Integration on \({\mathcal H_p}\times\mathcal H\) and arithmetic applications.

*(English)*Zbl 1035.11027The main purpose of the paper is to give a conjectural \(p\)-adic construction of points on elliptic curves, where the points are defined over certain ring class fields of real quadratic fields. This is an analogue of the classical theory of Heegner points, which uses the theory of complex multiplication to produce points defined over ring class fields of imaginary quadratic fields.

Let \(p\) be a prime and let \(N=pM\) with \(p\nmid M\). Let \(R\) be the ring of 2 by 2 matrices with entries in \(\mathbb Z[1/p]\) that are upper triangular mod \(M\). Let \(\Gamma\) be the image in PSL\(_2(\mathbb Z[1/p])\) of the elements of \(R^{\times}\) of determinant 1. Then \(\Gamma\) acts on the classical upper half plane \(\mathcal H\) and on its \(p\)-adic analogue \(\mathcal H_p=\mathbb P_1(\mathbb C_p)-\mathbb P_1(\mathbb Q_p)\). The quotient \((\mathcal H_p\times \mathcal H)/\Gamma\) has many properties analogous to a Hilbert modular surface.

Let \(f\) be a newform for \(\Gamma_0(N)\) with rational Fourier coefficients, so \(f\) is attached to an elliptic curve \(E\) over \(\mathbb Q\). The author uses \(f\) to define a certain double multiplicative integral. Let \(K=\mathbb Q\times \mathbb Q\) or a real quadratic field and let \(\Psi\) be a \(\mathbb Q\)-algebra embedding of \(K\) into \(M_2(\mathbb Q)\). This integral is used to define a \(p\)-adic period \(I_{\Psi}\) attached to \(\Psi\). Let \(\Phi_{\text{Tate}}:\mathbb C_p^{\times}\to E(\mathbb C_p)\) be the Tate parameterization of \(E\), with the kernel of \(\Phi_{\text{Tate}}\) generated by \(q\in p\mathbb Z_p\). It is proved that for any \(p\)-adic logarithm, \({\text{ord}}_p(q)\log(I_{\Psi}) = {\text{ord}}_p(I_{\Psi})\log(q)\). In the case \(K=\mathbb Q\times \mathbb Q\), this a reformulation of a result of R. Greenberg and G. Stevens [Invent. Math. 111, 407–447 (1993; Zbl 0778.11034)]. It is conjectured that \(I_{\Psi}\in q^{\mathbb Z}\).

Now assume that \(p\) is inert in the real quadratic field \(K\). The double multiplicative integral is used to define (assuming certain conjectures) a period \(J_{\Psi}\in K_p^{\times}/q^{\mathbb Z}\). Let \(\mathfrak f_K\) be the conductor of the order \(\Psi(K)\cap R\) and let \(H^+\) be the narrow ring class field of \(K\) of conductor \(\mathfrak f_{\Psi}\). Then the author conjectures that the local point \(\Phi_{\text{Tate}}(J_{\Psi})\in E(K_p)\) is actually a (global) point in \(E(H^+)\). Moreover, a conjectural reciprocity law is formulated giving the Galois action on these points, much in the spirit of the classical Shimura reciprocity law. Numerical evidence for the conjectures is contained in a paper of H. Darmon and P. Green [Exp. Math. 11, 37–55 (2002; Zbl 1040.11048)], where an analogue of the work of B. Gross and D. Zagier [Invent. Math. 84, 225–320 (1986; Zbl 0608.14019)] is also investigated.

Let \(p\) be a prime and let \(N=pM\) with \(p\nmid M\). Let \(R\) be the ring of 2 by 2 matrices with entries in \(\mathbb Z[1/p]\) that are upper triangular mod \(M\). Let \(\Gamma\) be the image in PSL\(_2(\mathbb Z[1/p])\) of the elements of \(R^{\times}\) of determinant 1. Then \(\Gamma\) acts on the classical upper half plane \(\mathcal H\) and on its \(p\)-adic analogue \(\mathcal H_p=\mathbb P_1(\mathbb C_p)-\mathbb P_1(\mathbb Q_p)\). The quotient \((\mathcal H_p\times \mathcal H)/\Gamma\) has many properties analogous to a Hilbert modular surface.

Let \(f\) be a newform for \(\Gamma_0(N)\) with rational Fourier coefficients, so \(f\) is attached to an elliptic curve \(E\) over \(\mathbb Q\). The author uses \(f\) to define a certain double multiplicative integral. Let \(K=\mathbb Q\times \mathbb Q\) or a real quadratic field and let \(\Psi\) be a \(\mathbb Q\)-algebra embedding of \(K\) into \(M_2(\mathbb Q)\). This integral is used to define a \(p\)-adic period \(I_{\Psi}\) attached to \(\Psi\). Let \(\Phi_{\text{Tate}}:\mathbb C_p^{\times}\to E(\mathbb C_p)\) be the Tate parameterization of \(E\), with the kernel of \(\Phi_{\text{Tate}}\) generated by \(q\in p\mathbb Z_p\). It is proved that for any \(p\)-adic logarithm, \({\text{ord}}_p(q)\log(I_{\Psi}) = {\text{ord}}_p(I_{\Psi})\log(q)\). In the case \(K=\mathbb Q\times \mathbb Q\), this a reformulation of a result of R. Greenberg and G. Stevens [Invent. Math. 111, 407–447 (1993; Zbl 0778.11034)]. It is conjectured that \(I_{\Psi}\in q^{\mathbb Z}\).

Now assume that \(p\) is inert in the real quadratic field \(K\). The double multiplicative integral is used to define (assuming certain conjectures) a period \(J_{\Psi}\in K_p^{\times}/q^{\mathbb Z}\). Let \(\mathfrak f_K\) be the conductor of the order \(\Psi(K)\cap R\) and let \(H^+\) be the narrow ring class field of \(K\) of conductor \(\mathfrak f_{\Psi}\). Then the author conjectures that the local point \(\Phi_{\text{Tate}}(J_{\Psi})\in E(K_p)\) is actually a (global) point in \(E(H^+)\). Moreover, a conjectural reciprocity law is formulated giving the Galois action on these points, much in the spirit of the classical Shimura reciprocity law. Numerical evidence for the conjectures is contained in a paper of H. Darmon and P. Green [Exp. Math. 11, 37–55 (2002; Zbl 1040.11048)], where an analogue of the work of B. Gross and D. Zagier [Invent. Math. 84, 225–320 (1986; Zbl 0608.14019)] is also investigated.

Reviewer: Lawrence C. Washington (College Park)