Selmer Groups and Galois Representations

Selmer Groups and Galois Representations Edray Herber Goins July 31, 2016

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Contents 1 Galois Cohomology 7 1.1 Notation........................................... 8 2 Galois Groups as Topological Spaces 9 2.1 Abstract Algebra Review................................. 9 2.1.1 Commutative Groups............................... 9 2.1.2 Fields........................................ 10 2.1.3 Galois Theory.................................... 10 2.2 Profinite Groups as Topological Spaces.......................... 11 2.2.1 Absolute Galois Groups.............................. 11 2.2.2 Krull Topology................................... 11 2.2.3 Profinite Groups.................................. 12 2.2.4 Example: l-adic Numbers............................. 13 3 G-Modules and X-Torsors 15 3.1 Galois Modules....................................... 15 3.1.1 Abelian Groups with Continuous Galois Action................. 15 3.1.2 Example: Additive Group G a and Multiplicative Group G m.......... 15 3.1.3 Example: Elliptic Curves............................. 16 3.1.4 Tate Modules.................................... 17 3.1.5 Isogenies of Galois Modules............................ 19 3.1.6 Sheaves of Galois Modules............................. 20 3.1.7 Principal Homogeneous Spaces.......................... 22 3.1.8 Torsors via Translation Maps........................... 24 3.1.9 Example: Quadratic Curves............................ 25 3.1.10 Example: Cubic Curves.............................. 27 3.2 Selmer s Cubic....................................... 27 3.2.1 Example: Quartic Curves............................. 29 3.2.2 Elliptic Curves: 2-Isogeny............................. 32 3.2.3 Elliptic Curves: 2-Torsion............................. 32 4 Galois Cohomology 35 4.1 Continuous Maps...................................... 35 4.1.1 Sections....................................... 35 4.1.2 Cochain Complexes................................ 36 3

4.1.3 Where are these maps coming from?....................... 38 4.1.4 Example: Free G-Modules............................. 38 4.1.5 Cohomology Groups................................ 38 4.2 Weil-Châtalet Groups................................... 39 4.2.1 Example: Hilbert s Theorem 90.......................... 41 4.3 Cup Product........................................ 41 4.4 Long Exact Sequence for Cohomology.......................... 43 4.4.1 Example: Kummer Theory............................ 45 4.4.2 Example: Elliptic Curve.............................. 45 4.4.3 Example: Weil Pairing............................... 45 4.5 Pushforward on Cohomology............................... 46 4.5.1 Example: Decomposition and Inertia Groups.................. 48 I Selmer Groups 51 4.6 History........................................... 53 4.6.1 Lind and Reichardt s Quartic........................... 53 4.7 Selmer and Shafarevich-Tate Groups........................... 56 4.7.1 Classical Definitions................................ 56 4.7.2 Elliptic Curves................................... 57 4

List of Figures 3.1 A Geometric Proof of the Associativity of the Group Law............... 18 5

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Chapter 1 Galois Cohomology 1.1 Notation k, F, K, L Fields k I G, Γ, U A separable, algebraic closure of k A partially ordered set Absolute Galois group, namely Galk/k, with composition : G G G A profinite group Open subgroup of G U, V, W Open sets in G σ, τ, ν, ω Elements of G X,, G-module with maps : X X X and : G X X P, Q, R O Y,, Elements of X Identity element of X Principal homogeneous space with maps : X Y Y and : G Y Y Q, R Elements of Y f g {Y/X} Bijection X Y from G-module to a principal homogeneous space Bijection Y Z between principal homogeneous spaces Equivalence class of principal homogeneous spaces Z Y E, E Elliptic curves φ, φ Dual m-isogenies φ : E E and φ : E E C n G, X All continuous maps G G X ξ, α, β Elements of C n G, X n Boundary map C n G, X C n+1 G, X Z n G, X B n G, X Kernel of n, i.e., the n-cocycles Image of n 1, i.e., the n-coboundaries H n G, X nth cohomology group, as Z n G, X/B n G, X 8 W CE/k Weil-Châtalet group of E, as H 1 Gal k/k, Ek f Pushforward H n G, X H n G, Y of the map f : X Y δ n Connecting homomorphism H n G, X H n+1 G, X ϕ Pullback H n G, X H n Γ, X of the map ϕ : Γ G

Chapter 2 Galois Groups as Topological Spaces 2.1 Abstract Algebra Review 2.1.1 Commutative Groups Let X be a set. We say that X is a commutative group or an abelian group if there exists a well-defined binary operation : X X X such that Associativity P Q R = P Q R for all P, Q, R X. Commutativity P Q = Q P for all P, Q X. Identity There exists a unique O X such that O P = P for all P X. Inverses For each P X there exists a unique P X such that P P = O. Say that X,, Y,, and Z, are commutative groups. A map f : X Y is a homomorphism if fp Q = fp fq for all P, Q X. Define the kernel, image, and cokernel of a homomorphism as, respectively, ker f = { P X fp = O } = X[f] im f = { Q Y Q = fp for some P X } = fx 2.1.1 coker f = { Q = Q im f Q Y } = Y/im f If g : Y Z is another homomorphism, we say that the composition of maps X f Y g Z 2.1.2 is an exact sequence if im f = ker g as subsets of Y. The following is a well-known result about homomorphisms between commutative groups which we state without proof. Proposition 2.1.1. Say that f : X Y is a homomorphism of commutative groups X and Y. 1. fo = O and f P = fp for all P X. 2. ker f,, im f,, and coker f, are commutative groups. 9

3. The following is an exact sequence: {O} ker f X f Y coker f {O} 2.1.3 For example, Z, + is a commutative group under addition with identity 0. For any integer n, the map f : Z Z defined by a n a is a homomorphism. The kernel and image are ker f = {0} and im f = n Z, respectively. In particular, coker f = Z/n Z is a commutative group under addition. 2.1.2 Fields Let k be a set. We say that k is a field if there are well-defined binary operations + : k k k and : k k k such that Addition G a k = k is a commutative group under addition + with identity 0. Multiplication G m k = k {0} is a commutative group under multiplication with identity 1. Distributivity a b + c = a b + a c for all a, b, c k. The sets of rational numbers Q, real numbers R, complex numbers C, and residue classes modulo a prime F p = Z/p Z are each examples of fields. For any field k, exists an integer n such that the ideal n Z is the kernel of the morphism 1 } + {{ + 1 } if m > 0; Z k defined by m m times 1 + + 1 }{{} m times 0 if m = 0. if m < 0; and 2.1.4 We call n the characteristic of k. For example, the sets of rational numbers Q, real numbers R, complex numbers C each have characteristic 0, whereas the set of residue classes modulo a prime F p has characteristic p. 2.1.3 Galois Theory Let k be a field. The polynomial ring R = k[x 1,..., x n ] is a unique factorization domain. We say that a field K is a normal, separable extension of k if 1 k K and K is the splitting field of a family of polynomials in R not having repeated roots. Define Aut and Emb! Let L be a normal, separable extension of k. Denote G = GalL/k = AutL/k = EmbL/k as its Galois group. Say that K is a subfield of L containing k; then K is a separable extension of k. Let H denote all σ G such that σa = a for all a K. Then H is a subgroup of G, and H = GalL/K. Conversely, say that H is a subgroup of G. Let K denote all a L such that σa = a for all σ H. Then K is a subfield of L, and K = L H. In either case, K is a normal extension of k if and only if H is a normal subgroup of G. 10

2.2 Profinite Groups as Topological Spaces 2.2.1 Absolute Galois Groups Let k be a field, and denote k as a separable, algebraic closure of k. In practice, we will choose k as either the rational numbers Q, a completion Q v, or a finite field F p. We define G = Gal k/k as follows. Let I k denote the category of finite, normal, separable extensions K of k with the morphisms being set inclusion: F K L. We have a directed family of groups given by restriction of the action to subfields: proj L/F : GalL/k proj L/K GalK/k proj K/F GalF/k 2.2.1 Define the projective limit as Gal k/k = lim GalK/k = K..., σ K,... GalK/k K I k proj L/Kσ L = σ K 2.2.2 It is easy to check that this is a group under composition. Since G = Gal k/k is the projective limit of finite groups GalK/k we say that G is a profinite group. It may be useful to think of this using the following commutative diagram: Gal k/k 2.2.3 GalL/k proj L proj L/K proj K GalK/k proj F proj K/F GalF/k 2.2.2 Krull Topology We may turn G into a topological space using the Krull topology: We say that a subgroup U G is open if it has finite index in G. One may think of this as an open ball centered at the origin. In general, we say that a subset V G is open if for each σ V we can find a subgroup U σ of finite index in G such that the coset σ U σ is contained in V. One may think of this coset as an open ball centered at σ. Here is a standard result: Theorem 2.2.1. For any field k, denote its absolute Galois group as G = Gal k/k. 1. Both and G are open. The union α V α of arbitrarily many open sets is open. The intersection α V α of finitely many open sets is open. 2. Let U G be a subgroup of finite index. Then each coset σ U is both open and closed. 3. The maps G G G and G G defined as σ, τ σ τ and σ σ 1, respectively, are continuous. 4. G is a topological group which is Hausdorff, totally disconnected, and compact. Proof. Clearly and G are open. Fix σ in an arbitrary union V = α V α. Then σ V α for some α, so that σ U σ V α for some open subgroup U σ. As σ U σ V, we see that that V must be open. Similarly, fix σ in a finite intersection V = α V α. Then σ V α for each α, so that σ U σ,α V α 11

for some open subgroups U σ,α. Denote U α = α U σ,α. As the index G : U σ α G : U σ,α is finite and σ U σ V, we see that the intersection of finitely many open sets must be open as well. Let U G is an open subgroup, and denote V = σ U. As τ U = V for each τ V, we see that V is open. Since U has finite index in G, we have a finite disjoint union G = n i=1 σi U. As V = σ j U for some j, the compliment G V = i j σi U is the union of open sets, we see that V is closed. We show that multiplication and inverses are continuous maps. Fix an open set W G. Fix σ, τ in the inverse image V = { σ, τ G G σ τ W }. We can find an open subgroup U σ τ in W such that σ τ U σ τ W. Denote the open subgroup U σ,τ = τ U σ τ τ 1 U σ τ in G G. It is easy to see that σ, τ U σ,τ V, so that V is indeed open. Similarly, fix σ in the inverse image V = { σ G σ 1 W }. We can find an open subgroup U σ 1 in W such that σ 1 U σ 1 W. Denote the open subgroup U σ = σ 1 U σ 1 σ in G. It is easy to see that σ U σ V, so that V is indeed open. Clearly G is a topological group. We show that G is both Hausdorff and totally disconnected. Choose distinct σ 1 and σ 2 in a subset S G. As σ1 1 σ 2 1, there exists a finite, normal, separable extension K of k such that σ1 1 σ 2 does not lie in the kernel U = ker projk of the canonical projection proj K : G GalK/k. As σ 1 U σ 2 U = we see that σ 1 and σ 2 can be housed ofh into disjoint open sets. Similarly, as S = σ S α G/U α is a nontrivial disjoint union of subsets S α = σ α U S, we see that S is disconnected in the subspace topology. Finally, we show that G is compact. As each finite group GalK/k is compact in the finite topology, Tychonoff s Compactness Theorem states that the product K I k GalK/k is also compact in the product topology. It suffices then to show that G K I k GalK/k is closed, which we do by showing that every convergent sequence in G has a limit in G. To this end, let I be a totally ordered set. We say that a subset {σ α α I} G is a Cauchy sequence if, given an open subgroup U G, we can find N U I such that σ 1 β σ α U whenever α, β N U. In particular, σ α β N U σβ U is contained in the intersection of closed sets, so that the subsequence {σ α α N U } is contained in a closed subset of G. Hence must have an accumulation point σ = lim α σ α in G. Any subgroup U of finite index in G contains a normal subgroup U = τ G τ 1 U τ also of finite index. Hence we may as well define an open subgroup as one which is a normal subgroup of finite index. In this case, the quotient group G/U GalK/k for some finite, normal, separable extension K of k, which shows that by definition G lim G/U. Often we abuse notation and U say U = Galk/K. 2.2.3 Profinite Groups In general, let I be some partially ordered set, and say that we have a collection of finite groups { Γα α I }. This is a directed family of groups whenever we have a collection of compatible group homomorphisms proj γ,β proj γ,α : Γ γ Γ β whenever α β γ. Define the projective limit as { Γ = lim α Γ α =..., g α,... α I proj β,α Γ α 2.2.4 Γ α proj β,α g β = g α } 2.2.5 12

Using the ideas from above, it is easy to check that this is a group under composition. As Γ is the projective limit of finite groups, we say that Γ is a profinite group. In fact, Γ is a topological group which is Hausdorff, totally disconnected, and compact. Serre, in his Galois cohomology, shows the converse: If Γ is a topological group which is Hausdorff, totally disconnected, and compact, then G is the projective limit of of finite groups. 2.2.4 Example: l-adic Numbers Fix a prime l. Denote I = {1, 2,... } = Z >0 as the collection of positive integers. For each positive integer α I, the subset l α Z is an additive subgroup of the integers Z which is also closed under multiplication. The quotients Γ α = Z/l α Z are then finite abelian groups. We have projection maps proj β,α : Z/l β Z Z/l α Z which sends a mod l β a mod l α 2.2.6 whenever α β. We define the set { Z l = lim Z/l α Z =..., a α α,... } Z/l α Z a β a α mod l α α I 2.2.7 It is easy to check that Γ α = Z/l α Z Z l /l α Z l for any α I. 2.2.8 It is easy to show that Z l is a ring; we call it the collection of l-adic integers. The quotient field Q l is the collection of l-adic numbers. 13

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Chapter 3 G-Modules and X-Torsors 3.1 Galois Modules 3.1.1 Abelian Groups with Continuous Galois Action Continue to denote G = Gal k/k as above. Let X be a set. In practice, X is a subset of the multiplicative group k or a subset of Ek for some elliptic curve defined over k. We say that X is a G-module or a Galois module over k if there exist binary operations : X X X and : G X X such that Associativity P Q R = P Q R and σ τ P = σ τ P for all σ, τ G and P, Q, R X. Commutativity P Q = Q P for all P, Q X. Distributivity σ P Q = σ P σ Q for all σ G and P, Q X. Identity There exists a unique O X such that O P = 1 P = P for all P X. Inverses For each P X there exists a unique P X such that P P = O. Continuity For each P X, the stabilizer V = { σ G σ P = P } of P is open. To be more precise, we may also say X is a Z[G]-module on which G acts continuously, where we view X as a topological space via the discrete topology. Condition Continuity needs further explanation: The set U V G is the kernel of the permutation representation G AutX defined as that map which sends σ to the homomorphism P σ P. Hence U is a normal subgroup of G; the condition Continuity assumes its index is finite. As each P X is contained in some subset X U = { P X σ P = P for all σ U } 3.1.1 corresponding to some normal, open subgroup U G, we have X = U XU. 3.1.2 Example: Additive Group G a and Multiplicative Group G m Say that X = G a k = k under = + or X = G m k = k {0} under =. The absolute Galois group G = Gal k/k acts in the canonical way; denote this action by the binary operation : G X X. We show that X is a G-module. 15

It is easy to see that axioms Associativity, Commutativity, Distributivity, Identity, and Inverses are satisfied. It suffices to show that axiom Continuity is safisfied: Choose P X, and let V G denote its stabilizer. As this is an element of k, it lies in some finite, normal separable extension K of k. Denote the open subgroup U = ker proj K as the kernel of the canonical projection proj K : G GalK/k. It is easy to see that the coset σ U acts trivially on P for any σ V. Hence σ U V, so we see that V is indeed open. Note that X U = G a K = K or X U = G m K = K {0}, respectively, so that X = U XU. 3.1.3 Example: Elliptic Curves Fix coefficients a 1, a 2, a 3, a 4, a 6 k, and consider the homogeneous cubic polynomial fx 1, x 2, x 0 = x 2 2 x 0 + a 1 x 1 x 2 x 0 + a 3 x 2 x 2 0 x 3 1 + a 2 x 2 1 x 0 + a 4 x 1 x 2 0 + a 6 x 3 0. 3.1.2 For each normal, separable extension K of k we denote the set { } EK = x 1 : x 2 : x 0 P 2 K fx 1, x 2, x 0 = 0 { } x, y A 2 K y2 + a 1 x y + a 3 = x 3 + a 2 x 2 + a 4 x + a 6 {O} 3.1.3 in terms of O = 0 : 1 : 0. The only solution over k to the simultaneous polynomial equations f = f = f = f = 0 is the trivial solution x 1 = x 2 = x 0 if and only if the discriminant x 1 x 2 x 0 = a 4 1 a 2 a 2 3 8 a 2 1 a 2 2 a 2 3 16 a 3 2 a 2 3 + a 3 1 a 3 3 + 36 a 1 a 2 a 3 3 27 a 4 3 + a 5 1 a 3 a 4 + 8 a 3 1 a 2 a 3 a 4 + 16 a 1 a 2 2 a 3 a 4 30 a 2 1 a 2 3 a 4 + 72 a 2 a 2 3 a 4 + a 4 1 a 2 4 + 8 a 2 1 a 2 a 2 4 + 16 a 2 2 a 2 4 96 a 1 a 3 a 2 4 64 a 3 4 a 6 1 a 6 12 a 4 1 a 2 a 6 48 a 2 1 a 2 2 a 6 64 a 3 2 a 6 3.1.4 + 36 a 3 1 a 3 a 6 + 144 a 1 a 2 a 3 a 6 216 a 2 3 a 6 + 72 a 2 1 a 4 a 6 + 288 a 2 a 4 a 6 432 a 2 6 is nonzero. In this case, we say that E : y 2 + a 1 x y + a 3 = x 3 + a 2 x 2 + a 4 x + a 6 is an elliptic curve defined over k. Proposition 3.1.1. For any field k, denote its absolute Galois group as G = Gal k/k. Let E be an elliptic curve defined over k. Then X = Ek is a G-module. Sketch of Proof. The absolute Galois group G = Gal k/k acts in the canonical way; denote this action by the binary operation : G X X. We construct a binary operation : X X X with certain properties: Consider two points P = p 1 : p 2 : p 0 and Q = q 1 : q 2 : q 0 in X. Draw a projective line through them, say a x 1 + b x 2 + c x 0 = 0 in terms of the projective point p 2 p 0 q 2 q 0 : p 0 p 1 q 0 q 1 : p 1 p 2 q 1 q 2 a : b : c = f f f P : P : P x 1 x 2 x 0 16 if P Q, or if P = Q. 3.1.5

Denote P Q as the point of intersection between the line a x 1 + b x 2 + c x 0 = 0 and the curve fx 1, x 2, x 0 = 0. We define : X X X by P Q = P Q O. This is the so-called Group Law for elliptic curves. We must show that six properties hold for the binary operations : X X X and : G X X. Commutativity holds because P Q = Q P. Distributivity holds because σ P Q = σ P σ Q and σ O = O. Identity holds for O because the line through P O and O goes through one other point, namely P, so that P O = P O O = P. Similarly, Inverses holds for P = P O because the line through P and P O goes through one other point, namely O, so that P P = P P O = O O = O. As for Continuity, choose P X and let V G denote its stabilizer. Then the coordinates of P = p 1 : p 2 : p 0 lie in some finite, normal separable extension K of k. Denote the open subgroup U = ker proj K as the kernel of the canonical projection proj K : G GalK/k. It is easy to see that the coset σ U acts trivially on P for any σ V. Hence σ U V, so we see that V is indeed open. Note that X U = EK, so that X = U XU. It suffices then to show that Associativity holds for : X X X. For this, it suffices to show that P Q R = P Q R for all P, Q, R X. We omit a rigorous proof, and instead give a geometric one for k R. See Figure 3.1. 3.1.4 Tate Modules Let X and Y be G-modules, and say that we have a G-module homomorphism f : X Y. We say that f is an isogeny if either fp = O for all P X or f has the following properties: Homomorphism fp Q = ϕp ϕq for all P, Q X. G-Map σ fp = fσ P for all σ G and P X. Kernel The kernel X[f] = { P X fp = O } is a finite group. Image The image fx = Y. That is, for each Q Y, there exists at least one P X such that fp = Q. If f : X Y is an isogeny, define the separable degree as the integer deg f = #X[f]. For each integer m, define the multiplication-by-m map [m] : X X as that map which sends P } {{ P } if m > 0; [m] P = m times P P }{{} m times O if m = 0. if m < 0; and 3.1.6 This is a G-module homomorphism. Let m = l α be a power of a prime l; then X[l α ] is a G- submodule of X. We have projection maps proj β,α : X[l β ] X[l α ] which sends P [l β α ] P 3.1.7 17

3 P*Q 2.5 R Q 2 Q*R 1.5 P 1 0.5 Figure 3.1: A Geometric Proof of the Associativity of the Group Law 18-4.4-4 -3.6-3.2-2.8-2.4-2 -1.6-1.2-0.8-0.4 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4-0.5-1 -1.5 Q+R P+Q*R = P*Q+R -2-2.5 P+Q -3

whenever α β. We define the l-adic Tate module for X as the projective limit { T l X = lim X[l α ] =..., P α α,... } X[l α ] [lβ α ] P β = P α α I 3.1.8 This is a Z l [G]-module. To see why, define : Z l [G] T l X T l X by [ ] mσ σ P =..., σ [m α σ] P α,.... 3.1.9 σ G σ G Each m α Z/l α Z. If m α = n α + a α l α, then [m α ] P α = [n α ]P α [a α ] [l α ]P α = [nα ]P α O = [n α ]P α. 3.1.10 Define V l X = T l X Zl Q l as a Q l -module. Since G acts continuously on V l X, we have a continuous representation ρ X,l : Gal k/k GL V l X. 3.1.11 We define this as the l-adic representation associated to X. 3.1.5 Isogenies of Galois Modules Let X, Y, and Z be G-modules. Say that we have G-module homomorphisms f : X Y and h : X Z. Whenever ker f ker h and im f = Y, there exists a G-module homomorphism g : Y Z which makes the following diagram commute: X h Z 3.1.12 f Y g Let f : X Y be an isogeny, and denote m = deg f as its separable degree. There exists a G-module homomorphism f : Y X such that f f = [m] is the multiplication-by-m map. Proof. Write ϕ 1 Q = R f 1 O for some R f 1 Q; then #f 1 Q = #f 1 O = m. For the composition f f : X X we have f f P = R = fp = 3.1.13 R f fp 1 R f 1 O For the composition f f : Y Y we have f f Q = ϕ R = R f 1 Q R f 1 Q ϕr = Hence f f = [m] is indeed the multiplication-by-m map. 19 R f 1 Q Q = [m] Q. 3.1.14

3.1.6 Sheaves of Galois Modules Continue to denote G = Gal k/k and X as a G-module, both as above. Given any element ν G, define the subset X ν = { P X σ P = P for all σ V } 3.1.15 ν V as the union over those open subsets V G containing ν. We call X ν the stalk over ν, and any element P ν X ν a germ of the stalk. Given any nonempty open set V G, define U V = τ G τ 1 V τ as the largest normal, open subgroup contained in the group generated by V ; for V being empty, define U = G. We wish to consider the subset X V =..., Pν,... for each ν V, there exists a normal, X ν open subgroup U ν G such that ν U ν V ν V, and σ P ν ω = P ν ω for all σ U V and ω U ν 3.1.16 where we set X = {O}. We may identify each point P =..., P ν,... in X V as a morphism P : V X ν defined by ν P ν. 3.1.17 ν V Proposition 3.1.2. For any field k, denote its absolute Galois group as G = Gal k/k. Let X be a G-module, X ν X be a stalk for each ν G, and X V as above for each open set V G. 1. X ν is a G-module. 2. X is a sheaf of G-modules. Proof. We begin by showing that each stalk X ν is a G-module. Since X ν = ν U XU is the intersection over normal, open subgroups U containing ν, it suffices to show that each X U is a G-module. First, we show that there are restrictions : X U X U X U and : G X U X U. Each subset X U is closed under addition because σ P Q = σ P σ Q = P Q for any σ U and P, Q X U. As U is a normal subgroup in G, the conjugate σ = τ 1 σ τ is also in U for any τ G. Hence σ τ P = τ σ P = τ P, showing that τ P is also in X U for any τ G. Hence properties Associativity, Commutativity, Distributivity, and Continuity hold. Property Identity holds because σ O σ O = σ O O = σ O, so that σ O = O for any σ G. Property Inverses holds because σ P σ P = σ P P = σ O = O, so that σ P = σ P for any σ G. In particular, σ P = P for any σ U and P X U. Proposition 2.2.1 shows that G is a topological space. In order to show that X is a sheaf, we must show that the following three properties hold: G-Modules X V is a G-module for every open set V G, where X = {O}. We define binary operations : X V X V X V and : G X V X V componentwise: P =..., P ν,... } { P Q =..., Pν Q ν,... Q =..., Q ν,... = τ P =..., τ P ν,... 3.1.18 20

We explain why these are well-defined. Fix P, Q X V. For each ν V, we can find normal, open subgroups U ν,p, U ν,q G such that ν U ν,p, ν U ν,q V and σ P ν ω = P ν ω, σ Q ν ω = Q ν ω for all σ U V and ω U ν,p, ω U ν,q. Recall that for V nonempty we set U V = τ G τ 1 V τ, and U = G otherwise. Define U ν = U ν,p U ν,q ; this is a normal, open subgroup such that ν U ν V. Property Distributivity for the stalks X ν ω implies that σ P ν ω Q ν ω = σ Pν ω σ Qν ω = Pν ω Q ν ω for all σ U V and ω U ν, showing that P Q is in X V. As U V is a normal subgroup, the conjugate σ = τ 1 σ τ is also in U V for any τ G. Then σ τ P ν ω = τ σ P ν ω = τ Pν ω for all σ U V and ω U ν, showing that τ P is in X V. It is easy to see that properties Associativity, Commutativity, and Distributivity hold. As σ O = O and σ P ν = σ P ν for all σ G, the elements O =..., O,... and P =..., P ν,... are in X V ; hence properties Identity and Inverses hold. It remains to show that property Continuity holds. Fix a point P =..., P ν,... in X V, and consider an element σ of the stabilizer { σ G σ P = P } = { } ν V σ G σ P ν = P ν. This group contains the open set σ U V, so that it is indeed open. Hence X V is a G-module for every open set V G. Restriction Morphisms Whenever we have open sets U V W, there exist G-module homomorphisms such that res V/V = 1. res W/U : X W res W/V X V res V /U X U 3.1.19 Define the map res W/V : X W X V as that which takes a tuple P =..., P ν,... for ν W to that tuple res W/V P =..., P ν,... for ν V which is formed by removing those coordinates corresponding to ν W V. We explain why this is well-defined for V nonempty. For each ν W, there exists a normal, open subgroup U ν G such that ν U ν W and σ P ν ω = P ν ω for all σ U W and ω U ν. As V is open, we can find a normal, open subgroup U ν G such that ν U ν V. Denote U ν = U ν U ν ; this is a normal, open subgroup such that ν U ν V. As V W, we have U V U W recall that V is assumed nonempty so that σ P ν ω = P ν ω for all σ U V and ω U ν. This shows that res W/V P is indeed in X V. As σ O = O, it is clear that σ [ res W/V P ] [ ] = res W/V σ P and resw/v P Q = res W/V P res W/V Q for all σ G and P, Q X W ; hence res W/V is a G-module homomorphism. Similarly, it is clear that res W/U = res V/U res W/V and res V/V = 1 whenever U V W. Gluing Property Say that V = α V α is a covering of open sets. Whenever we have a collection of points P α X V α such that res Vα/Vα V β P α = res Vβ /V α V β P β for all α and β, then there exists a unique P X V such that res V/Vα P = P α. V res V /Vα P res V /Vβ 3.1.20 V α V β P α P β V α V β res V/Vα V β P 21

Say that V = α V α is an open cover, and that we have a collection of points P α =..., P α,ν,... in X V α such that res Vα/Vα V β P α = res Vβ /V α V β P β for all α and β. We must show that there is a unique P X V such that res V/Vα P = P α. If V is empty, we must have P = O because X = {O}, so assume V is nonempty. Each ν V lies in some V αν, so denote P =..., Pαν,ν,... as that tuple constructed by choosing the ν-coordinate of P αν X V αν. We call X the sheafification of the G-module X. We view X as a contravariant functor from the category of open sets V in G to the category of G-submodules X V of X. Note that when V is nonempty, U = τ G τ 1 V τ is an open, normal subgroup. In fact, since G/U GalK/k for some finite, normal separable extension K of k, we may identify X V as the K-rational points of X. 3.1.7 Principal Homogeneous Spaces Continue to denote G = Gal k/k and X as a G-module, both as above. Let A be a set. In practice, A is a subset of C d k for some quartic curve associated to an elliptic curve defined over k. We say that A is an X-torsor or a principal homogeneous space for X if there exist binary operations : X A A and : G A A such that Associativity P Q R = P Q R and σ τ R = σ τ R for all σ, τ G, P, Q X and R A. Distributivity σ P Q = σ P σ Q for all σ G, P X, and Q A. Identity O Q = 1 Q = Q for all Q A. Inverses For each Q, R A there exists a unique P X such that P Q = R. Continuity For each Q A, the stabilizer V = { σ G σ Q = Q } of Q is open. Note that X is a principal homogeneous space of itself. In a sense, A looks like X, yet we do not fix an identity O. Indeed, we even have A = U AU. Proposition 3.1.3. For any field k, denote its absolute Galois group as G = Gal k/k. Let A be a principal homogeneous space for a G-module X. 1. Both X and G act continuously on A in the discrete topology. Moreover, X acts transitively, and the stabilizer of any element via this action is trivial. 2. There exists a bijection Ξ : X A defined over a finite, normal, separable extension K of k such that ΞP = P ΞO for all P X. 3. Define : A A X as that map which sends R, Q to the unique P such that P Q = R. For σ, τ G and Q A, we have the following identity in X: [ σ τ Q Q ] = [ σ Q Q ] [ σ τ Q Q ]. 3.1.21 22

4. Define the relation A B when B is a principal homogeneous space for X and there is a bijection f : A B, commuting with the action by G, satisfying fp Q = P fq for all P X and Q A. Then is an equivalence relation. Moreover, for σ G, Q A, and R B, we have the following identity in X: [ σ Q Q ] = [ σ R R ] [ σ P P ] 3.1.22 where P = fq R is in X. Three principal homogeneous spaces A, B, and C for a G-module X are said to be equivalent when we can find maps f : A B and g : B C which make the following diagram commute: X X X Ξ A Ξ B Ξ C A f B g C 3.1.23 In fact, it is easy to verify that top row is a translation map P P O B,A P O C,A for all P X, in terms of O B,A = Ξ B 1 f Ξ A O, etc. Milne, in his Étale Cohomology, denotes P SHX/k as the set of equivalence classes {A/X} of such principal homogeneous spaces for X. Proof. The action of X on A is Q P Q for any fixed P X. As X, is an abelian group with identity O, properties Associativity and Identity show that both X and G do indeed act on A. Property Inverses shows that the only P = O if the only element such that P Q = Q; hence the stabilizer of any Q A is the trivial group {O} X. As one point sets are open in the discrete topology, this shows that the group action is continuous. Similarly, property Inverses shows that any R A is in the orbit of a given Q A, so this action is transitive. Fix Q A. Define Ξ : X A by ΞP = P Q. Clearly this is well-defined and bijective. Let V be the stabilizer of Q. As G acts on A continuously, we can find an open normal subgroup U V G, so that G/U GalK/k for some finite, normal, separable extension K of k. Clearly, Ξ is defined over K. Continue to fix Q A. For each σ G, denote ξσ = σ Q Q as that unique element such that ξσ Q = σ Q. It suffices to show that ξσ τ = ξσ σ ξτ. We have the identity ξσ τ Q = σ τ Q = σ τ Q = σ ξτ Q = σ ξτ σ Q = [ σ ξτ ξσ ] Q, 3.1.24 where we have used properties Associativity and Distributivity. As ξσ is unique, we must have ξσ τ = ξσ σ ξτ as desired. We show that is an equivalence relation. Clearly A A since we may choose f = 1. Assume that A B. Denote f 1 : B A as the inverse of the map f : B A. Choose P A and R B, and denote Q = f 1 R A. Then we have f 1 P R = f 1 P fq = f 1 fp Q = P Q = P f 1 R. 3.1.25 Hence B A. Now assume that A B and B C. Denote f : A B and g : B C as bijections such that for all P X, Q A, and R B we have fp Q = P fq and gp R = P gr. The composition h = g f : A C is also a bijection. We have the identity hp Q = g fp Q = g P fq = P g fp = P hq. 3.1.26 23

Hence A C. This shows that is indeed an equivalence relation. Fix σ G, Q A, and R B. Define ξ A σ, ξ B σ, P X via the relations ξ A σ Q = σ Q, ξ B σ R = σ R, and P R = fq. We have the identity ξa σ P fq = P [ ξ A σ fq ] = P [ f ξ A σ Q ] = P f σ Q = P σ fq = P σ [ P R ] = P σ P σ R = P σ P ξ B σ R = ξ B σ σ P P R = ξ B σ σ P fq. 3.1.27 This shows that ξ A σ P = ξ B σ σ P, so that ξ A σ = ξ B σ [ σ P P ]. 3.1.8 Torsors via Translation Maps If A is a principal homogeneous space for X, then there exists a bijection Ξ : X A. The following result gives a partial converse: if there exists a certain type of bijection Ξ : X A, then there is a binary operation : X A A such that A is a principal homogeneous space for X. Proposition 3.1.4. For any field k, denote its absolute Galois group as G = Gal k/k. Let X and Y be G-modules, f : X Y be a G-module homomorphism, and A and B be sets on which G acts continuously. Assume that we have bijections Ξ A : X A and Ξ B : Y B such that for each σ G there exists ξ B σ = f ξ A σ Y for ξ A σ X satisfying σ Ξ A P = Ξ A σ P ξa σ σ Ξ B Q = Ξ B σ Q ξb σ for all P X, Q Y. 3.1.28 1. The map : X A A defined by P Q = Ξ A P Ξ 1 A Q makes A a principal homogeneous space for X. Moreover, Ξ A P = P Ξ A O for all P X. 2. Then there is a map f : A B such that f P Q = fp f Q and σ f Q = f σ Q for all σ G, P X, and Q A. Moreover, the following diagram commutes: ξ A G ξ B 3.1.29 X f Y Ξ A A f Ξ B B This proposition explains how the existence of a bijection Ξ A : X A which depends on a map ξ A : G X constructs homogeneous spaces. We say that A is an f-descendant of B whenever the conditions above hold. Note that if ξ B = O, then B Y as G-modules and im ξ A ker f. In this case, we say A an f-cover for Y, with f : A Y being the covering map. Proof. We show that : X A A satisfies the following properties: Associativity P Q R = P Q R for all P, Q X and R A. 24

Distributivity σ P Q = σ P σ Q for all σ G, P X, and Q A. Identity O Q = Q for all Q A. Inverses For each Q, R A there exists a unique P X such that P Q = R. Translation Ξ A P = P Ξ A O for all P X. For Associativity, we have P Q R = ΞA P Q Ξ 1 A R [ = Ξ A P Ξ 1 A ΞA Q Ξ 1 A R] = P Q R. 3.1.30 For Distributivity, σ P Q = σ Ξ A P Ξ 1 A Q = f σ P [ σ Ξ 1 A Q] ξ A σ [ = Ξ A σ P σ Ξ 1 A Q ξσ] = Ξ A σ P Ξ 1 A = σ P σ Q. [ σ Q ] 3.1.31 For Identity, we have O Q = Ξ A O Ξ 1 A Q = Ξ A Ξ 1 A Q = Q. For Inverses, let Q, R A be given, and define P = Ξ 1 A R Ξ 1 A Q. It is easy to see that P X is that unique element such that P Q = R. For Translation, let Q = Ξ A O. Then Ξ A P = Ξ A P O = ΞA P Ξ 1 A Q = P Q = P Ξ A O. Define f : A B by the composition f = Ξ B f Ξ 1 A. For any P X and Q A, we have f P Q = ΞB f P Ξ 1 A Q = Ξ B [fp 1 f Ξ ] A Q = fp f Q. 3.1.32 Now choose σ G and Q A, and set P = Ξ 1 A Q. Since σ Q = Ξ A σ P ξa σ, we have f σ Q = Ξ B f σ P ξ A σ = Ξ B [f σ P f ξ A σ ] [ ] σ = Ξ B fp ξb σ = σ Ξ B fp 3.1.33 = σ f Q. By construction, the diagram commutes. 3.1.9 Example: Quadratic Curves The following proposition explains how to determine conic sections as principal homogeneous spaces for Pell s equation x 2 D y 2 = 1. Proposition 3.1.5. Let k be a field of characteristic different from 2, and denote its absolute Galois group as G = Gal k/k. Fix elements a i k such that the determinants a 11, D = a 11 a 12 a 11 a 12 a 13 a 12 a 22 and a 12 a 22 a 23 3.1.34 a 13 a 23 a 33 are nonzero. 25

1. Denote X as the collection of k-rational points P = x, y satisfying x 2 D y 2 = 1. Then X is a G-module via the binary operation : X X X defined by x 1, y 1 x 2, y 2 = x 1 x 2 + D y 1 y 2, x 1 y 2 + x 2 y 1. 3.1.35 2. Denote A as the collection of k-rational points Q = z, w on the conic section a 11 z 2 + 2 a 12 z w + a 22 w 2 + 2 a 13 z + 2 a 23 w + a 33 = 0. 3.1.36 Then A is a principal homogeneous space for X. Proof. Consider the injective group homomorphism [ ] x D y X SL 2 k defined by x, y. 3.1.37 y x The obvious map X k defined by P x + D y is a homomorphism of G-modules, but it is not injective! As SL 2 k is G-module, the induced structure turns X into a G-module as well. Note that x, y 1, 0 = x, y, while O = 1, 0 is the identity and P = x, y is the inverse. Consider the bijection Ξ : X A defined by Ξx, y = z 0 + x a 12 y, w 0 + a 11 y z d d 0 = a 12 a 23 a 13 a 22 a 11 a 22 a 2, 12 where ΞO = z 0 + 1 in terms of, w 0 w 0 = a 12 a 13 a 11 a 23 d a 11 a 22 a 2. 12 3.1.38 This is defined over the finite, normal, separable extension K = k d of k, where a d = a 11 a 12 a 11 a 12 a 13 11 a 12 a 22 a 12 a 22 a 23 a 13 a 23 a 33 1 a 11 =. 3.1.39 a 13 z 0 + a 23 w 0 + a 33 Geometrically, the point Q 0 = z 0, w 0 is the center of the conic section; it is not actually on the curve. It is easy to check that σ [ ΞP ] = Ξ σ P ξσ { 1, 0 if σ d = d; where ξσ = O if σ d = + 3.1.40 d. Proposition 3.1.4 states that A is a principal homogeneous space for X. For example, the binary operation : X A A defined by x, y z, w = f x, y f 1 z, w = z 0 + x z z 0 y a 12 z + a 22 w + a 23, w 0 + x w w 0 + y a 11 z + a 12 w + a 13 26 3.1.41

which may also be realized via the injective group homomorphism Y SL 2 k defined by a 12 z + a 22 w + a 23 a 11 z + a 12 w + a 13 a z, w 13 z 0 + a 23 w 0 + a 33 z z. 0 w w 0 a 13 z 0 + a 23 w 0 + a 33 3.1.42 The proposition follows. 3.1.10 Example: Cubic Curves 3.2 Selmer s Cubic Proposition 3.2.1. Let k be a field of characteristic different from 2 and 3. Fix D k, and consider the cubic curve C : a u 3 + b v 3 + c w 3 = 0 for a, b, c k satisfying D = a b c. 1. C is a principal homogeneous space for the elliptic curve E : y 2 = x 3 432 D 2. In particular, E acts continuously and transitively on C via the map : E C C given by x, y u : v : w [ 2 w y + x 2 17 z = x 2 + 17 + 34 x z 2, w x 4 17 2] 34 z [ x w z x 2 17 + y 2 x + 17 z 2 + x 2 z 2 ] [ x 2 + 17 + 34 x z 2] 2. 3.2.1 2. Denoting the elliptic curve E : y 2 = x 3 + 27 D 2 x, there are rational maps ϕ : E E and g : C E defined over k which make the following diagram commute: E ϕ E 3.2.2 Moreover, gp Q = ϕ P gq for all P on E and Q on C. f C 3. If C has a k-rational point Q 0 = u 0 : v 0 : w 0, then C and E are birationally equivalent over k. Ernst Selmer considered this family of cubic curves for k = Q. In particular, he showed that C : 3 u 3 + 4 v 3 + 5 w 3 = 0 has a Q v -rational point for every place v of Q, yet it has no Q-rational point. We will see later that we can use properties of the Selmer group to better understand this phenomenon. Proof. Denote X = Ek as the collection of k-rational points P = x, y on the cubic curve E : y 2 = x 3 432 D 2. As its discriminant is a nonzero = 2 12 3 9 D 4, we see that E is an elliptic curve defined over k. Recall that X is a G-module by Proposition 3.1.1. Denote Y = Ck as the collection of k-rational points Q = u, v, w the quartic curve C : a u 3 + b v 3 + c w 3 = 0. The map f : E C defined by fx, y = 6 b x 3 d : 36 a b c y 3 d 2 : 36 a b c + y d 27 g where fo = 0 : 3 d : 1 3.2.3

is a bijection from X to Y which is defined over the finite, normal, separable extension K = k 3, 3 d of k for d = c/b. Using the identity x, y 0, 0 = 17/x, 17 y/x 2 for the group law on E, one checks that for any σ in G = Gal k/k we have the relation σ [ fp ] = f σ P ξσ where ξσ = { 0, 0 if σ 2 = 2; O if σ 2 = + 2. 3.2.4 Proposition 3.1.4 states that Y = Ck is a principal homogeneous space for X = Ek. The map : X Y Y above is defined by x, y u : v : w = f x, y f 1 u : v : w. We have seen before that there is a 2-isogeny ϕ : E E defined by x ϕ 2 + 17 x, y =, y x2 17 x x 2 = ϕ f 1 2 z, w = z 2, 4 w z 3. 3.2.5 Since im ξ = { 0, 0, O } = E[ϕ ], the second statement in the proposition above follows from Proposition 3.1.4. Finally, say that Q 0 = u 0 : v 0 : w 0 is a k-rational point on C. Since x = 1 17 z2 z 2 0 + 2 w w 0 z z 0 2 y = 34 z z 0 z 2 w 0 + z 2 0 w 2 w + w 0 z z 0 3 3.2.6 if and only if z = z 0 + 2 w 0 y + 17 z0 3 x + 17 z 0 x 2 + 34 z0 2 x + 17 w = w 0 34 z 0 z 2 0 x 2 + 2 x + 17 z0 2 y + w0 3 z 2 0 x 3 + 34 z0 4 x2 + x 2 + 17z0 2 x + 17 x 2 + 34 z0 2 x + 17 2 3.2.7 we see that the quartic curve 2 w 2 = 1 17 z 4 is birationally equivalent over k to the cubic curve y 2 = x 3 + 17 x. Proposition. Let k be a field of characteristic different from 3 containing a cube root ζ 3 of unity. In practice, we choose k = Q 3 since we want a number field. Choose k-rational numbers A, B, C, and D such that = 27 A B C D 3 A B C 3 is nonzero. We want to consider the projective curves C : A z 3 1 + B z 3 2 + C z 3 0 = 3 D z 1 z 2 z 0. These are called Desboves Curves. An example would be 3 z1 3 + 4 z3 2 + 5 z3 0 = 0. This family of curves has complex multiplication, i.e., for a primitive cube root ζ 3 of unity, we have an automorphism of order 3 defined by [ζ 3 ] : Ck Ck which sends z1 : z 2 : z 0 ζ3 z 1 : ζ 2 3 z 2 : z 0. 3.2.8 28

First, we show why this is a principal homogeneous space for an elliptic curve. Consider the map f : E C defined by 3 B x 3 D x + 1 ζ 3 y + 1 ζ3 2 fx, y = 3 : D3 A B C 3 d d 2 where fo = 0 : 3 D x + 1 ζ3 2 : y + 1 ζ 3 D 3 A B C d 3 d : 1 3.2.9 for the elliptic curve E : y 2 + 3 D x y + D 3 A B C y = x 3 and d = C/B. This is defined over the finite, normal, separable extension K = k 3 d of k, and one checks that σ [ fp ] = f σ P ξσ 0, 0 if σ 3 d = ζ 3 3 d; where ξσ = 0, b if σ 3 d = ζ 3 3 2 d; 3.2.10 O if σ 3 d = 3 d. Hence C is indeed a principal homogeneous space for E. Second, we show that if C has a rational point P 0 = a 1 : a 2 : a 0, then C is birationally equivalent to a different elliptic curve. 3.2.1 Example: Quartic Curves The following proposition explains how to determine quartic curves as principal homogeneous spaces for elliptic curves. Proposition 3.2.2. Let k be a field of characteristic different from 2, and denote its absolute Galois group as G = Gal k/k. Consider a quartic Qz = c 4 z 4 + c 3 z 3 + c 2 z 2 + c 1 z + c 0 with coefficients in k and a nonzero discriminant DiscQ = c 2 1 c 2 2 c 2 3 4 c 0 c 3 2 c 2 3 4 c 3 1 c 3 3 + 18 c 0 c 1 c 2 c 3 3 27 c 2 0 c 4 3 4 c 2 1 c 3 2 c 4 + 16 c 0 c 4 2 c 4 + 18 c 3 1 c 2 c 3 c 4 80 c 0 c 1 c 2 2 c 3 c 4 6 c 0 c 2 1 c 2 3 c 4 + 144 c 2 0 c 2 c 2 3 c 4 3.2.11 Define the coefficients 27 c 4 1 c 2 4 + 144 c 0 c 2 1 c 2 c 2 4 128 c 2 0 c 2 2 c 2 4 192 c 2 0 c 1 c 3 c 2 4 + 256 c 3 0 c 3 4. a 1 = c3 3 4 c 2 c 3 c 4 + 8 c 1 c 2 4 8 c 2 4 a 2 = 3 c2 3 8 c 2 c 4 4 c 4 a 4 = 3 c4 3 16 c 2 c 2 3 c 4 + 16 c 2 2 c2 4 + 16 c 1 c 3 c 2 4 64 c 0 c 3 4 16 c 2 4 c 3 a 6 = 3 4 c 2 c 3 c 4 + 8 c 1 c 2 2 4 64 c 3 4 3.2.12 1. The polynomial P x = x 3 + a 2 x 2 + a 4 x + a 6 in terms of is a resolvent cubic for Qz such that DiscP = DiscQ. 2. The quartic curve C : w 2 = Qz is a principal homogeneous space for the elliptic curve E : y 2 = P x. If C has a k-rational point Q 0 = z 0, w 0, then C and E are birationally equivalent over k. 29

3. Assume that a 1 = 0. In particular, the quartic curves w 2 = 1 z 4 and 2 w 2 = 1 17 z 4 are principal homogeneous spaces for the cubic curves y 2 = x 3 + 4 x and y 2 = x 3 + 17 x, respectively, as well as g-covers for the cubic curves y 2 = x 3 x and y 2 = x 3 68 x, respectively. We will see later that the former quartic curve has Q-rational points, whereas the latter quartic curve does not. Proof. Using the factorization Qz = c 4 z e 1 z e 2 z e 3 z e 4 over k, we find the factorization e 1 + e 2 e 3 e 4 P x = [x 2 ] e 1 e 2 + e 3 e 4 + c 4 [x 2 ] e 1 e 2 e 3 + e 4 + c 4 [x 2 ] + c 4 4 4 4 3.2.13 also over k. One readily verifies that DiscP = DiscQ. Denote X = Ek as the collection of k-rational points P = x, y satisfying y 2 = P x, and A = Ck as the collection of k-rational points Q = z, w satisfying w 2 = Qz. The discriminant of the cubic curve is = 2 4 DiscQ. As this is nonzero, we see that E is indeed an elliptic curve over k; Proposition 3.1.1 states that X is indeed a G-module. The birational transformation Ξ : X A defined by Ξx, y = c 3 x + 2 c 4 a 1 + 2 d y d 2 a 6 + a 4 x x 3 +2 c 4 a 1 y, 4 c 4 x 4 c 4 x 2 3.2.14 is defined over the finite, normal, separable extension K = k d of k, where d = c 4. It is easy to check that σ [ ΞP ] = Ξ σ P ξσ 0, a 1 d if σ d = d; where ξσ = O if σ d = + 3.2.15 d. Proposition 3.1.4 states that A is a principal homogeneous space for X. Say that Q 0 = z 0, w 0 is a k-rational point on C. Then the invertible substitution x = b 1 z z 0 + b 2 b 3 + z z 2 w w 0 0 z z0 y = b 4 z z 0 2 + b 5 z z 0 + b 6 z z0 2 + b 7 z z 0 + b 8 z z0 3 w w 0 3.2.16 in terms of the coefficients b 1 = 2 c 4 z 2 0 + c 3 z 0 b 2 = 4 c 4 z 3 0 + 3 c 3 z 2 0 + 2 c 2 z 0 + c 1 b 3 = 2 w 0 b 4 = w 0 4 c4 z 0 + c 3 b 5 = 2 w 0 6 c4 z0 2 + 3 c 3 z 0 + c 2 b 6 = 2 w 0 4 c4 z0 3 + 3 c 3 z0 2 + 2 c 2 z 0 + c 1 b 7 = 4 c 4 z0 3 + 3 c 3 z0 2 + 2 c 2 z 0 + c 1 b 8 = 4 w0 2 3.2.17 shows that the relation y 2 = P x holds if and only if w 2 = Qz holds. Hence the curves C and E are birationally equivalent over k. 30

Proposition 3.2.3. Let k be a field of characteristic different from 2, and denote its absolute Galois group as G = Gal k/k. Fix c i k such that the discriminants c 4 0, c 3 3 4 c 2 c 3 c 4 + 8 c 1 c 2 4 = 0, c 2 1 c 2 2 c 2 3 4 c 0 c 3 2 c 2 3 4 c 3 1 c 3 3 + 18 c 0 c 1 c 2 c 3 3 27 c 2 0 c 4 3 4 c 2 1 c 3 2 c 4 3.2.18 + 16 c 0 c 4 2 c 4 + 18 c 3 1 c 2 c 3 c 4 80 c 0 c 1 c 2 2 c 3 c 4 6 c 0 c 2 1 c 2 3 c 4 + 144 c 2 0 c 2 c 2 3 c 4 27 c 4 1 c 2 4 + 144 c 0 c 2 1 c 2 c 2 4 128 c 2 0 c 2 2 c 2 4 192 c 2 0 c 1 c 3 c 2 4 + 256 c 3 0 c 3 4 0. 1. 2. C : w 2 = c 4 z 4 + c 3 z 3 + c 2 z 2 + c 1 z + c 0 3.2.19 E : y 2 = x 3 + c 2 x 2 + c 1 c 3 4 c 0 c 4 x + c0 c 2 3 + c 2 1 c 4 4 c 0 c 2 c 4 E : y 2 = x 3 + c 2 x 2 + 5 c 2 c 2 3 20 c2 2 c 4 + 14 c 1 c 3 c 4 + 64 c 0 c 2 4 4 c 4 x 3.2.20 g z, w = + 3 c2 2 c2 3 + 12 c3 2 c 4 6 c 1 c 2 c 3 c 4 + 32 c 0 c 2 3 c 4 24 c 2 1 c2 4 64 c 0 c 2 c 2 4 4 c 4 16 c 2 4 z 2 + 8 c 3 c 4 z + 4 c 2 c 4 c 2 3 Moreover, the following diagram commutes: 4c 4, 2Y c 3 + 4Xc 4. 3.2.21 E ξ A f G E ξ B O g E 3.2.22 Ξ A Ξ B C f C g Proof. Denote X = E k, Y = Ek, and Z = E k. We have an isogeny g : Y Z defined by x 2 e x + e 2 4 c 0 c 4 gx, y =, y x2 2 e x + 4 c 0 c 4 x e x e 2 where e = c2 3 4 c 2 c 4 4 c 4. 3.2.23 Similarly, denote A = C k and B = Ck. In the proof of Proposition 3.2.2, we exhibited a birational transformation Ξ B : Y A such that σ [ Ξ B P ] = Ξ B σ P ξb σ { }, where ξ B σ e, 0, O = ker g for all σ G. Proposition 3.1.4 states that the map g = g Ξ 1 B makes the diagram above commute. 31

3.2.2 Elliptic Curves: 2-Isogeny Let E : y 2 = x 3 + a x 2 + b x be an elliptic curve over a field k having characteristic different from 2, and denote X = Ek as the collection of k-rational points P = x, y. Recall that X, is an abelian group with identity O = 0 : 1 : 0: x 1, y 1 x 2, y 2 y1 y 2 2 = x 1 x 2 a, x 1 x 2 x 1 + 2 x 2 + a y 1 x 2 + 2 x 1 + a y 2 x 1 x 2 y1 y 3 2. x 1 x 2 3.2.24 For example, x, y 0, 0 = b/x, b y/x 2. Note that 0, 0 is a point of order 2, i.e., [2] 0, 0 = O. For each d k, consider the quartic curve C d : w 2 = d 2 a z 2 + a 2 4 b/d z 4, and denote Y = C d k as the collection of k-rational points Q = z, w. The map f : E C d defined by fx, y = d y x 2 + a x + b, d x 2 b x 2 + a x + b where fo = 0, d 3.2.25 is a bijection from X to Y which is defined over the finite, normal, separable extension K = k d of k. You can find these formulas on page 294 of Silverman s The Arithmetic of Elliptic Curves. In fact, one checks that σ [ fp ] = f σ P ξσ { 0, 0 if σ d = d; where ξσ = O if σ d = + 3.2.26 d. Proposition 3.1.4 states that Y = C d k is a principal homogeneous space for X = Ek. We abuse notation and say C d is a principal homogeneous space for E. Rather explicitly, the map : E C d C d is given by x, y z, w = f x, y f 1 z, w d 2 x 2 + a x + b [ w x 2 b 2 a y z ] d w y + d x 2 b z = d x 2 + a x + b a 2 4 b x z 2, + d a 2 4 b z [ x z w x 2 b + 2 y d x + b z 2 + x 2 z 2 ] [ d x 2 + a x + b a 2 4 b x z 2] 2. 3.2.3 Elliptic Curves: 2-Torsion 3.2.27 Here s a slightly more advanced example using the same elliptic curve. Assume that E[2] Ek, so that we can write E : y 2 = x e 1 x e 2 x e 3 in terms of the k-rational roots e 1 = a + a 2 4 b 2, e 2 = a a 2 4 b, and e 3 = 0. 3.2.28 2 For each d 1, d 2 k, consider the quadric intersection H d1,d 2 : x A x = x B x = 0 defined in terms of the 4 4 matrices d 1 0 0 0 d 1 0 0 0 A = 0 d 2 0 0 0 0 0 0 and B = 0 0 0 0 0 0 d 1 d 2 0. 3.2.29 0 0 0 e 1 0 0 0 e 2 32

When x = u : v : w : 1 is an affine point, we may express this in terms of the affine equations d 1 u 2 d 2 v 2 = e 1 and d 1 u 2 d 1 d 2 w 2 = e 2. Denote Y = H d1,d 2 k as the collection of k-rational points x. The map f : E H d1,d 2 defined by x 2 e 1 e 2 fx, y = 2 d 1 y : x2 2 e 1 x + e 1 e 2 2 : x2 2 e 2 x + e 1 e 2 d 2 y 2 : 1 d 1 d 2 y 3.2.30 1 1 1 where fo = : : : 0 d1 d2 d1 d 2 is a bijection from X to Y which is defined over the finite, normal, separable extension K = k d 1, d 2 of k. It is a bit tedious, but one checks that σ [ fp ] = f σ P ξσ where e 1, 0 if σ d 1 = d 1, σ d 2 = + d 2 ; e 2, 0 if σ d 1 = d 1, σ d 2 = d 2 ; ξσ = e 3, 0 if σ d 1 = + d 1, σ d 2 = d 2 ; O if σ d 1 = + d 1, σ d 2 = + d 2. 3.2.31 Proposition 3.1.4 states that H d1,d 2 is also a principal homogeneous space for E. Rather explicitly, the map : E H d1,d 2 H d1,d 2 is given by x, y u : v : w : 1 = f x, y f 1 u : v : w : 1 =? :? :? :?. 3.2.32 I find it unsettling that no one has ever written down these expressions before when it seems obvious to do so... Elliptic Curves: 3-Isogeny Let E : y 2 + a x y + b y = x 3 be an elliptic curve over a field k having characteristic different from 3 as well as a primitive cube root of unity ζ 3, and denote X = Ek as the collection of k-rational points P = x, y. It is easy to check that x, y 0, 0 = b y/x 2, b 2 y/x 3 and x, y [2] 0, 0 = b x/y, b x 3 /y 2. Note that 0, 0 is a point of order 3, i.e., [3] 0, 0 = O. For each d k, consider the cubic curve C d : w 3 = d + 3 a z w + a 3 27 b/d z 3, and denote Y = C d k as the collection of k-rational points Q = z, w. The map f : E C d defined by 3 x fx, y = d 2 a x + 1 ζ3 2 y + 1 ζ 3 b, 3 d a x + 1 ζ 3 y + 1 ζ3 2 b a x + 1 ζ3 2 y + 1 ζ 3 b where fo = 3 0, d 3.2.33 is a bijection from X to Y which is defined over the finite, normal, separable extension K = k 3 d of k. Recall that a primitive cube root ζ 3 of unity is assumed to be an element of k. One checks that σ [ fp ] = f σ P ξσ 0, 0 if σ 3 d = ζ 3 3 d; where ξσ = 0, b if σ 3 d = ζ 3 3 2 d; 3.2.34 O if σ 3 d = 3 d. 33

Proposition 3.1.4 states that C d is a principal homogeneous space for E. Rather explicitly, the map : E C d C d is given by x, y z, w = f x, y f 1 z, w =?,?. 3.2.35 34

Chapter 4 Galois Cohomology 4.1 Continuous Maps 4.1.1 Sections Let G = Gal k/k, and X be a G-module as above. Let C 0 G, X = X; and for each positive integer n, let C n G, X denote the collection of continuous maps ξ : G G X. That is, for each normal, open subgroup U G we have maps ξ U : G/U G/U s s G G ξ X U 4.1.1 in terms of sections which are continuous maps s : G/U G of profinite groups. We explain why such sections exist. To this end, we will show the following: Proposition 4.1.1. Let G = Gal k/k. Given closed subgroups U and W satisfying W U G, there exists a continuous map s such that the composition is the identity map. G/U s G/W G/U 4.1.2 Proposition 2.2.1 shows that every open subgroup U is also a closed set. Similarly, W = {1} is a closed set: Choose σ G W. Then σ U G W for any nontrivial open subgroup U G. Proof. We follow the ideas in Serre s Galois Cohomology. Consider the set { } V is a closed subgroup of U containing W I = V, s. 4.1.3 and s : G/U G/V is a continuous section First we show that I contains a maximal element V, s. We may place a partial ordering on I by saying V α, s α V β, s β whenever V β V α and the following diagram is commutative: G/V 4.1.4 G/U s s β G/V β G/U s α G/V α 35