Arakelov self-intersection numbers of minimal regular models of modular curves $$X_0(p^2)$$ X 0 ( p 2 )
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Mathematische Zeitschrift
Arakelov self-intersection numbers of minimal regular models of modular curves X0 (p2 ) Debargha Banerjee1 · Diganta Borah1 · Chitrabhanu Chaudhuri1 Received: 26 March 2018 / Accepted: 13 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 (Banerjee and Chaudhuri in Isr J Math, 2020). Keywords Arakelov theory · Heights · Eisenstein series Mathematics Subject Classification Primary 11F72; Secondary 14G40 · 37P30 · 11F37 · 11F03 · 11G50
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . 2 Arakelov intersection pairing . . . . . . . . . . . 3 Canonical Green’s functions and Eisenstein series 4 Spectral expansions of automorphic kernels . . . . 5 Minimal regular models of Edixhoven . . . . . . . 6 Algebraic part of self-intersection . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
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Debargha Banerjee was partially supported by the SERB Grant YSS/2015/001491. Diganta Borah author was partially supported by the DST-INSPIRE Grant IFA-13 MA-21. Chitrabhanu Chaudhuri was partially supported by the DST-INSPIRE grant IFA-16 MA-88 during the course of this research.
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Diganta Borah [email protected] Debargha Banerjee [email protected] Chitrabhanu Chaudhuri [email protected]
1
Indian Institute of Science Education and Research (IISER), Pune 411008, India
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D. Banerjee et al.
1 Introduction In this article, we derive an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of the minimal regular model over Z for the modular curve X 0 ( p 2 ) in terms of its genus g p2 for a prime p. For odd, square-free N ∈ N, this quantity was computed for the congruence subgroups Γ0 (N ) [1], Γ1 (N ) [28] and recently for the principal congruence subgroups Γ (N ) [13]. We generalize our work to semi-stable models of these modular curves in a companion article [6]. We use the computation regarding the infinite part of the Arakelov self-intersection for the modular curve X 0 ( p 2 ) done in the present paper to compute the Arakelov self-intersection numbers for semi-stable models in [6]. From the viewpoint of Arakelov theo
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