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PRINCIPAL HOMOGENEOUS SPACES. SELMER GROUP and SHAFAREVICH-TATE GROUP. Introduction the Goal of the Paper Is to Introduce the Se

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PRINCIPAL HOMOGENEOUS SPACES. SELMER GROUP AND SHAFAREVICH-TATE GROUP. DIMITAR P. JETCHEV Abstract. A proof of the ¯niteness of the weak Mordell-Weil groups is presented as a motiva- tion for the de¯nitions of Selmer and Shafarevich-Tate groups. The two groups are interpreted geometrically in terms of principal homogeneous spaces. We prove the ¯niteness of the Selmer group and then explain how to compute it for the case of elliptic curves. For higher genus curves, it is di±cult to describe explicitly the homogeneous spaces in terms of equation and so we present a slightly di®erent method for computing Selmer groups in terms of functions on the Jacobian of the curve. Several speci¯c examples are considered at the end. Introduction The goal of the paper is to introduce the Selmer group and Shafarevich-Tate group by using the proof of the weak Mordell-Weil theorem as a motivation. This approach allows us to present a more geometric interpretation of the two groups in terms of principal homogeneous spaces and their relation to Galois cohomology. In section 1, we introduce the Kummer pairing and prove its main properties. We compute the left and right kernels and then view the pairing in terms of the coboundary map in a long exact sequence in Galois cohomology. In section 2, we study in details the properties of the right kernel of the pairing, which turns out to be a ¯nite index subgroup of the Galois group Gal(K=K). Several classical results from algebraic number theory are assumed, such as the Dirichlet's S-unit theorem and standard facts about unrami¯ed ¯eld extensions of a number ¯eld. Section 3 is devoted to the classical techniques for computing the weak Mordell-Weil by con- structing a pairing b : E(K)=mE(K) E[m] K¤=K¤m out of the Weil pairing and using explicit description of principal homogeneous£spaces.!In section 4, we illustrate these techniques for the case m = 2, which is known as complete 2-descent. In section 5, we de¯ne Selmer group and Shafarevich-Tate group using Galois cohomology and then interpret the two groups geometrically, in terms of rational points on homogeneous spaces. We state one of the big open problems in number theory - the conjecture about the ¯niteness of X. In section 6, we prove the ¯niteness of the Selmer group and explain the main techniques for computing this group for elliptic curves. In the next section, we illustrate these techniques for the case of 2-isogenies. In the last section, we introduce another method for computing the Selmer group which works in a greater generality and uses rational functions on the Jacobian of the curve. 1. Weak Mordell-Weil Group and Kummer Pairing via Galois Cohomology Suppose that E=K is an elliptic curve over a number ¯eld and m 2 is an integer, such that E[m] E(k). We de¯ne the weak Mordell-Weil group for E=K to¸be the quotient group E(K)=mE(K), where E(K) is the group of rational points on the elliptic curve E. This group is an interesting object to study for each m, since it contains a lot of information about the full Mordell-Weil group E(K). In fact, De¯ne a pairing · : E(K) Gal(K=K) E[m]; £ ! 1 2 DIMITAR P. JETCHEV in the following way: for each P E(K) choose Q E, such that [m]Q = P and let ·(P; ) := Q Q. 2 2 First¡ of all, this pairing is well-de¯ned. Indeed, suppose that Q0 is another point, such that [m]Q = [m]Q0 = P . We need to check that Q0 Q0 = Q Q. But [m](Q0 Q) = 0, i.e. 0 0 ¡ ¡ ¡ Q Q E[m] E(K), which means that Q Q is ¯xed by the action of Gal(K=K). Hence, (Q¡0 Q)2 = Q0 Q, or Q0 Q0 = Q Q. We¡often call the pairing · the Kummer pairing. The¡ basic prop¡erties of · ¡are summarized¡ in the following proposition: Proposition 1.1. The pairing · is bilinear, with left kernel equal to mE(K) and right kernel equal to Gal(K=L), where L is a ¯eld extension of K obtained by adjoining the coordinates of all points in [m]¡1E(K) (or L = K([m]¡1E(K))). In particular, · induces a perfect bilinear pairing E(K)=mE(K) Gal(L=K) E[m]: £ ! Proof: Bilinearity of · is obvious from the de¯nition. Suppose that P E(K) is in the left 2 kernel of ·. Choose Q E(K), such that [m]Q = P . We will show that Q E(K) and thus, it will follow that P mE(K2). But this is clear from the de¯nition, since ·(P;2) = 0 means precisely that Q is ¯xed2by . Conversely, any P mE(K) is in the left kernel of ·. 2 Let Gal(K=K)(K) be in the right kernel. In this case it su±ces to show that ¯xes the ¯eld extension2L=K. Let P E(K) and Q be a point, such that [m]Q = P . Then ·(P; ) = 0 implies 2 ¡ Q = Q. Since this is true for any point in [m] 1E(K), then L is ¯xed by , i.e. Gal(K=L). 2¡ Conversely, any Gal(K=L) is in the right kernel, because it ¯xes the points in [m] 1E(K). We obtain the p2erfect bilinear pairing by moding out by the left and right kernels of ·. ¤ Next, our goal is to describe the Kummer pairing in terms of Galois cohomology. To begin with, consider the short exact sequence of Gal(K=K)-modules for a ¯xed integer m > 1 [m] 0 E[m] E(K) E(K) 0: ! ¡¡¡¡! ¡¡¡¡! ! This short exact sequence gives a long exact sequence on cohomology [m] 0 H0(Gal(K=K); E[m]) H0(Gal(K=K); E(K)) H0(Gal(K=K); E(K)) ¡¡¡¡! ¡¡¡¡! ¡¡¡¡! [m] ± H1(Gal(K=K); E[m]) H1(Gal(K=K); E(K)) H1(Gal(K=K); E(K)): ¡¡¡¡! ¡¡¡¡! ¡¡¡¡! But H0(G; M) = M G for any group G and a G-module M, so we rewrite the above sequence as [m] 0 E(K)[m] E(K) E(K) ¡¡¡¡! ¡¡¡¡! ¡¡¡¡! [m] ± H1(Gal(K=K); E[m]) H1(Gal(K=K); E(K)) H1(Gal(K=K); E(K)): ¡¡¡¡! ¡¡¡¡! ¡¡¡¡! Next, we obtain a short exact sequence out of this long exact sequence, using the fact that ker ± = mE(K). We call this short exact sequence the Kummer sequence: 0 E(K)=mE(K) ± H1(Gal(K=K); E[m]) H1(Gal(K=K); E(K))[m] 0: ! ¡¡¡¡! ¡¡¡¡! ! Since the left kernel of the pairing · is mE(K), then · induces a homomorphism ± : E(K)=mE(K) Hom(Gal(K=K); E[m]): E ! Using the Galois cohomology discussion from above, the homomorphism ±E is precisely the con- necting homomorphism ± for the long exact sequence, constructed above. PRINCIPAL HOMOGENEOUS SPACES. SELMER GROUP AND SHAFAREVICH-TATE GROUP 3 2. Properties of the Field Extension L = K([m]¡1E(K))=K After introducing the Kummer pairing in the previous section, we will to study in a more detail the ¯eld extension L = K([m]¡1E(K)), which appeared in proposition 1.1 in the previous section. The main result that we prove is that this extension is abelian of exponent m, which is unrami¯ed outside of a ¯nite set of places º. Then, using a general result from algebraic number theory, we prove that L=K is a ¯nite extension. The main properties of the ¯eld extension L=K are summarized in the following Proposition 2.1. (i) The ¯eld extension L=K is an abelian extension of exponent m. In other words, the Galois group Gal(L=K) is abelian and every element has order dividing m. (ii) If S is the ¯nite set of places, at which E has bad reduction, together with the in¯nite places and the places º, for which º(m) = 0, then L=K is unrami¯ed at each º = S. 6 2 The following lemma will be used in the proof of the proposition: Lemma 2.2. Suppose that º is a discrete valuation, such that º(m) = 0 and E=K has a good reduction at º. Then the reduction map E(K)[m] E~ (k ) is injective. ! º º Proof: This is proved in [Sil-1, VIII. 1]. ¤ x We are now ready to prove the proposition: Proof of proposition 2.1: (i) This is a consequence of proposition 1.1. Indeed, the map ·(; ) is an injection Gal(L=K) Hom(E(K); E[m]). Therefore every element of Gal(L=K) is 7!abelian¢ and of order dividing m, !since every homomorphism of Hom(E(K); E[m]) has order dividing m. (ii) Take a point Q [m]¡1E(K) and let P = [m]Q. Consider the extension L = K(Q) over K. It su±ces to show that2this extension is unrami¯ed at each º = S. Let º 0 be an extension of º in K(Q) 2 and Dº0=º and Iº0=º be the inertia and the decomposition groups, respectively. We will be done if we show that each element of Iº0=º acts trivially on K(Q). Indeed, every element of Iº0=º acts triv- ~ 0 0 0 » ~ ~ ~ ially on Eº (kº0 ), where kº0 denote the reduction of K(Q) at º . Therefore (Q Q) = Q Q = 0 ¡1 ¡ ¡ for all Iº0=º . But Q Q E[m], because Q [m] E(K).
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