15   Galois Cohomology  Let G be a group and A be a G module  an abelian group with an action of G via group homomorphisms   G module means exactly the same thing as     G   module   Definition   H 0   Define      H 0   G   A     A G     a   A       a     a       G         Definition   Cochains   Define      C 1   G   A       maps G   A        called  cochains     Definition   Cocycles   Define      Z 1   G   A         a         G   a           a       a             G        called  cocycles     Definition   Coboundaries   Define      B 1   G   A           b   b       G   b   A        called  coboundaries     Note   C 1   G   A     Z 1   G   A     B 1   G   A     Then we can define   Definition   H 1   Define      H 1   G   A     Z 1   G   A   B 1   G   A         Remark   If G acts trivially on A  then      H 1   G   A     Hom   G   A         Theorem 15 1  Assuming that   we have a short exact sequence of G modules     0   A     B     C   0       Then  it gives rise to a long exact sequence of abelian groups       0   A G     B G     C G     H 1   G   A         H 1   G   B         H 1   G   C         Proof  Omitted     Definition   d e l t a   Let c   C G  Then b   B such that     b     c  Then            b   b       c   c   0       G     so   b   b       a     for some a     A   Can check    a         G   Z 1   G   A     Then     c     class of   a         G in  H 1   G   A     Theorem 15 2  Assuming that   A is a G module  H   G a normal subgroup  Then  there is an inflation restriction exact sequence      0   H 1   G   H   A H     inf H 1   G   A     res H 1   H   A       Proof  Omitted     Let K be a perfect field  Then Gal   K     K   is a topological group with basis of open subgroups being the Gal   K     L   for   L   K         If G   Gal   K     K   then we modify the definition of  H 1   G   A   by insisting    1    The stabiliser of each a   A is an open subgroup of G    2    All cochains G   A are continuous  where A is given the discrete topology   Then      H 1   Gal   K     K     A     lim   L L   K finite Galois H 1   Gal   L   K     A Gal   K     L            direct limit is with respect to inflation maps    Theorem   Hilbert s Theorem 90   Assuming that   L   K a finite Galois extension  Then      H 1   Gal   L   K     L       0       Proof  Let G   Gal   L   K    Let    a         G   Z 1   G   L      Distinct automorphisms are linearly independent  so there exists y   L such that            G a     1     y       x   0       Then         x           G     a       1       y     a         G a       1       y       x     Then a         x   x for all     G   a           a     a         a       1   a   a       1    So    a         G   B 1   G   L      Therefore  H 1   G   L       0     Corollary  H 1   Gal   K     K     K         0   Application   Assume char K   n  There is a short exact sequence of Gal   K     K   modules     0     n   K       K       0 x   x n     Long exact sequence      K     K     H 1   Gal   K     K       n     H 1   Gal   K     K     K           0 by Hilbert 90 x   x n     Therefore  H 1   Gal   K     K       n     K       K     n   If   n   K then     Hom cts   Gal   K     K       n     K       K     n     Finite subgroups of Hom cts   Gal   K     K       n   are of the form Hom   Gal   L   K       n   for L   K a finite abelian extension of K of exponent dividing n   This gives another proof of Theorem 11 2   Notation  H 1   K       means  H 1   Gal   K     K           Let     E   E   be an isogeny of elliptic curves over K  Short exact sequence of Gal   K     K   modules     0   E         E     E     0     has long exact seqeucne      E   K       E     K       H 1   K   E           H 1   K   E         H 1   K   E           We get a short exact sequence   Now take K a number field   For each place v  fix an embedding K     K   v  Then Gal   K   v   K v     Gal   K     K     Definition   Selmer group   We define the   Selmer  group     S         E   K     ker   H 1   K   E             v H 1   K v   E             H 1   K   E           res v         Im     v     v        the map   H 1   K   E             v H 1   K v   E   is as in the commutative diagram above    Definition   Tate Shafarevich group   The Tate Shafarevich group  is      T S   E   K     ker   H 1   K   E       v H 1   K v   E           We get a short exact sequence      0   E     K     E   K     S         E   K     T S   E   K       K     0       Taking       n   gives      0   E   K   n E   K     S   n     E   K     T S   E   K     n     0       Reorganising the proof of Mordell Weil gives  Theorem 15 3  S   n     E   K   is finite   Proof  For L   K a finite Galois extension    Therefore by extending our field  we may assume  E   n     E   K   and hence by the Weil pairing   n   K   Therefore  E   n       n     n as a Gal   K     K   module  Then     H 1   K   E   n       H 1   K     n     H 1   K     n     K       K     n   K       K     n     Let      S     primes of bad reduction for E       v   n          a finite set of places    Define the subgroup of  H 1   K   A   unramified outside of S as      H 1   K   A   S     ker   H 1   K   A       v   S H 1   K v nr   A           There is a commutative diagram with exact rows    The bottom map   n is surjective   v   S  see Theorem 9 8     Therefore     S   n     E   K     H 1   K   E   n     S     H 1   K     n   S     H 1   K     n   S       But     H 1   K     n   S     ker   K     K     n     v   S   K v nr       K v nr     n     K   S   n       which is finite by Lemma 11 4     Remark   S   n     E   K   is finite and effectively computable   It is conjectured that    T S   E   K          This would imply that rank E   K   is effectively computable