## 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.

### MSC:

 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

Zbl 0882.11034
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