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A rigid analytic Gross-Zagier formula and arithmetic applications. (With an appendix by B. Edixhoven). (English) Zbl 1029.11027
Let \(f\) be a newform of weight 2 and squarefree level \(N\). Its Fourier coefficients generate a ring \({\mathcal O}_f\) whose fraction field \(K_f\) has finite degree over \(\mathbb{Q}\). Fix an imaginary quadratic field \(K\) of discriminant prime to \(N\), corresponding to a Dirichlet character \(\varepsilon\). The \(L\)-series \(L(f/K,s)= L(f,s) L(f\otimes \varepsilon,s)\) of \(f\) over \(K\) has an analytic continuation to the whole complex plane and a functional equation relating \(L(f/K,s)\) to \(L(f/K,2-s)\). Assume that the sign of this functional equation is 1, so that \(L(f/K,s)\) vanishes to even order at \(s=1\). This is equivalent to saying that the number of prime factors of \(N\) which are inert in \(K\) is odd. Fix any such prime, say \(p\).
The field \(K\) determines a factorization \(N= N^+N^-\) of \(N\) by taking \(N^+\), resp. \(N^-\) to be the product of all the prime factors of \(N\) which are split, resp. inert in \(K\). Given a ring-class field extension \(H\) of \(K\) of conductor \(c\) prime to \(N\), write \(H_n\) for the ring-class field of conductor \(cp^n\).
Let \(J\) be the Jacobian of \(X\), \({\mathcal J}_n\) the Néron model of \(J\) over \(H_n\), and \(\Phi_n\) the group of connected components at \(p\) of \({\mathcal J}_n\). More precisely, \(\Phi_n:= \bigoplus_{{\mathfrak p}\mid p}\Phi_{\mathfrak p}\), where \(\Phi_{\mathfrak p}\) is the group of connected components of the fiber at \({\mathfrak p}\) of \({\mathcal J}_n\) and the sum is extended over all primes \({\mathfrak p}\) of \(H_n\) above \(p\). Define a Heegner divisor \(\alpha_n:= (P_n)- (w_NP_n)\), where \(w_N\) is the Atkin-Lehner involution denoted \(w_{N^+p, N^-/p}\). We view \(\alpha_n\) as an element of \({\mathcal J}_n\), and let \(\overline{\alpha}_n\) be its natural image in \(\Phi_n\). We have found that the position of \(\overline{\alpha}_n\) in \(\Phi_n\) is encoded in the special values of the \(L\)-functions attached to cusp forms of weight 2 on \(X\) twisted by characters \(\chi\) of \(\Delta:= \text{Gal} (H/K)\).
More precisely, observe that the Galois group \(\text{Gal} (H_n/K)\) acts on \(J(H_n)\) and on \({\mathcal J}_n\). Since the primes above \(p\) are totally ramified in \(H_n/H\), the induced action on \(\Phi_n\) factors through \(\Delta\). Define \(e_\chi:= \sum_{g\in\Delta} \chi^{-1}(g) g\in \mathbb{Z}[\chi] [\Delta]\), and let \(\overline{\alpha}_n^\chi:= e_\chi \overline{\alpha}_n\). The ring \(\mathbb{T}\) generated over \(\mathbb{Z}\) by the Hecke correspondences on \(X\) acts in a compatible way on \(J(H_n)\), \({\mathcal J}_n\) and \(\Phi_n\). Write \(\varphi_f: \mathbb{T}\rightarrow{\mathcal O}_f\) for the homomorphism associated to \(f\) by the Jacquet-Langlands correspondence, and let \(\pi_f\in \mathbb{T}\otimes K_f\) be the idempotent corresponding to \(\varphi_f\). Fix \(n_f\in{\mathcal O}_f\) so that \(\eta_f:= n_f\pi_f\) belongs to \(\mathbb{T}\otimes{\mathcal O}_f\), and define \(\overline{\alpha}_n^{f,\chi}:= \eta_f \overline{\alpha}_n^\chi\).
The group \(\Phi_n\) is equipped with a canonical monodromy pairing \([\;,\;]_n: \Phi_n\times \Phi_n\rightarrow \mathbb{Q}/\mathbb{Z}\), which we extend to a Hermitian pairing on \(\Phi_n\otimes{\mathcal O}_f[\chi]\) with values in \(K_f[\chi]/ {\mathcal O}_f[\chi]\), denoted in the same way by abuse of notation. Our main result is:
Theorem A. Suppose that \(\chi\) is a primitive character of \(\Delta\). Then \[ [\overline{\alpha}_n^\chi, \overline{\alpha}_n^{f,\chi}]_n= \frac{1}{e_n} \frac{L(f/K,\chi,1)} {(f,f)} \sqrt{d}\cdot u^2\cdot n_f\pmod {{\mathcal O}_f[\chi]}, \] where \((f,f)\) is the Petersson scalar product of \(f\) with itself, and \(d\) denotes the discriminant of \({\mathcal O}\).
The proof is based on Grothendieck’s description of \(\Phi_n\), on the work of Edixhoven on the specialization map from \({\mathcal J}_n\) to \(\Phi_n\) given in the appendix of this paper, and on a slight generalization of Gross’ formula for special values of \(L\)-series (which we assume in this paper and which will be contained in [H. Daghigh, Ph.D. thesis]). Theorem A can be viewed as a \(p\)-adic analytic analogue of the Gross-Zagier formula, and it was suggested by the conjectures of Mazur-Tate-Teitelbaum type formulated in [M. Bertolini and H. Darmon, Invent. Math. 126, 413-456 (1996; Zbl 0882.11034)]. It is considerably simpler to prove than the Gross-Zagier formula, as it involves neither derivatives of \(L\)-series nor global heights of Heegner points.
The above formula has a number of arithmetic applications. Let \(A_f\) be the Abelian variety quotient of \(J\) associated to \(\varphi_f\) by the Eichler-Shimura construction. Following the methods of Kolyvagin, we can use the Heegner points \(\alpha_n\) to construct certain cohomology classes in \(H^1(H, (A_f)_{e_n})\), whose local behaviour is related via Theorem A to \(L(A_f/K,\chi,1)= \prod_\sigma L(f^\sigma/ K,\chi,1)\), where \(\sigma\) ranges over the set of embeddings of \(K_f\) in \(\overline{\mathbb{Q}}\). This can be used to study the structure of the \(\chi\)-isotypical component \(A_f(H)^\chi:= e_\chi A_f(H)\subset A_f(H)\otimes \mathbb{Z}[\chi]\) of the Mordell-Weil group \(A_f(H)\). In particular, we show:
Theorem B. If \(L(A_f/K,\chi,1)\) is nonzero, then \(A_f(H)^\chi\) is finite.
When \(\chi= \overline{\chi}\), this result also follows from the work of Gross-Zagier and Kolyvagin-Logachev, but if \(\chi\) is nonquadratic the previous techniques cannot be used to study these questions.
Theorem B allows us to control the growth of Mordell-Weil groups over anticyclotomic \(\mathbb{Z}_\ell\)-extensions, addressing a conjecture of Mazur. Let \(f\) and \(K\) be as at the beginning of this section. Let \(\ell_1,\dots, \ell_k\) be primes not dividing \(N\), and let \(K_\infty\) denote the compositum of all the ring-class field extensions of \(K\) of conductor of the form \(\ell_1^{n_1}\dots \ell_k^{n_k}\), where \(n_1,\dots, n_k\) are nonnegative integers. Thus, the Galois group of \(K_\infty/K\) is isomorphic to the product of a finite group by \(\mathbb{Z}_{\ell_1}\times \cdots\times \mathbb{Z}_{\ell_k}\).
Theorems A and B provide a technique to study “analytic rank-zero situations” in terms of Heegner points of conductor divisible by powers of a prime \(p\) of multiplicative reduction for \(A_f\) and inert in \(K\). What makes this possible, ultimately, is a “change of signs” phenomenon: If \(L(f/K,s)\) vanishes to even order, and \(\chi\) is an anticyclotomic character of conductor \(cp^n\) with \(c\) prime to \(N\), then \(L(f/K,\chi,s)\) vanishes to odd order, and there are Heegner points on \(A_f\) defined over the extension cut out by \(\chi\). The previous applications of the theory of Heegner points, such as the analytic formula of Gross-Zagier and the methods of Kolyvagin, occur in situations where \(L(f/K,s)\) and \(L(f/K,\chi,s)\) both vanish to odd order.

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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