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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, Oesterlé 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].

##### MSC:
 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
##### Keywords:
modular curve; Mumford curve; theta functions