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Kronecker’s polynomial, supersingular elliptic curves, and \(p\)-adic periods of modular curves. (English) Zbl 0863.14015
Mazur, Barry (ed.) et al., \(p\)-adic monodromy and the Birch and Swinnerton-Dyer conjecture. A workshop held August 12-16, 1991 in Boston, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 165, 135-148 (1994).
The modular curve \(X_0(p)\) \((p\geq 5\) a prime) has bad reduction at \(p\): The special fibre \(\overline {X_0(p)}\) of the (semi-) stable model consists of two non-singular rational curves that intersect transversally in the \(g+1\) points \(e_0,\dots,e_g\) corresponding to the supersingular elliptic curves \((g\) is the genus of \(X_0(p))\). Thus \(X_0(p)\) is a Mumford curve, i.e. its analytification can be represented as the quotient of some open dense domain in \(\mathbb{P}^1(K)\) by a Schottky group \(\Gamma\subset \text{PGL}_2(K)\) \((K\) a suitable extension of \(\mathbb{Q}_p)\). According to Manin and Drinfeld, the Jacobian of \(X_0(p)^{an}\) can analytically be constructed as the quotient of \(\operatorname{Hom}(\Gamma,K^*)\) by the period lattice \(q(\Gamma)\); they defined the period pairing \(Q:\Gamma^{ab} \times\Gamma^{ab} \to K^*\) (which gives \(q(\Gamma))\) with the help of \(p\)-adic theta functions. Using the \(j\)-invariant of the supersingular curves, OesterlĂ© has defined a pairing on the free abelian group \(\mathbb{Z}[S]\) on the set \(S=\{e_0, \dots, e_g\}\). Interpreting \(\Gamma^{ab}\) as the homology group of the intersection graph of \(\overline {X_0(p)}\) one can identify \(\Gamma^{ab}\) with the augmentation subgroup of \(\mathbb{Z}[S]\). In Math. Ann. 303, No. 3, 457-472 (1995), the author proved that Oesterle’s pairing \(\langle , \rangle\) (essentially) agrees with the period pairing up to \(p\)-adic units. In the present paper he finds an alternate formula for the diagonal values \(\langle e_i,e_i\rangle\) in terms of special values of Kronecker’s polynomial (which gives the relation between \(j(\tau)\) and \(j(p\tau)\), and thus an affine equation for \(X_0(p))\).
For the entire collection see [Zbl 0794.00016].

14G35 Modular and Shimura varieties
14H20 Singularities of curves, local rings
11G18 Arithmetic aspects of modular and Shimura varieties
14G20 Local ground fields in algebraic geometry