Arakelov self-intersection numbers of minimal regular models of modular curves $$X_0(p^2)$$

Abstract

We compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve X 0 (p 2 ) over Q. The computation of the self-intersection numbers are used to prove an effective version of the Bogolomov conjecture for the semi-stable models of modular curves X 0 (p 2 ) and obtain a bound on the stable Faltings height for those curves in a companion article [6].

The intersection matrices of $X_0(p^r)$ and some applications

“…In a similar spirit, Grados-von Pippich [13] computed this asymptotic expression for the case of modular curves X(N ) with some restriction on N . Recently, Banerjee-Borah-Chaudhuri [4] proved this asymptotic expression for curves X 0 (p 2 ) with a prime number p by following mostly the lines of proof in [1]. Banerjee-Chaudhuri [5] proved an effective Bogomolov conjecture and found an asymptotic expression for the stable Faltings heights for the modular curves of the form X 0 (p 2 ) with a prime number p.…”

“…Till now for all modular curves (cf. [1] and [25] for X 0 (N ), [23] for X 1 (N ), and [4] for X 0 (p 2 )), the leading term in the asymptotics for the Arakelov self-intersection number of the relative dualizing sheaf of the minimal regular model over Z for the modular curve X 0 (N ) is 3g N log N . In all these instances of modular curves, g N log N comes from the geometric part, and 2g N log N comes from the analytic part.…”

“…To derive the asymptotics for the geometric part of the Arakelov self-intersection number of the relative dualizing sheaf, we follow the line of proof from [4]. For the modular curve X 0 (p r ) with r ∈ {3, 4}, we compute the intersection matrices of the special fiber of the Edixhoven's regular model [10] for the modular curve X 0 (p r ).…”

“…Then using the Faltings-Hriljac [12], we prove that the leading term in the geometric part of the Arakelov self-intersection of the relative dualizing sheaf of X 0 (p r ) is g p r log(p r ). Note that upto r = 2 [4], these vertical divisors are obtained using only one carefully chosen component. For r ≥ 3, these carefully chosen vertical divisor (needed to apply the theorem of Faltings-Hriljac) are not supported at one irreducible component.…”

Self-intersection of the relative dualizing sheaf on modular curves X(N)

“…Namely, if h NT denotes the Néron-Tate height on the Jacobian [5], then for all ε > 0 and sufficiently large N , the set {x ∈ X 0 (N )(É) | h NT (x) ≤ (2/3 − ε) log(N )} is finite, whereas the set {x ∈ X 0 (N )(É) | h NT (x) ≤ (4/3 + ε) log(N )} is infinite. Recently, Banerjee-Borah-Chaudhuri [4] removed the square-free condition on N and proved that (1) and the Bogomolov conjecture hold for curves X 0 (p 2 )/É, with p a prime number. By its very definition, the self-intersection of the relative dualizing sheaf on modular curves is the sum of a geometric part that encodes the finite intersection of divisors coming from the cusps, and an analytic part which is given in terms of the Arakelov Green's function evaluated at these cusps.…”