arXiv subjects

Gabriele Bogo

Publications and source records attributed to Gabriele Bogo.

8 recordsLinked to original sources

arXiv2026

Partial Hasse invariants for genus zero curves in Hilbert modular varieties

We construct characteristic-zero lifts of partial Hasse invariants for genus zero non-compact curves in Hilbert modular varieties. The construction is based on recent results on the associated Picard-Fuchs differential equations. As an application, we relate the size of the non-ordinary locus of the modulo $p$ reduction of these curves to the dimension of spaces of (twisted) modular forms. We compute it explicitly for several Teichm\"uller curves, obtaining Deuring-like formulae. Moreover, we study the modulo $p$ reduction of (twisted) modular forms on not necessarily arithmetic genus-zero Fuchsian groups with modular embedding.

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arXiv2026

Atkin polynomials for families of abelian varieties with real multiplication

Generalizing the work of Atkin and Kaneko-Zagier in the elliptic case, we describe the non-ordinary locus of a genus-zero non-compact curve $Y$ in a Hilbert modular variety in terms of the zeros of generalized Atkin's orthogonal polynomials. The argument relies on the recent construction of lifts of partial Hasse invariants for $Y$. We further describe these orthogonal polynomials as denominators of Pad\'e approximants to the logarithmic derivatives of solutions of the Picard-Fuchs differential equations associated with $Y$. This provides a new link between Pad\'e approximation and the geometry of the non-ordinary locus, extending a classical observation of Igusa for the Legendre family and applying, in particular, to situations where the Picard-Fuchs equations do not admit modular solutions. As applications, we determine the three-term recurrence relations for Atkin polynomials attached to triangle curves via hypergeometric identities, and compute the supersingular locus of a double cover of the Teichm\"uller curve $W_{17}$. In the latter case, we conjecture that the associated supersingular polynomial is self-reciprocal, implying that supersingular points occur in pairs.

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arXiv2025

Integrality of Picard-Fuchs differential equations of Kobayashi geodesics and applications

We prove that the holomorphic solutions of Picard-Fuchs differential equations associated with one-parameter families of abelian varieties with real multiplication admit power series expansions with $S$-integral coefficients at a maximal unipotent monodromy point. This extends classical integrality results for hypergeometric functions and Bouw-M\"oller's work on Teichm\"uller curves. The integral solutions are related to the non-ordinary locus of the modulo $p$ reduction of the family, whose cardinality we bound in terms of the Euler characteristic and Lyapunov exponents of the base curve. In some cases, the non-ordinary locus can be recovered by truncating the integral solutions, as in Igusa's classical observation for the Legendre family. We also establish $S$-integrality of expansions of modular forms at cusps in terms of a modular function for (not necessarily arithmetic) Fuchsian groups with modular embeddings, and deduce congruences. These results are applied in subsequent work to construct lifts of partial Hasse invariants for rational curves in Hilbert modular varieties.

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arXiv2023

Ramanujan systems of Rankin-Cohen type and hyperbolic triangles

In the first part of the paper we characterize certain systems of first order nonlinear differential equations whose space of solutions is an $\mathfrak{sl}_2(\mathbb{C})$-module. We prove that such systems, called Ramanujan systems of Rankin-Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin-Cohen structure. In the second part of the paper we consider triangle groups $Δ(n,m,\infty)$. By means of modular embeddings, we associate to every such group a number of systems of non linear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on $Δ(n,m,\infty)$ are generated by the solutions of one such system (which is of Rankin-Cohen type). As a corollary we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non classical setting, we construct the space of integral weight twisted modular form on $Δ(2,5,\infty)$ from solutions of systems of nonlinear ODEs.

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arXiv2023

Laurent expansions of meromorphic modular forms

In this paper, we study the Laurent coefficients of meromorphic modular forms at CM points by giving two approaches of computing them. The first is a generalization of the method of Rodriguez-Villegas and Zagier, which expresses the Laurent coefficients as constant terms of a family of polynomials obtained through recursion. The second applies to meromorphic modular forms that are regularized theta lifts, and expresses their Laurent coefficients in terms of Fourier coefficients of harmonic Maass forms.

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arXiv2022

Span of Restriction of Hilbert Theta Functions

In this paper, we study the diagonal restrictions of certain Hilbert theta series for a totally real field $F$, and prove that they span the corresponding space of elliptic modular forms when the $F$ is quadratic or cubic. Furthermore, we give evidence of this phenomenon when $F$ is quartic, quintic and sextic.

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arXiv2022

Accessory Parameters for Four-Punctured Spheres

We study the accessory parameter problem for four-punctured spheres from the point of view of modular forms. The value of the accessory parameter giving the uniformization is characterized as the unique zero of a system of equations. This gives an effective method to compute the uniformizing differential equation. As an application, we compute numerically and study the local expansion of the real-analytic function associating to a four-punctured sphere the value of its uniformizing parameter, and make some observations on its coefficients.

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arXiv2021

Modular forms, deformation of punctured spheres, and extensions of symmetric tensor representations

Let~$X=\Po/Γ$ be an~$n$-punctured sphere, $n>3$. We introduce and study~$n-3$ deformation operators on the space of modular forms~$M_*(Γ)$ based on the classical theory of uniformizing differential equations and accessory parameters. When restricting to modular functions, we recover a construction in Teichmüller theory related to the deformation of the complex structure of~$X$. We describe the deformation operators in terms of derivations with respect to Eichler integrals of weight-four cusp forms, and in terms of vector-valued modular forms attached to extensions of symmetric tensor representations.

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