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Modular \(p\)-adic \(L\)-functions attached to real quadratic fields and arithmetic applications. (English) Zbl 1368.11052
This paper proves a vast generalization of a number of results on \(p\)-adic \(L\)-functions for real quadratic fields, and Stark-Heegner, or Darmon, points on elliptic curves to the setting of more general elliptic modular forms. More precisely, one fixes an integer \(N\geq 1\), a prime number \(p\nmid N\), and a normalized eigenform \(f=\sum_{n\geq 1}a_nq^n\in S_{k_0+2}(\Gamma_0(Np))\) which is new at \(N\), satisfying the conditions that \(a_p^2\neq p^{k_0+1}\) and \(\mathrm{ord}_p(a_p)<k_0+1\). Let \(F\) be Coleman’s family of eigenforms passing through \(f\). Let \(K\) be a real quadratic field and let \(\psi\) be an unramified character of \(\mathrm{Gal}(\bar K/K)\). Consider the algebraic part \(L^{\mathrm{alg}}(F_k^\sharp/K,\psi,k/2+1)\) of the special value of the complex \(L\)-function \(L(F_k^\sharp/K,\psi,k/2+1)\), where if \(F_k\) is the \(k\)-specialization (where \(k\geq 2\) is an even integer) of the Coleman family \(F\) and \(F_k\) is \(p\)-old, then \(F_k^\sharp\) is the form of level \(N\) whose \(p\)-stabilization is \(F_k\) (note that all forms verity this except possibly one of them, but the constancy of the slope).
The first result is (under mild technical assumptions) the existence of a \(p\)-adic analytic \(L\)-function \(\mathcal{L}_{F/K,\psi}\) defined on the weight space \(\mathcal{W}=\mathrm{Hom}(\mathbb Z_p^\times,\mathbb Z_p^\times)\) which interpolates \(k\mapsto L^{\mathrm{alg}}(F_k^\sharp/K,\psi,k/2+1)\) for even integers \(k\geq 2\), up to some explicit Euler-type factors. This is done in all cases when these values are not forced to vanish for sign reasons.
The second main result of the paper considers the case of genus characters of \(K\). When \(\psi\) is a such a character, we have a factorization of \(\mathcal{L}_{F/K,\psi}\) into two Mazur-Kitagawa \(p\)-adic \(L\)-functions (due to the fact that the classical \(L\)-series which are interpolated by these \(p\)-adic \(L\)-functions have a similar factorization).
The third result is a relation between these \(p\)-adic \(L\)-functions and global cycles, with applications to the rationality conjecture for Darmon cycles. For this, one has to assume in addition that \(F\) has a \(p\)-new specialization in weight \(k_0+2\). Under natural parity hypotheses, the authors relate the \(p\)-adic derivatives of each of the Mazur-Kitagawa factors of \(\mathcal{L}_{F/K,\psi}\) (when \(\psi\) is a genus character) at \(k_0\) to Bloch-Kato logarithms of Heegner cycles. On the other hand, the derivatives of \(\mathcal{L}_{F/K,\psi}\) encodes the position of the so called Darmon cycles, higher weight analogues of Stark-Heegner points on elliptic curves. As an application, the authors obtain striking rationality results for these Darmon cycles.

11F85 \(p\)-adic theory, local fields
11F11 Holomorphic modular forms of integral weight
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11S40 Zeta functions and \(L\)-functions
Full Text: DOI
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