On the \(p\)-adic periods of \(X_ 0 (p)\)
type: Item
instance of: scholarly article
title: On the \(p\)-adic periods of \(X_ 0 (p)\)
zbMATH Open document ID: 0864.14014
DOI: 10.1007/BF01461000
author: Ehud de Shalit
published in: Mathematische Annalen
publication date: 1997-06-30T00:00:00Z
full work available at URL: https://eudml.org/doc/165377
review text: 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 \(q:\Gamma^{ab} \cong\mathbb{Z} [S]_0\to \Hom (\mathbb{Z} [S]_0,K^*)\) 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 \(L\mapsto \Hom (\mathbb{Z}[S]_0,L^*)/q (\mathbb{Z}[S]_0)\). 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 \textit{E. de Shalit} [``Kronecker's polynomial, supersingular elliptic curves, and \(p\)-adic periods of modular curves…
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 840195
zbMATH Keywords: \(p\)-adic periods
zbMATH Keywords: \(p\)-adic uniformization
zbMATH Keywords: modular curve
reviewed by: Dino Lorenzini
MaRDI profile type: Publication
cites work: Q4314331
cites work: Q5635548
OpenAlex ID: W2083856327
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description: scientific article; zbMATH DE number 840195