International Congress of Mathematicians

Elliptic Curves and Class Field Theory 189for every ±7 £ ±Y, the element c®c^ £ 'H®^Dii 'H^T' > is independent of the choiceof Heegner generator, and is therefore a totally canonical generator of the free, rankone A anti -module % ®A„ ti %^ •Definition 7. The Heegner F-function (for the triple (E,K,p)running hypotheses) is the element£ := h(c® C (T) ) G F cyc i ® Zp A anti .Conjecture 8. F cyc i ® cha,r(U/H) 2 = A anti £ inside F cyc i ® A anti .satisfying ourOne sees easily that F cyc i ® char(U/H) 2 D A ant j£, and that Conjecture 8 isequivalent to the Height Conjecture (Conjecture 6).4. The analytic theoryThe ( "two-variable" ) p-adic F-function for F over K is an element L £ Aconstructed by Haran [3] and by a different, more general, method by Hida [4] (seealso the papers of Perrin-Riou [13, 14]). The F-function L is characterized by thefact that it interpolates special values of the classical Hasse-Weil F-function of twistsof F over K. More precisely, embedding Q both in C and Q p , if x '• T —¥ Z xis a character of finite order thenx(L) = C( X ) Lda 7 a :; i !^X' 1) (4.1)where L c i aiSS j cai i(E/ K , X, s ) ' 1S the Hasse-Weil F-function of the twist of E/ K by x,c(x) is an explicit algebraic number (cf. [13] Théorème 1.1), fp is the modular formon Y'o(N) corresponding to F, and ||/#|| is its Petersson norm.Projecting L £ A to the cyclotomic or the anticyclotomic line via the naturalprojections A —t A cyc i and A —t A ant ;, we get "one-variable" p-adic F-functionsL 1-^ F cyc i £ A cyc i and L 1-^ F ant ; £ A ant ;.It follows from the functional equation satisfied by L ([13] Théorème 1.1) and theHeegner Hypothesis that F ant i = 0. In other words, viewing A = A ant ;[[r cyc i]] as thecompleted group ring of F cyc i with coefficients in A ant ;, we have that the "constantterm" of L £ A ant ; [[F cyc i]] vanishes. We now consider its "linear term."There is a canonical isomorphism of (free, rank one) A ant ;-modulescZ pxIcycl ®Z P A ant i — I if >nti /IK%which sends 7 1 £ F cyc i ® Zp A anti to 7 - 1 £I K^/12ifantiConjecture 9 (A-adic Gross-Zagier Conjecture). Let L' denote the image ofL under the map Isanti/I^„ ti ^y F cyc i ®z p A ant i. ThenL' = d- x £where d is the degree of the modular parametrization X 0 (N) —t E.