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A theory of ``special functions" and ``periods" emerged in the framework of function fields of positive characteristic after the early works of Carlitz in the 1930s and later, in the hands of Anderson, Goss, Hayes, Thakur and others. In this theory, the role of the ring of relative numbers, prominent in the classical arithmetic theory, is played by the ring A=k[T] with k a finite field; in other words, the ring of polynomials in an indeterminate T with coefficients in a finite field k. Instead of the real line, one then looks at the ``Carlitz line", that is, the local field completion of the fraction field of A with respect to the infinite place. One of the most important features available here is the possibility to see all our modules as k-vector spaces. My interest in these topics began around 2005. The first transcendental special function ever considered in this framework is the Carlitz exponential function. Later, the theory was developed along lines parallel to the classical theory of ``special functions" and ``periods," including, for instance, zeta- and L-functions, modular forms, Galois representations, etc. In most developments, the analogy real line/Carlitz line is assumed to be a basic point of view. All I did since then, is to try to overcome this analogy which I find superficial, looking for more significative and deeper structures.
Recent stuff (last five papers, for a complete list see here)
F. Pellarin, with an appendix of G. H. Ferraro. Zeta functions over curves. In preparation (it will be for the proceedings of a conference in Osaka).
L. Di Vizio, F. Pellarin. The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function. arXiv:2508.21237 (2025).
N. Green, F. Pellarin. Non-commutative factorizations of higher sine functions in positive characteristic, arXiv:2503.13295 (2025).
F. Pellarin. The analytic theory of vectorial Drinfeld modular forms. Memoirs AMS, Volume 312, Number 1581 (2025).
F. Pellarin. Carlitz operators and higher polylogarithm identities. Proc. London Math. Soc., 130: e70028 (2025). hal-04125119v1.
Editorial boards:
Managing Editor of the Journal of Number Theory
Associate Editor of Research in Number Theory
Associate Editor of Confluentes Mathematici
Forthcoming activities:
2025-12 and 2026-4 Corso della Prof.ssa Lucia di Vizio (UVSQ Versailles-CNRS) per il dottorato in Matematica di Sapienza Università di Roma dal titolo: “Introduzione alla teoria di Galois differenziale, alle differenze, e applicazioni.” This is a course of lectures to doctoral students of the PhD of the University of Rome. The course will consist of 16 one-hour lectures: the first part will take place from December 8 to 19, 2025 and the second part from April 18 to May 1, 2026. More information will appear in this webpage. If you are interested contact me.
2026-03 With Nathan Green, Masanobu Kaneko and Pavel Guerzhoy, I'm organizing the Meeting in the Middle: Conference on characteristic 0 and characteristic p multiple zeta values (and other topics) (University of Hawai'i at Manōa, 2026 March 16-20)
2026-04 Minicourse on Drinfeld modular forms in the framework of a Trimestre Temático at ICMAT Madrid (12-18 April 2026).
Recent:
2025-2 17th MSJ-SI `Developments of multiple zeta values'
2025-02-25 Seminar in Keio University (Yokohama).
2025-04-01 Séminaire Différentiel, Paris
2025-06-03 Séminaire AG Versailles
2025-06 Rencontres Arithmétiques de Caen (p-adic and modulo-p aspects) (June 18-20).
Ph. Students
Quentin Gazda (Saint-Etienne 2018-2021)
Giacomo Hermes Ferraro (Rome 2021-2024)
Leonardo Carofiglio, co-supervision with Andrea Ferraguti (Torino 2025- )
Luigi De Filpo co-supervision with Ignazio Longhi (Roma 2025- )
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