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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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