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Federico Pellarin

Publications and source records attributed to Federico Pellarin.

At least 19 recordsLinked to original sources
arXiv2025

The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function

We prove a differential transcendence result of type "Ax-Lindemann-Weierstrass" for Euler's gamma function. Given meromorphic functions $\zeta_1,\dots,\zeta_n$ of a complex variable $\nu$ that are pairwise distinct modulo $\mathbb Z$ and algebraic over the field $k$ of meromorphic $1$-periodic functions, the functions $ \Gamma(\nu-\zeta_1(\nu)),\dots,\Gamma(\nu-\zeta_n(\nu))$ are differentially independent over the field $k(\nu)$. We determine the structure of certain difference field extensions related to the torsion of an avatar of the Carlitz module over meromorphic functions. These extensions are abelian and purely transcendental, the latter property being crucial in our main result, and obtained applying a criterion of differential algebraicity of Hardouin and Singer.

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arXiv2025

Noncommutative factorizations of higher sine functions in positive characteristic

In this paper we describe new noncommutative factorizations of functions related to $d$-th tensor powers of Carlitz's $\mathbb F_q[\theta]$-module for $d\geq 1$, called higher sine functions. In recent work by the second author, factorizations of this type have been constructed for operators which are combinations of powers of a Frobenius endomorphism with coefficients ``in $\operatorname{End}(\operatorname{End}(\mathbb G_a^d))$''. In the present paper we succeed in determining factorizations with coefficients ``in $\operatorname{End}(\mathbb G_a^d)$'' which are not easily deducible from previous work. One key ingredient in obtaining this is an application of a ``motivic pairing'' that the first author introduced in recent work. Another key ingredient is the notion of ``$\Delta$-matrix'' which comes into play in the analysis of the coefficients of the factorizations. Our results can be applied to explicitly describe analogues of shuffle $q^n$-powers for multiple polylogarithms at one, and to multiple zeta values of Thakur. All the identities we prove occur at the finite level.

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arXiv2021

Universal Families of Eulerian Multiple Zeta Values in Positive Characteristics

We study positive characteristic multiple zeta values associated to general curves over $\mathbb F_q$ together with an $\mathbb F_q$-rational point $\infty$ as introduced by Thakur. For the case of the projective line these values were defined as analogues of classical multiple zeta values. In the present paper we first establish a general non-commutative factorization of exponential series associated to certain lattices of rank one. Next we introduce universal families of multiple zeta values of Thakur and show that they are Eulerian in full generality. In particular, we prove a conjecture of Lara Rodriguez and Thakur arXiv:2003.12910. One of the main ingredients of the proofs is the notion of L-series in Tate algebras introduced by the third author arXiv:1107.4511 in 2012.

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arXiv2021

The analytic theory of vectorial Drinfeld modular forms

In this paper we generalise the notion of Drinfeld modular form for the group $Γ$ := GL2(Fq[$θ$]) to a vector-valued setting, where the target spaces are certain modules over positive characteristic Banach algebras over which are defined what we call the 'representations of the first kind'. Under quite reasonable restrictions, we show that the spaces of such modular forms are finite-dimensional, are endowed with certain generalisations of Hecke operators, with differential operators{à} la Serre etc. The crucial point of this work is the introduction of a 'field of uniformisers', a valued field in which we can study the expansions at the cusp infinity of our modular forms and which is wildly ramified. Examples of such modular forms are given through the construction of Poincar{é} and Eisenstein series. After this the paper continues with a more detailed analysis of the special case of modular forms associated to a restricted class of representations $ρ$ * $Σ$ of $Γ$ which has more importance in arithmetical applications. More structure results are given in this case, and a harmonic product formula is obtained which allows, with the help of three conjectures on the structure of an Fp-algebra of A-periodic multiple sums, to produce conjectural formulas for Eisenstein series. Some of these formulas can be proved.

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arXiv2020

From the Carlitz exponential to Drinfeld modular forms

This paper contains the written notes of a course the author gave at the VIASM of Hanoi in the Summer 2018. It provides an elementary introduction to the analytic naive theory of Drinfeld modular forms for the simplest 'Drinfeld modular group' GL 2 (Fq[$θ$]) also providing some perspectives of development, notably in the direction of the theory of vector modular forms with values in certain ultrametric Banach algebras.

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arXiv2019

On extremal quasi-modular forms after Kaneko and Koike

Kaneko and Koike introduced the notion of extremal quasi-modular form and proposed conjectures on their arithmetic properties. The aim of this note is to prove a rather sharp multiplicity estimate for these quasi-modular forms. The note ends with discussions and partial answers around these conjectures and an appendix by G. Nebe containing the proof of the integrality of the Fourier coefficients of the normalised extremal quasimodular form of weight 14 and depth 1.

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arXiv2017

A sum-shuffle formula for zeta values in Tate algebras

We prove a sum-shuffle formula for multiple zeta values in Tate algebras (in positive characteristic), introduced in \cite{PEL3}.This follows from an analog result for double twisted power sums, implying that an ${\mathbb{F}\_p$-vector space generated by multiple zeta values in Tate algebras is an ${\mathbb{F}\_p$-algebra.

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arXiv2016

On twisted $A$-harmonic sums and Carlitz finite zeta values

In this paper, we study various twisted A-harmonic sums, named following the seminal log-algebraicity papers of G. Anderson. These objects are partial sums of new types of special zeta values introduced by the first author and linked to certain rank one Drinfeld modules over Tate algebras in positive characteristic by Anglès, Tavares Ribeiro and the first author. We prove, by using techniques introduced by the second author, that various infinite families of such sums may be interpolated by polynomials, and we deduce, among several other results, properties of analogues of finite zeta values but inside the framework of the Carlitz module. In the theory of finite multi-zeta values in characteristic zero, finite zeta values are all zero. In the Carlitzian setting, there exist non-vanishing finite zeta values, and we study some of their properties in the present paper.

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arXiv2016

On certain generating functions in positive characteristic

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit connection with certain deformations of the Carlitz logarithm introduced by M. Papanikolas and involve the Anderson-Thakur function and the Carlitz exponential function. They collect certain functional identities in families for a new class of L-functions introduced by the first author. This paper also deals with specializations at roots of unity of these generating functions, producing a link with Gauss-Thakur sums.

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arXiv2015

Arithmetic of positive characteristic L-series values in Tate algebras

The second author has recently introduced a new class of L-series in the arithmetic theory of function fields over finite fields. We show that the value at one of these L-series encode arithmetic informations of certain Drinfeld modules defined over Tate algebras. This enables us to generalize Anderson's log-algebraicity Theorem and Taelman's Herbrand-Ribet Theorem.

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arXiv2015

Anderson-Stark units for $\mathbb F_q[θ]$

We investigate the arithmetic of special values of a new class of $L$-functions recently introduced by the second author. We prove that these special values are encoded in some particular polynomials which we call Anderson-Stark units. We then use these Anderson-Stark units to prove that $L$-functions can be expressed as sums of polylogarithms.

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arXiv2013

Universal Gauss-Thakur sums and L-series

In this paper we study the behavior of the function omega of Anderson-Thakur evaluated at the elements of the algebraic closure of the finite field with q elements F_q. Indeed, this function has quite a remarkable relation to explicit class field theory for the field K=F_q(T). We will see that these values, together with the values of its divided derivatives, generate the maximal abelian extension of K which is tamely ramified at infinity. We will also see that omega is, in a way that we will explain in detail, an universal Gauss-Thakur sum. We will then use these results to show the existence of functional relations for a class of L-series introduced by the second author. Our results will be finally applied to obtain a new class of congruences for Bernoulli-Carlitz fractions, and an analytic conjecture is stated, implying an interesting behavior of such fractions modulo prime ideals of A=F_q[T].

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arXiv2012

On the generalized Carlitz module

The aim of this note is to gather formal similarities between two apparently different functions; {\em Euler's function} $Γ$ and {\em Anderson-Thakur function} $ω$. We discuss these similarities in the framework of the {\em generalized Carlitz's module}, a common structure which can be helpful in framing the theories of both functions. We further analyze several noticeable differences while investigating the {\em exponential functions} associated to these structures.

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arXiv2011

Values of certain L-series in positive characteristic

We introduce a family of L-series specialising to both L-series associated to certain Dirichlet characters over F_q[T] and to integral values of Carlitz-Goss zeta function associated to F_q[T]. We prove, with the use of the theory of deformations of vectorial modular forms, a formula for their value at 1, as well as some arithmetic properties of other values at positive integers

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arXiv2011

tau-recurrent sequences and modular forms

In this paper we deal with Drinfeld modular forms, defined and taking values in complete fields of positive characteristic. Our aim is to study a sequence of families of Drinfeld modular forms depending on a parameter t that produces, for certain values of t, several kinds of Eisenstein series considered by Gekeler. We obtain formulas involving these functions. To obtain our results, we introduce and discuss tau-linear recurrent sequences and deformations of vectorial modular forms in this setting.

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