International Congress of Mathematicians

Elliptic Curves and Class Field Theory 187(To simplify notation and to avoid some complications, we will often work withQp-vector spaces instead of natural Z p -modules; in particular we have tensored theusual Iwasawa ring with Q p .) For every (finite or infinite) extension F of K in K œwe also defineA F := Z p [[Gal(F/K)]] ® Zp Q p , I F := ker{A -» A F }.Then IK is the augmentation ideal of A, and if [F : K] is finite then Ap is just thegroup ring Q p [Gal(F/K)]. If Gal(F/K) is Z p or Z 2 , and M is a finitely generatedtorsion Af-module, then charA F (M) will denote the characteristic ideal of M. Inparticular charA F (M) is a principal ideal of Ap.There is a Q p -projective line of Z p -extensions of K, all contained in K^.Among these are two distinguished Z p -extensions:• the cyclotomic Z p -extension K^ci , the compositum of K with the unique(cyclotomic) Z p -extension of Q (write F cyc i = Gal(K^d/K),A cyc i = A^-cyd),• the anticyclotomic Z p -extension K%£ u , the unique Z p -extension of K thatis Galois over Q with non-abelian, and in fact dihedral, Galois group (writeFanti = Gal(F^J 1 /K), A a „ti = A K^)-Then T = Y cyci ® F anti and A = A cyd ® Zp A anti .Complex conjugation r : K —¥ K acts on F, acting as +1 on F cyc i and ~1 onF ant i. This induces nontrivial involutions of A and A ant i, which we also denote by r.If M is a module over A (or similarly over A ant i), let M^ denote the module whoseunderlying abelian group is M but where the new action of 7 £ T on TO £ M^ isgiven by the old action of 7 T on rn.Our A-modules will usually come with a natural action of Gal(K 0O /Q). Theseactions are continuous and Z p -linear, and satisfy the formula f (7-771) = 7 T -f(ro) forevery lift f of r to Gal(K 0O /Q). Thus the action of any lift f induces an isomorphismAf ^ M( T \ We will refer to such A or A a „ti-modules as semi-linear r-modules.If M is a semi-linear r-module and is free of rank one over A ant j, we define the signof M to be the sign ±1 of the action of r on the one-dimensional Q p -vector spaceM ®A„ti A if. Such an M is completely determined (up to isomorphism preservingits structure) by its sign.Definition 2. If M and A are semi-linear T-modules, then a (A-bilinear) A-valuedr-Hermitian pairing IT is a A-module homomorphism ix : M ® A M^ —^ A suchthat for every lift f of r to Gal(K 0O /Q)n(m ® n) = n(n ® m) T= n(fn ® fro).3. Universal normsDefinition 3. If K C F C Koo, the universal norm module U(F) is the projectivelimitU(F):=Q p ® Hm (E(L)®Z p )KCLCF