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On the \(p\)-adic periods of \(X_ 0 (p)\). (English) Zbl 0864.14014
Let \(K\) be the quadratic unramified extension of \(\mathbb{Q}_p\). The modular curve \(X_0(p)\) can be uniformized over \(K\) in the sense that it can be, rigid analytically, expressed as the quotient of an upper-half space \({\mathfrak H}_\Gamma\) by a discrete subgroup \(\Gamma\) of \(PGL_2(K)\). Let \(S=\{e_0, \dots,e_g\}\) denote the set of supersingular elliptic curves in characteristic \(p\). The abelianization \(\Gamma^{ab}\) of \(\Gamma\) can be canonically identified with the augmentation subgroup \(\mathbb{Z}[S]_0\) of the free abelian group \(\mathbb{Z}[S]\). To construct the \(p\)-adic uniformization of the jacobian \(J_0(p)\) of \(X_0(p)\), Manin and Drinfeld define a bilinear symmetric pairing \(Q:\Gamma \times\Gamma \to K^*\) using the automorphy factors of explicit automorphic functions. Let \(q:\Gamma^{ab} \cong\mathbb{Z} [S]_0\to \operatorname{Hom} (\mathbb{Z} [S]_0,K^*)\) denote the associated linear map \((q(\alpha) =Q(\alpha,-))\). The set \(q(\mathbb{Z} [S]_0)\) is the set of \(p\)-adic periods of \(X_0(p)\), and \(J_0(p)\) can be rigid analytically identified with the functor \(L\mapsto \operatorname{Hom} (\mathbb{Z}[S]_0,L^*)/q (\mathbb{Z}[S]_0)\). The pairing \(Q\) takes values in \(\mathbb{Q}^*_p\). Let \(U_1(\mathbb{Q}_p)\) denote the subgroup of principal units. Oesterlé conjectured in 1985 formulae for the values of \(Q(e_i,e_j)\) modulo \(U_1(\mathbb{Q}_p)\) in terms of the \(j\)-invariants \(j(e_i)\). For instance, when \(p\equiv 1\pmod{12}\), the conjectures states that \(Q(e_i,e_j) \equiv (j(e_i)- j(e_j))^{p+1}\) if \(i\neq j\), and \(Q(e_i,e_i) \equiv p \prod_{k \neq i} (j(e_i)- j(e_k))^{-(p+1)}\).
In this paper, the author proves Oesterlé’s conjecture when \(p\equiv 1\pmod 4\), and proves it when \(p\equiv 3\pmod 4\) up to a \(\pm\) sign in the expression for \(Q(e_i,e_i)\). A different expression for the values of \(Q\) has been given by E. de Shalit [“Kronecker’s polynomial, supersingular elliptic curves, and \(p\)-adic periods of modular curves”, in: \(p\)-adic monodromy and the Birch and Swinnerton-Dyer conjecture, Boston 1991, Contemp. Math. 165, 135-148 (1994; Zbl 0863.14015)].

MSC:
14G20 Local ground fields in algebraic geometry
14H25 Arithmetic ground fields for curves
14G35 Modular and Shimura varieties
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References:
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