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Abdellah Sebbar

Publications and source records attributed to Abdellah Sebbar.

17 recordsLinked to original sources
arXiv2026

Replication Descent and $J$-Finality of Replicable Functions

We show that replication carries congruence symmetry down an explicit level tower. If a normalized replicable function $f$, holomorphic on the upper half-plane, is invariant under $\Gamma_0(N)$, then its $n$th replicate is invariant under $\Gamma_0(N/(N,n))$. Thus every replicate whose index is divisible by $N$ is the normalized modular invariant $J=j-744$. This gives a direct and classification-free proof of $J$-finality for congruence-invariant replicable functions. The argument uses only the replication identities and an elementary generation theorem for congruence subgroups; it requires neither complete replicability nor arithmetic hypotheses on the Fourier coefficients. We then apply the descent law to completely replicable functions of finite replication order. Their replication towers have a canonical terminal replicate, and the only possible terminal functions are $J$, $q^{-1}$, and $q^{-1}+q$. Moreover, a finite-order completely replicable function is $J$-final precisely when its terminal replicate is $J$, and any symmetry beyond translations forces this alternative.

math.NT↗
arXiv2025

Jacobi pole divisors and supersingular lifts

We study the reducible parameter values of a second-order modular differential equation for the full modular group and show that their pole data is controlled by a single polynomial linking three a priori different structures. The residue conditions for the associated weight two form are exactly the Stieltjes equilibrium equations for four shifted Jacobi families. This determines the poles uniquely and produces a canonical divisor on $X(1)$ supported on the elliptic arc. After the standard eta-normalization, with $\ell=6r$, the differential equation takes the Kaneko-Zagier form, and the same Jacobi polynomial is precisely the polynomial factor in its distinguished modular solution. When $\ell$ is prime, this polynomial is an $\ell$-integral characteristic-zero lift of the non-elliptic supersingular polynomial. Consequently, its splitting field over $\Q_\ell$ is unramified of degree at most two, with Frobenius cycle structure determined by the supersingular $j$-invariants. Complete splitting occurs exactly when every supersingular $j$-invariant in characteristic $\ell$ is defined over $\F_\ell$, equivalently when the Fricke quotient $X_0^+(\ell)$ has genus zero. These are Ogg's primes, which are exactly the prime divisors of the order of the Monster. A common singular Sturm-Liouville equation further gives strict interlacing for consecutive reducible parameters. Finally, we prove that the canonical representatives of the poles are transcendental although their $j$-invariants are algebraic, and determine their limiting distribution and endpoint scales on the modular arc.

math.NT↗
arXiv2025

P-adic Period Conjectures for 1-motives: Integration and Linear Relations

We develop a $p$-adic theory of periods for 1-motives, extending the classical theory of complex periods into the non-archimedean setting. For 1-motives with good reduction over $p$-adic local fields, we construct a $p$-adic integration pairing that generalizes the Colmez--Fontaine--Messing theory for abelian varieties. This pairing is bilinear, perfect, Galois-equivariant, and compatible with the Hodge filtration, taking values in a quotient of the de Rham period ring. Building on this construction, we introduce a stratified formalism for $p$-adic periods, defining period spaces at various depths that capture increasingly refined relations among periods, and formulating conjectures that mirror the Grothendieck period conjecture in this new context. The classical period conjecture for 1-motives over $\bar{Q}$, previously resolved via the Huber--Wustholz analytic subgroup theorem, is recovered at depth 1 in our framework. For 1-motives over number fields with good reduction at $p$, we identify canonical $Q$-structures on the $p$-adic realizations arising from rational points of their associated formal $p$-divisible groups. Relative to these structures, we establish the conjectures at depths 1 and 2. A key tool is the development of a $p$-adic analytic subgroup theorem tailored to 1-motives, providing an analogue of Wustholz's classical result. Our work not only yields a $p$-adic counterpart to the Kontsevich--Zagier conjecture for 1-motives but also opens new pathways for the study of linear relations among $p$-adic periods and their transcendence properties.

math.NT↗
arXiv2025

Higher Schwarzian, quasimodular forms and equivariant functions

The Schwarzian derivative plays a fundamental role in complex analysis, differential equations, and modular forms. In this paper, we investigate its higher-order generalizations, known as higher Schwarzians, and their connections to quasimodular forms and equivariant functions. We prove that a meromorphic function is equivariant if and only if its higher Schwarzians are quasimodular forms of prescribed weight and depth, thereby extending classical results and linking projective differential operators to the structure of modular and quasimodular forms.

math.NT↗
arXiv2024

On level 2 Modular differential equations

In this paper, we explore the modular differential equation $\displaystyle y'' + F(z)y = 0$ on the upper half-plane $\mathbb{H}$, where $F$ is a weight 4 modular form for $\Gamma_0(2)$. Our approach centers on solving the associated Schwarzian equation $\displaystyle \{h, z\} = 2F(z)$, where $\{h, z\}$ represents the Schwarzian derivative of a meromorphic function $h$ on $\mathbb{H}$. We derive conditions under which the solutions to this equation are modular functions for subgroups of the modular group and provide explicit expressions for these solutions in terms of classical modular functions. Key tools in our analysis include the theory of equivariant functions on the upper half-plane and the representation theory of level 2 subgroups of the modular group.

math.NT↗
arXiv2024

Hypergeometric solutions to Schwarzian equations

In this paper we study the modular differential equation $y''+s\,E_4\, y=0$ where $E_4$ is the weight 4 Eisenstein series and $s=\pi^2r^2$ with $r=n/m$ being a rational number in reduced form such that $m\geq 7$. This study is carried out by solving the associated Schwarzian equation $\{h,\tau\}=2\,s\,E_4$ and using the theory of equivariant functions on the upper half-plane and the 2-dimensional vector-valued modular forms. The solutions are expressed in terms of the Gauss hypergeometric series. This completes the study of the above-mentioned modular differential equation of the associated Schwarzian equation given that the cases $1\leq m\leq 6$ have already been treated in the litterature.

math.NT↗
arXiv2021

On the modularity of solutions of certain differential equations of hypergeometric type

The purpose of this paper is to provide answers to some questions raised in a paper by Kaneko and Koike about the modularity of the solutions of a differential equations of hypergeometric type. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in other cases. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases.

math.NT↗
arXiv2021

Equivariant solutions to modular Schwarzian equations

For every positive integer $r$, we solve the modular Schwarzian differential equation $\{h,τ\}=2π^2r^2E_4$, where $E_4$ is the weight 4 Eisenstein series, by means of equivariant functions on the upper half-plane. This paper supplements previous works \cite{forum, ramanujan}, where the same equation has been solved for infinite families of rational values of $r$. This also leads to the solutions to the modular differential equation $y''+r^2π^2E_4\,y=0$ for every positive integer $r$. These solutions are quasi-modular forms for $\mbox{SL}_2(\mathbb Z)$ if $r$ is even or for the subgroup of index 2, $\mbox{SL}_2(\mathbb Z)^2$, if $r$ is odd.

math.NT↗
arXiv2020

Modular groups and planar maps

In this paper we give an explicit formula for the number of subgroups of the modular group of a given index that are genus zero and torsion-free and a formula for their conjugacy classes. We do so by exhibiting a correspondence between these groups and the trivalent maps on a sphere. We focus on the particular case of the subgroups of index 18 which have some interesting geometric properties.

math.NT↗
arXiv2020

Automorphic Schwarzian equations and integrals of weight 2 forms

In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,τ\}=sE_4(τ)$ where $E_4(τ)$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit solutions for each $s=2π^2(n/6)^2$ with $n\equiv 1\mod 12$. These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation $\displaystyle y''+\frac{π^2n^2}{36}\,E_4\,y=0$ for each $n\equiv 1\mod 12$ generalizing the work of Hurwitz and Klein on the case $n=1$. Our investigation relies on the theory of equivariant functions on the complex upper half-plane. This paper supplements a previous work where we determine all the parameters $s$ for which the above Schwarzian equation has a modular solution.

math.NT↗
arXiv2020

Automorphic Schwarzian equations

This paper concerns the study of the Schwarz differential equation $\{h,τ\}=s\,E_4(τ)$ where $E_4$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we determine all the values of $s$ for which the solutions $h$ are modular functions for a finite index subgroup of $\mbox{SL}_2(\mathbb Z)$. We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of $\mbox{SL}_2(\mathbb Z)$. This also leads to the solutions to the Fuchsian differential equation $y''+s\,E_4\,y=0$.

math.NT↗
arXiv2017

Elliptic Zeta functions and equivariant functions,

In this paper we establish a close connection between three notions at- tached to a modular subgroup. Namely the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

math.NT↗
arXiv2014

Vector-valued automorphic forms and vector bundles

While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group $Γ$ and an arbitrary representation $R$ of $Γ$ in GL$(n,{\mathbb C})$, their existence has been established in the literature only when restrictions are imposed on both $Γ$ and $R$. In this paper, we prove the existence of $n$ linearly independent vector-valued automorphic forms for any Fuchsian group $Γ$ and any $n$-dimensional complex representation $R$ of $Γ$. To this end, we realize these automorphic forms as global sections of a special rank $n$ vector bundle built using solutions to the Riemann-Hilbert problem over various noncompact Riemann surfaces and Kodaira's vanishing theorem.

math.NT↗
arXiv2013

Equivariant functions and vector-valued modular forms

For any discrete group $Γ$ and any 2-dimensional complex representation $ρ$ of $Γ$, we introduce the notion of $ρ-$equivariant functions, and we show that they are parameterized by vector-valued modular forms. We also provide examples arising from the monodromy of differential equations.

math.NT↗
arXiv2011

Eisenstein series and modular differential equations

The purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions and equivariant forms.

math.CA↗
arXiv2005

Non-commutative ADE geometries as holomorphic wave equations

Borrowing ideas from the relation between classical and quantum mechanics, we study a non-commutative elevation of the ADE geometries involved in building Calabi-Yau manifolds. We derive the corresponding geometric hamiltonians and the holomorphic wave equations representing these non-commutative geometries. The spectrum of the holomorphic waves is interpreted as the quantum moduli space. Quantum A_1 geometry is analyzed in some details and is found to be linked to the Whittaker differential equation.

hep-th↗