Sel is itself a finite group of exponent 2 and hence has the structure of a finite-dimensional F 2 -vector space. Moreover, we have Lemma 4.1.1. One has the following containments: Sel (2) (Q, J) ⊆ Sel ⊆ Sel Proof. As we noted above, Sel = Sel ∩ res − 1 (J∞). So the second containment is clear. Also, having already proved in Lemma 3.4.2 that Sel( 2 ) (Q, J) ⊆ Sel, to prove the first containment, it is enough to prove that res ∞ (ξ) ∈ J ∞ whenever ξ ∈ Sel ( 2 ) (Q, J ). But this obviously follows from the definiton of Sel(2)(Q, J) in Theorem 3.2.4. ● Since Sel consists of elements in Sel satisfying an extra condition at infinity, we can rightfully expect the upper bound on dim Sel( 2 )( Q , J ) obtained by approximating by dim Sel to be more precise.
4.2
Restriction map at infinity
In this section, we will look more closely at the map res ∞ and see that it actually agrees with the signature map sgntr We restrict ourselves to the case where f ( x ) has n ≥ 3 real roots and is of degree n. In this case, the algebra L defined by f(x) is said to be totally real. So if we write 2 L 1 × · · × L m then L is a totally real number field for each i = 1, an embeddings of the algebra L, via τ i (α) = τ i (α 1 ,α 2 ,α . . . ,α m ) := τ i (α j ), where (α 1 , . . . ,α m ) ∈ L , and τ i : L j ↔ R is an embedding of the real number field L j with α j ∈ L j Note that given α = (α 1 , . . . ,α m ) ∈ L , for each i = 1 , . . , n , there exists a unique α j ∈ L j such that τ is an embedding of L j . So the map is well-defined. In particular, for each i = 1,...,m there are [ L i : Q]-many real embeddings of L i among all the embeddings τ 1 , . . . , τ n so that the number of all embeddings τ 1 , . . . , τ n of the algebra L . Finally, we can, and do, order the embeddings τ 1 , . . . , τ n in accordance with the ordering of the number fields L 1 , . . . , L m Since f has all real roots and is of degree n, we have as R-algebras. Also,
Hence, by definition, (R n )× /((R n )× )² → {±1}), 。: H = ker(N : L × /(L × )² → Q × /(Q × )² ) → H ∞ = ker(N ∞ where N ∞ is induced by the norm map Suppose α = (αι, . so that as required. We extend the domain of res ∞ to L × /( L × ) 2 and still call the resulting map res ∞ That is, we have defined res。 : L × /(L × )² Consider now the signature homomorphism given by We obtain Proposition 4.2.1. Il follows that res ∞ agrees with sgntr in the sense that for any α ∈ L × /(L × ) 2, ifã E L× denotes its lift to L×, then res∞ (α) = sgntr(∂ Proof. Let α ∈ L × /(L × )² be arbitrary. Note first that if α and α are two different lifts of α to L×, then sgntr(ā) = sgntr(ā). So, if i ∈ {1, . . . , n} is arbitrary, then we want to show that
But τ i (α)  mod ( R × )² ∈ {±1}. Hence this is obvious by a sign analysis. Now, let Ož,+ ⊆ Ož be the subgroup of totally positive units of O L . These are by definition the elements α ∈ O ξ such that ri(a) > 0 for all i = 1,...,n. From Proposition 4.2.1, it follows that
Definition 4.2.2. We put and j∞ = dim ( res∞ (O ξ /(O ξ )²) ∩ J∞ ) In particular, if every totally positive unit of L is a square in L, then we have ρ∞ = 0. Proposition 4.2.3. We have dim ( O ξ /( O ξ )² ∩ resﾆ¹ ( J∞ )) ＝ J∞ + P∞ Proof.  Considering the surjective group homomorphism res∞ | o ∈ /(o ∈ )² : O ∈ /(o ∈ )² ∩ res− 1(J∞ ) → res∞ (O ∈ /(O
Remark 4.2.4. In [4], the above proposition and construction is only done for number fields Hence, we have generalised this to algebras. This will allow us to generalise the refinement of the bound on the rank they obtained in [4, Proposition 2.25] to algebras as well. See Theorem 4.5.1.
4.3
A new bound on the dimension of the 2- Selmer group
As a piece of notation, we begin by putting Lj∞ = res∞ (J∞) Sel = {ξ ∈ L j ∞ | val(ξ) ∈ G } I(L)/I(L)² and compute the As in §3.4, we will define certain subspaces W 0 ⊆ L J ∞ and W dimension of Sel with the help of these subspaces. Indeed, we put
and
W = G ∩ val( L J ∞ ) ⊆ I( L )/ I( L )².
Observe that the subspaces W 0 and W are analogues of the ones we have defined in §3.4. Observe also that dim( W 0 ) = dim( W 0 ∩ L J ∞ ) ≤ dim( and dim( W ) = dim( W ∩ val( L J ∞ )) ≤ dim( W ) The following two lemmas should therefore look familiar: Lemma 4.3.1. Let the subspaces Wo and W be defined as above. dim Sel = dim W 0 + dim Proof. Consider the group homomorphism Sel → G ∩ val( L J ∞ ), ξ → val(ξ) We claim that W 0 = ker( β ). First observe that φ is surjective by construction. Moreover, the containment ker( β ) ⊆ W 0 is a triviality. Conversely, if ξ ∈ W 0 , to show that ξ ∈ ker( β ), we only need to show val( ξ ) ∈ G. But we have already showed this in Lemma 3.4.5. Thus, we conclude that ker( β ) = Wo ● Before we state the next lemma, we need to introduce some notation. Let U = (O ξ /(O ξ )²)∩ L J ∞ and Cl(L J ∞ ) = I(L)/P(L J ∞ ), where P(L J ∞ ) is the group of principal fractional ideals 21 = (α) with a generator α ∈ L such that res∞(α) ∈ J∞. Moreover, we put Cl(L J ∞ )[2] = {[2] ∈ Cl(L J ∞ ) | ≥ 2 = (α) for some α ∈ L Note that then we have an exact sequence 0 → O ζ ∩ resﾆｭ(J∞) → L × ∩ resﾆｭ(J∞) → I(L) → Cl(LJ∞) → 0. We obtain Lemma 4.3.2. There exists a short exact sequence In particular, dim W 0  = dim U + dim Cl( L J ∞ )[2]. Proof. Similarly to Lemma 3.4.4, one applies the Snake Lemma to the diagram
to get an exact sequence
which induces a short exact sequence 0 → O ξ ∩ resﾆ1 (J∞)/(O ξ ∩ resﾆ1 (J∞))² → W 0 → Cl(L J∞) Therefore, it remains to see that O ξ ∩ res − 1 (J∞)/(O ξ ∩ res − 1 (J∞))² ≒ O ξ /(O ξ )² ∩ res − 1 (J∞) To this end, consider the map O ξ ∩ res− 1 (J∞) → O ξ /(O ξ )² ∩ res− 1 (J∞), u → π Note that if u ∈ O ζ ∩ res ζ 1 (J ∞ ), then we have u ∈ O ζ and res ζ 1 ( π ) ∈ J ∞ . So the map is well-defined and is also clearly surjective. Moreover, it is also clear that the kernel of the map is ( O ξ ∩ res − 1 ( J ∞ ))². Therefore, we obtain the desired isomorphism and hence the exact sequence in the theorem. The claim on dimensions is immediate from the exact sequence. ● Let P(L) C I(L) be the subgroup of principal ideals of L, P+(L) C P(L) be the subgroup of P(L) consisting of principal ideals generated by totally positive elements and P(Lj 2 ) be as above We then have Lemma 4.3.3. One has P+(L) C P(Ljc) C P(L). In particular Cl(L)[2] ⊆ Cl(L J ∞ )[2] ⊆ Cl + (L)[2], where Cl+(L) denotes the narrow class group of L. Proof. Note that the second containment follows from the definition. So we only prove the first containment that P+(L) ⊆ P(L J ∞ ) To this end, assume that α ∈ L × is a totally positive element. Let α be its class in L × /( L × ) 2 We then have res ∞ ( ∞ ) = (1, . . , 1)  This implies that α ∈ ker(res∞) ⊆ res∞ (J∞) Thus, res(α) ∈ J ∞ and the principal ideal (α) generated by the totally positive element α ∈ L × is in P(L J ∞ )  Thus, P + (L) ⊆ P(L J ∞ ) Finally, the containments P+(L) ⊆ P(Lj ∞ ) ⊆ P(L) clearly imply the desired containments of 2-torsion subgroups of the class groups.口The following theorem is in some sense the analogue of Theorem 3.4.7 from the previous chapter. Theorem 4.3.4. We have dimSel( 2 ) (Q, J ) ≤ dim ( O ξ /( O ξ )² ∩ res − 1 (J ∞ )) + dim Cl + (L )[2] + dim G ∩ val( L J ∞ ), Proof. The results we have obtained in this section clearly imply dim Sel ( ² ) (Q, J) ≤ dim Sel = dim W 0 + dim W = dim U + dim Cl( L j ∞ )[2] + dim W dim (O ξ /(O ξ )² ∩ res − 1 (J ∞ )) + dim Cl(L J∞ )[2] + dim G ∩ val(L J∞ ) ≤ dim( O ξ /( O ξ )² ∩ res − 1( J∞ )) + dim Cl + ( L )[2] + dim G ∩ val( L J∞ ). Remark 4.3.5. The bound in the above theorem is not immediately comparable with Stoll's bound (Theorem 3.4.7). However, we observe, for example, that dim( O ξ /( O ξ )² ∩ res − 1 ( J∞ )) ≤ m∞ − 1. To see this, it is enough to find an element in OL/(OL)2 which does not belong to the intersection O ξ /(O ζ )² ∩ res ⇒ 1(J ∞ ). But res ∞ (−1) does the job. Indeed, since J ∞ is contained in the kernel of the norm map N ∞ : L ∞ /(L ∞ ) 2 → R × /(R × ) 2 , we have N ∞ (j) = 1 for any j ∈ J ∞ . However, N(res ∞ (−1)) = N(−1) = −1, as deg(f) is odd. Thus, res ∞ (−1) f res ∞ (J∞) and the result follows. ●
4.4
Totally positive units
In this section, we study totally positive units in number fields as a preliminary to the next section, where we will obtain some refinements of the bound we obtained in Theorem 4.3.4. In particular, we will be able to get the term Cl+(L) in the bound under control. Lemma 4.4.1. Let L be a totally real number field of odd degree n ≥ 3. Define ρ = dim = 2 Cl(L)/2 Cl(L), ρ + = dim = 2 Cl + (L)/2 Cl + (L), and ρ ∞ = dim = 2 (O ζ ;+/(O ζ )²). Then there exists an exact sequence 0 → {±1} n / res∞ ( O ξ / ( O ξ ) 2 ) → Cl + ( L ) → Cl ( L ) → O In particular, ρ+ ≤ ρ∞ + ρ Proof.  Recall that P(L) denotes the group of principal ideals in L and P+(L) the group of principal ideals generated by totally positive elements. We then have the isomorphisms: Cl(L) ≥ I(L)/P(L) ≥ (I(L)/P+(L))/(P(L)/P+(L)) ≥ Cl+(L)/(P(L)/P+(L)), yielding the following exact sequence relating the narrow and usual class groups:
Also, the map α → (α) that sends α ∈ L × to the principal ideal (α) it generates yields the exact sequence 1 → O ξ → L× → P(L) → 0. Let L ×,+ ⊂ L × be the subgroup of totally positive elements. By definition, P + ( L ) is the image of L×;+ in P(L). This gives an isomorphism P(L)/P+(L) ≥ L×/O÷ L×,+ So the exact sequence ( 8 ) can be wrtitten as
Now, consider the surjective homomorphism {±1} n → L×/O ξ L×,+, mapping an n-tuple (ε i ) i =1,...,n ∈ {±1} n to the class of an element α ∈ L × that satisfies sgntr(α) = (ε);=1,...,n Note that if α,β ∈ L × such that sgntr(α) = sgntr(β), then since sgntr is a group homomorphism, we have sgntr(αβ−¹) = sgntr(α) sgntr(β)−¹ = 1 so that αβ−1 ∈ O ξ L×,+. Hence the map is well-defined. Moreover, the kernel of this homomor- phism is res ∞ ( O ξ /( O ξ )² ). Hence we get an isomorphism {±1} n / res∞ (O ξ /(O ξ )²) ≥ L×/O ξ L×,+ and the exact sequence ( 8 * ) can be written as
as needed. Finally, to see that ρ+ ≤ ρ∞ + ρ, first observe that dim{±1} n / res∞( O ξ /( O ξ )²) = n − dim res∞ ( O ξ /( O ξ )²) Considering now the surjective homomorphism res∞ | o ξ /( o ξ )² : O ξ /( o ξ )² → res∞ (o ξ /(o ξ )²), since dim ( O ξ /( O ξ )²) = n and ρ ∞ = dim ( O ξ : + /( O ξ ))² is the dimension of ker ( res∞ | O ξ /( O ξ )² ) it follows that dim{±1}"/res∞(O ξ /(O ξ )²) = n − (n − ρ∞) = ρ∞ Thus, ρ + ≤ ρ ∞ + ρ by exactness of the sequence ( *** ). Proof is complete. ● The proof of the following theorem is technical. So we only give a reference for it. The corollary following it will prove important in subsequent sections. Theorem 4.4.2. Let L/Q 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 the above theorem, we have ρ = ρ +. Proof. See [14]. ● Corollary 4.4.3. Let L be a cyclic number field of odd prime degree p, and suppose that the order of 2 in (Z/pZ)* is even. Then, ρ = ρ + . In particular, dim Cl(L)[2] = dim Cl + (L)[2]. Proof. By assumption Gal(L/Q) ≥ Z/nZ for n ≥ 2. Suppose that the order of 2 in (Z/nZ)× is even.  This is equivalent to −1 being congruent to a power of 2 modulo n.  Thus result follows from the previous theorem. ● We finally state the following proposition with its proof omitted. We restrict to the case where L is a cyclic number field of odd prime p ≥ 3 degree and we let O x , 1 denote the units of norm 1. Proposition 4.4.4. Let L be as above. Suppose that the polynomial φρ(x) = (xP − 1)/(x − 1) is irreducible over F2. Then, either ρ∞ = 0, or ρ∞ = p − 1, in which case every unit in O ξ :1 is totally positive. Proof  See [4, Proposition 2.22]. □
4.5
Improvement of the rank bound
Remember that an algebra L = Q[x]/(f(x)) defined by f(x) E Q[x] is totally real if and only if f(x) has deg(f)-many real roots. Recall also that in §4.2, we have introduced the following notations: ℓ∞ = dim (Ož ;+ /(Ož )²) and j∞ = dim(res∞(Ož /(Ož )²) ∩ J∞) With notations as above, we obtain Theorem 4.5.1. Let n be an odd number and let C : y 2 = f(x) be a hyperelliptic curve with f(x) of degree n (and hence of genus g = (n − 1)/2) such that L, the algebra defined by f(x), is totally real of degree n. Then we have dim Sel (2) (Q, J) ≤ j∞ + ℓ∞ + dim Cl + (L)[2] + dim G ∩ val(L J∞ ) Moreover, (i) P∞ + j∞ ≤ n − 1. (11) J∞ ≤ g (111) One has dim Sel (2) (Q, J) ≤ j∞ + 2 ρ∞ + dim Cl(L)[2] + dim G ∩ val(L J∞ ) (iv) If G Π val(L,J,) is trivial, then dim Sel ( 2 ) (Q, J ) ≤ j ∞ + p ∞ + dim Cl + (L) [2] ≤ j ∞ + 2 ρ ∞ + dim Cl (L) [2] . Proof. We have shown in Proposition 4.2.3 that dim ( O ξ /( O ξ )² ∩ res − 1 ( J∞ )) ＝ j∞ + ρ∞ Therefore, the first bound is basically the same as the one we obtained in Theorem 4.3.4. We prove the rest one by one: (i) Since
Lemma 3.4.6 implies that ρ∞ + j∞ ≤ dim O ξ /(O ξ )² = m∞ = n, where the last equality follows from the asumption that L is totally real. Moreover, res ∞ (−1) É J ∞ as shown before in Remark 4.3.5. Hence, the result follows. (ii) Recall that we have defined J ∞ = δ∞ (J(R)/2J(R)) and we know that δ∞ is injective by Theorem 3.2.2. This yields by Theorem 1.3.5 that dim(J∞) = dim J(R)/2J(R) = m∞ Hence the containment res ∞ (O ξ /(O ξ )²) ∩ J∞ ⊆ J∞ implies ≤ dim(J∞) = m∞ − 1 − g = n − 1
as desired. (iii) By Lemma 4.4.1, we have dim Cl + ( L )[2] ≤ ρ ∞ + dim Cl( L )[2]. Thus the result follows. (iv) Clear from the previous part. ● In the following, we finally obtain a neat bound on dim Sel( 2 ) (Q, J) under the assumption that L is a cyclic number field . Theorem 4.5.2. Suppose that L is a cyclic number field of odd degree n ≥ 3, such that the order of 2 in (Z/nZ)× is even. Then we have dim Sel²(Q, J) ≤ j∞ + ρ∞ + dim Cl(L)[2] + dim ker(G → Cl(L)/2 Cl(L)). Hence, if p∞ = 0, then dim Sel²(Q, J) ≤ g + dim Cl(L)[2] + dim ker(G → Cl(L)/2 Cl(L)). Proof. Clear from Corollary 4.4.3 and Theorem 4.5.1. ● Corollary 4.5.3. Let C : y 2 = f(x) be a hyperelliptic curve such that the algebra L defined by f(x) is a cyclic number field of odd degree n ≥ 3. Then we have the following bound on rank J(Q). rank J(Q) ≤ j∞ + ρ∞ + dim Cl(L)[2] + dim ker(G → Cl(L)/2 Cl(L)). Proof. Clear from Theorem 4.5.2 above. Note in particular that L being a number field means that f ( x ) is irreducible over Q so that dim J ( Q )[2] = 0 by Theorem 1.3.1. So the above bound ● cannot be made any better using Theorem 2.3.1.
We aim to compute the rank bounds of some families of hyperelliptic curves explicitely. We shall make use of the improvements on the bounds we obtained in the previous chapter. We will also obtain a lower bound for the rank of certain hyperelliptic Jacobians whenever the curve has sufficiently many rational points. We will finally include a very short discussion about the bounds on the 2-torsion of class groups and obtain some rank bounds involving discriminants.
A family of elliptic curves
5.1
Let n ≥ 1 be an integer such that D = 3n 2 + 3n + 1 is squareefree. We consider the following family of elliptic curves: En : y 2 = fn(x) = x 3 + (9n + 6) x 2 + (9n + 3) x − 1, which is due to F. Thaine, see [25]. In this section, we will prove that rank E n (Q) can be bounded only in terms of dim Cl( L n )[2] by using Theorem 4.5.2. We remark that by a result of T. Nagel, see [11], there are infinitely many n ≥ 1 such that D = 3n² + 3n + 1 is squarefree; i.e., the above family En is infinite. Since E n is an elliptic curve, it can be identified with its Jacobian J. We have disc(f n ( x )) = (27D) ². Also, f n ( x ) is irreducible over Q for any n . So the algebra L n defined by f n ( x ) is a cubic field. Since disc( f n (x)) > 0, we conclude that L n is a cyclic cubic field. Moreover, the order of 2 in ( Z /3Z )× is 2, which is even. Hence, we can use the bound in Theorem 4.5.2. Lemma 5.1.1. In the above set-up, we have p∞ = 0. Proof. Note that fn(x) has two negative roots and one positive root and roots are all units. Therefore, for any root α of f n ( x ), we must have N ( α ) = 1. However, α is not totally positive. Hence, we have found an element in O X , 1 which is not totally positive. Moreover the polynomial φ 3(x) = (x 3 − 1)/(x − 1) = x 2 + x + 1 is clearly irreducible in F 2 . Thus, the result follows by Proposition 4.4.4. ● With the following lemma, we find the images G v = val v ( J v ) ⊆ I v to be trivial for almost all places v ∈ S = {∞, 2} ∪ { v | v 2 divides disc( f n )} . More precisely, we have Lemma 5.1.2. In the above set-up, for any p E S \ {∞, 3}, fn(x) is irreducible over Qp Proof. Let p = 2. Then, reducing the polynomial fn(x) in F2[x], we have
n ≡ 0  mod 2 n ≡ 1  mod 2 In both cases, we see that f n ( x ) is irreducible over F 2 , hence it is irreducible over Q 2 Now, let p > 3 be a prime such that p 2 divides disc(f n ( x )); that is, p divides D. We compute that f n (x − (3n + 2)) = x ³ − 9Dx + 9D(2n + 1). Moreover, we have 4D − 3(2n + 1)² = 1, which implies gcd( D, 2n +1) = 1. Since D is squarefree by assumption, we see that f n ( x − ( 3n +2)) is irreducible over Zp by Eisenstein criterion applied with prime p. Hence, it follows that fn(x) is irreducible over Q p , as claimed. We obtain the following theorem on rank E n (Q): Theorem 5.1.3. Let E n , n ≥ 1, be the family of elliptic curves defined above. We have rank E n (Q) ≤ 3 + dim Cl( L n )[2] Proof. Fix an n ≥ 1 and consider By Lemma 3.3.4, dim I n,p = m n,p = 1, where m n,p is the number of irreducible factors of fn ( x ) in Q p . From Lemma 5.1.2, it follows that when p > 3, dim I n,p = 1 − 1 = 0 and hence, G n,p is trivial. On the other hand, when p = 3, we have
which implies that dim I n .3 = 3 − 1 = 2. Therefo dimker(G n → Cl(L n ) /2 Cl(L n ) ≤ dim G n = 2. Moreover, since f n ( x ) is irreducible over Q and E n identifies with its Jacobian, we have dim E n (Q)[2] = 1 − 1 = 0, by Theorem 1.3.1. Note finally that the genus g of E n (Q) is 1. Therefore, Theorem 2.3.1 and Theorem 4.5.2 imply
as claimed. □ Remark 5.1.4. Since fn(x) has all real roots, it follows that m = 3. Therefore, if we used the bound in Theorem 3.4.7 for the above family of elliptic curves, we would obtain the bound rank E n (Q) ≤ 4 + dim Cl( L n )[2], which is weaker than the bound we found in the above theorem.  The family En : y 2 = fn(x) is therefore an explicit example illustrating how Stoll's bound in Theorem 3.4.7 can be made stronger with the  "refined estimation". Remark 5.1.5. We finally remark that the bound we found above is optimal, meaning that there are c urves En from this family salisfying that rank En(Q) = 3 + dim Cl(Ln)[2]. Below picture created using Sage shows the optimal cases for varying n in the horizontal axis. Note that by as- sumption, we must only consider n ≥ 1 such that D = 3n² + 3n + 1 is squarefree. Indeed, it turns out that among such n E [1,100], when n = 4,40,58,67,76, the bound is optimal.
5.2
Joshi-Tzermias family of hyperelliptic curves
Let p ≥ 5 be a prime. In this section, we will consider the following family of hyperelliptic curves: where d ≥ 1 is an integer and a i are distinct modulo p . This family is special in the sense that it has many rational points. This will allow us to have a bound on its Jacobian rank from below. Following theorem is key to this section. We state its contrapositive as a corollary to emphasise that it yields a lower bound on the rank of a Jacobian. Theorem 5.2.1 (Coleman's Effective Chabauty). Let C/Q be a smooth projective curve of genus g ≥ 2 and let J be the Jacobian associated with C.  Let r denote the rank of the Jacobian J. Suppose that r ≤ g − 1. Then, for any prime p ≥ 2g + 1 for which C has good reduction, which is to be denoted by C, we have
Proof. See [2].
Corollary 5.2.2. Let C/Q be a smooth projective curve of genus g ≥ 2 and let J be the Jacobian associated with C. Let r denote the rank of the Jacobian J. Suppose that, for any prime p ≥ 2g + 1 for which C has good reduction, which is to be denoted by C, we have #C(F,) + 2g − 2 ≤ #C(Q). Then we have the following lower bound on the rank r: Lemma 5.2.3. Let be a curve from the Joshi-Tzermias family and let J/Q denote its Jacobian. Let g = (p — 1)/2 be its genus. Then (ï) rank J(Q) ≥ g, (ii) If f(x) is irreducible over Q, then rank J(Q) ≥ 2g. Proof.  (i) Since for all i = 0, − 1, a i 's are distinct modulo p, the reduction ζ of ζ modulo p is given by which can be easily checked to be non-singular.  Note that every affine F p -rational point of C is of the form (a,0), where a ∈ F p . Adding the point at infinity, we get =C(F p ) = p + 1 = 2g + 2. On the other hand, the points P i = ( a i , ±pd ) is in C(Q) for all i = 0,...,p − 1. Again, with the point at infinity counted, we have But then we get 4g + 3 2g + 2 + 2g − 2 = 4g Thus, result follows by Corollary 5.2.2. (ii) See [7, Theorem 1.1].
We have Lemma 5.2.4. For a fixed prime p ≥ 5 and an integer d ≥ 1, write with a, all distinct modulo p. Then, if ao is large enough, then fao(x) has p distinct real roots. Proof. Fix j ≥ 1. We have If we choose ao so large that then we see that f a 0 ( a j + 1/2) and f a 0 ( α j − 1/2) have distinct signs for j = 1,..,p − 1. This implies that f a 0 (x) has p distinct real roots by the Intermediate Value Theorem. ● Therefore, fixing a prime p ≥ 5 and an integer d ≥ 1, we have obtained an infinite family of Joshi-Tzermias hyperelliptic curves f ( x ) ∈ Q [ x ] such that f ( x ) is of degree p and has p distinct real roots so that the algebra L defined by f ( x ) is totally real. Therefore, Theorem 4.5.1 is applicable. Thus, we obtain Theorem 5.2.5. Let C : y 2 = f(x) be a hyperelliptic curve from the family described above so that it has genus g = (p − 1)/2. Let foo, foo and L J ∞ be as defined before (see f4.1 and f4.2). Then we obtain g ≤ rank J(Q) ≤ j∞ + ρ∞ + dim Cl + (L)[2] + dim G ∩ val(L j∞ ) Moreover, the results of Theorem 4.5.1 hold true in this case. Proof. Trivial from Lemma 5.2.3.
5.3
Obtaining bounds that involve the discriminants
Let C : y 2 = f ( x ) be a hyperelliptic curve with f ( x ) ∈ Q [ x ] monic, squarefree and of odd degree such that f(x) defines the algebra L. We will see that rank J(Q) can be bounded only in terms of disc( f ) as f ( x ) ∈ Q [ x ] runs through monic and squarefree polynomials of fixed odd degree n. When L is a number field, an improvement of the Brauer-Siegel Theorem on the size of Cl(L)[2] has been made in the article [1]. Namely, if L is a number field of degree n, then one has
for some δ n > 0. Since disc( L ) ≤ disc( f ), this implies that
Recall that when L = L 1 × . . × L m is an algebra with L i 's number fields for i = 1, . . . , m , we have defined  Cl(L ) = Cl(L 1 ) × · · × Cl(L m ) so  that  dim Cl(L)[2] = dim Cl(L 1 )[2] + · · · + dim Cl(L m )[2] This implies that (*) holds when L is an algebra as well. On the other hand, we can put a "trivial" bound on dimker(G → Cl(L)/2Cl(L)) in terms of disc( f ) without having to assume L to be a number field. Indeed,
where we have used the definition of G and the fact dim I p = m p − 1 ≤ n − 1 (see Lemma 3.3.4). By definition, S contains ∞, 2 and odd primes p such that p ² divides disc( f ). But p ² | disc( f ) implies p ≤ √ [disc(f) ] . We note that the Prime Number Theorem implies
Since n — 1 is considered as a constant, we obtain
We have the following proposition: Proposition 5.3.1. Let n ≥ 1 be an odd integer, and let C : y 2 = f(x) be a hyperelliptic curve with f(x) E Q[x] monic, squarefree and of degree n. Then we have Proof. Corollary 3.4.8 and the results we have obtained above above yield
as desired.
●

[1] Manjul Bhargava, Arul Shankar, Takashi Taniguchi, Frank Thorne, Jacob Tsimerman, Yongqiang Zhao. Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves. https://arxiv.org/abs/1701.02458, [v1], Tue, 10 Jan 2017. [2] Robert F. Coleman. Effective Chabauty. Duke Math J. 52 (3) 765-770, 1985. [3] Gary Cornell, Joseph H. Silverman. Arithmetic Geometry. Springer-Verlag New York, 1986. [4] Harris B. Daniels, Álvaro Lozano-Robledo, Erik Wallace. Bounds of the Rank of the Momell- Weil Group of Jacobians of Hyperelliptic Curves. https://arxiv.org/abs/1708.07896, [v4], Thu, 11 Jan 2018. [5] Robin Hartshorne. Algebraic Geometry . Graduate Texts in Mathematics, Springer-Verlag New York, 1977. [6] Marc Hindry, Joseph H. Silverman. Diophantine Geometry, An Introduction. Graduate Texts in Mathematics, Springer-Verlag New York, 2000. [7] Kirti Joshi, Pavlos Tzermias. On the Coleman-Chabauty Bound . Théorie des nombres, Série I, 459-463, 1999 [8] Arthur Mattuck. Abelian varieties over p-adic ground fields. Annals of Mathematics 62, 92-119, 1955. [9] James Milne. Abelian Varieties. http://www.jmilne.org/math/CourseNotes/AV.pdf, unprb- lished course notes. [10] David Mumford. Tata Lectures on Theta II. Birkhäuser Basel, 2007. [11] Trygve Nagel. Zur Arithmetik der Polynome. Abhandlungen aus dem Mathematischen Semi- nar der Universität Hamburg, Vol. 1, 179-194, 1992. [12] Jürgen Neukirch. Algebraic Number Theory. A Series of Comprehensive Studies in Mathemat- ics, Springer-Verlag Berlin Heidelberg, 1999. [13] Jürgen Neukirch. Class Field Theory, The Bonn Lectures. Edited by Alexander Schmidt. https://www.mathi.uni-heidelberg.de/ schmidt/Neukirch-en/, electronic version. [14] Bernard Oriat. Relation entre les 2-groupes des classes d'idéaux au sens ordinaire et restreint des certains corps de nombres. Bulletin de la S. M. F., tome 104, p. 301-307, 1976. [15] Bjorn Poonen. The Selmer Group, the Shafarevich-Tate Group, and the Weak Mordell-Weil Group . http://math.univ-lyon1.fr/ roblot/ihp/weakmw.pdf, unpublished notes. [16] Edward F. Schaefer. Computing a Selmer group of a Jacobian using functions on the curve. Mathematische Annalen, 310, 447-471, 1998. [17] Edward F. Schaefer. 2-Descent on the Jacobians of Hyperelliptic Curves. Journal of Number Theory, Vol. 51, Issue 2, 219-232, 1995.