# Federico Pellarin Home Page.

f e d e r i c o . p e l l a r i n @ u n i v - s t - e t i e n n e . f r

## Research activity.

This is a kind of research statement. 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. Later, in 2006, I participated to a working seminar in Paris (organized by Daniel Bertrand and Lucia Di Vizio) based on the seminal work of Matthew Papanikolas and, in March 2007, I gave a Bourbaki seminar and drawn a state of the art focused on that paper.
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. You can read my recent papers, mostly also due to my co-authors Bruno Anglès, Vincent Bosser, Rudy Perkins and Floric Tavares Ribeiro, to see how...

Associate editor of the Journal of Number Theory, of Mathematica Slovaca
and of Research in Number Theory. Please, send us your papers!

**Selected publications.**

*Sur une borne supérieure explicite pour un
degré d'isogénie liant des courbes elliptiques.
*Acta Arith.100, No.3, 203-243 (2001).
*On the arithmetic properties of complex values of
Hecke-Mahler series I*. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 5, No. 3, 329-374 (2006).
*Aspects de l'indépendance algébrique en caractéristique non nulle.*
Bourbaki seminar. Volume 2006/2007. Exposes 967--981. Paris: Société Mathématique de France. Astérisque 317, 205-242 (2008).
*Hyperdifferential Properties of Drinfeld Quasi-Modular Forms*. International Mathematics Research Notices, Vol. 2008, with V. Bosser.
*On certain families of Drinfeld quasi-modular forms*, with V. Bosser. Journal of Number Theory Volume 129, Issue 12, December 2009, Pages 2952-2990
*An introduction to Mahler's method for transcendence and algebraic independence.* To appear in the EMS proceedings of the conference "Hodge structures, transcendence and other motivic aspects"
September 2009 at the BIRS, Banff, Canada, edited by G. Boeckle, D. Goss, U. Hartl, et M. Papanikolas. arxiv.org/abs/1005.1216
*Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms.* J. für die reine und angew. Math. (Crelles J.). 2014, 687, 1-42. arxiv.org0907.4507
*Values of certain **L*-series in positive characteristic. Annals of Math. Vol. 176 (2012), 2055-2093. hal-00610418 se also the link to the journal's webpage
*On the generalized Carlitz module.* Journal of Number Theory Vol. 133, 2013, 1663-1692. arxiv.org1210.2490
*Functional identities for **L*-series values in positive characteristic, with B. Anglès. Journal of Number Theory Vol. 142, 2014, 223-251. l-series.pdf
*Universal Gauss-Thakur sums and **L*-series, with B. Anglès. Invent. Math. (2015) Vol. 200, 653-669. file
se also the link to the journal's webpage
*Arithmetic of positive characteristic **L*-series values in Tate algebras, with B. Anglès and F. Tavares Ribeiro. Preprint 2014. arXiv:1402.0120 To appear in Compositio Math.

**Preprints**.

**2. ***On certain generating functions in positive characteristic, with R. Perkins.* Preprint 2014. arxiv1409.6006
**3. ***Anderson-Stark units for ***F**_{q}(T), with B. Anglès and F. Tavares Ribeiro. Preprint 2015. arxiv1501.06804

**Some dates**.

I organize with F. Jouhet and T. Rivoal the *Séminaire tournant de théorie des nombres,* (Institut Camille Jordan and Institut Fourier). The 18th day is planned for Thursday, October 8, 2015 at the ICJ, Lyon. The programme can be found here
here
Imperial College, London, June, 22-26 2015. With A. Pál and L. Taelman, I organized the conference *Function fields, zeta functions and Drinfeld modular forms.*
link
Université Jean-Monnet, Saint-Etienne, September, 7-9 2015. *French-Japanese Workshop on multiple zeta functions and applications.*
Istituto de Giorgi, Pisa, September, 21-24 2015. *Terzo Incontro Italiano di Teoria dei Numeri.*
link
Visiting MPIM, Bonn, February, 1-28, 2016.
Visiting IHES, Bures-sur-Yvette, April, 5-26, 2016
Upcoming conference in Lyon, June, 27 to July 1, 2016 on function
field arithmetic (the dates have been changed!).
link
If you are interested in, please ask me.

Ancient Beijing observatory