Rubén Muñoz--Bertrand

Postdoc in mathematics

Curriculum vitae

I'm working in arithmetic geometry. More precisely, I study overconvergent de Rham–Witt cohomology and its coefficients. I've also become interested in algorithmic applications of these theories.

I have defended my PhD thesis in 2020.

Articles

1. Pseudovaluations on the de Rham–Witt complex, Bulletin de la Société Mathématique de France 150 (2022), no. 1, pp. 53–75, doi:10.24033/bsmf.2844

   For a polynomial ring over a commutative ring of positive characteristic, we define on the associated de Rham–Witt complex a set of functions, and show that they are pseudovaluations in the sense of Davis, Langer and Zink. To achieve this, we explicitly compute products of basic elements on the complex. We also prove that the overconvergent de Rham–Witt complex can be recovered using these pseudovaluations.

Preprints

1. Local structure of the overconvergent de Rham–Witt complex (2023), 54 pages, arXiv:2311.15449

   We give a general description of the structure of the relative de Rham–Witt complex on a polynomial ring, seen as an algebra over its integral part. After giving a control of the overconvergence of Lazard's morphism, we similarly give the structure of the overconvergent complex for a finite étale algebra over a perfect Noetherian ring. We then deduce a generalization of the usual decomposition as a direct sum, which is compatible with overconvergence on the projections.
2. Isocrystals and de Rham–Witt connections (2025), 28 pages, arXiv:2502.01902

   We introduce the notion of integrable connections for a sheaf of differential graded algebras on a topological space. We then describe them in the finite locally projective setting, when the sheaf is either the de Rham complex of a formal or a weakly formal scheme, or for the convergent or the overconvergent de Rham-Witt complex on a smooth scheme over a perfect field of positive characteristic. This enables us to give a new description of convergent and overconvergent isocrystals with a Frobenius structure.
3. Characterising properties of commutative rings using Witt vectors (2025), 11 pages, arXiv:2503.20115

   We give equivalences between given properties of a commutative ring, and other properties on its ring of Witt vectors. Amongst them, we characterise all commutative rings whose rings of Witt vectors are Noetherian. We define a new category of commutative rings called preduced rings, and explain how it is the category of rings whose ring of Witt vectors has no p-torsion. We then extend this characterisation to the torsion of the de Rham–Witt complex.
4. Faster computation of Witt vectors over a polynomial ring (draft), SageMath code

   We describe an algorithm which computes the ring laws for Witt vectors of finite length over a polynomial ring with coefficients in a finite field. This algorithm uses an isomorphism of Illusie in order to compute in an adequate product ring. We also give an implementation of the algorithm in SageMath, which turns out to be faster that Finotti's algorithm, which was until now the most efficient one for these operations.
5. Effective Artin–Schreier–Witt theory for curves (draft, joint with Christophe Levrat), SageMath code

   We present an algorithm which, given a connected smooth projective curve X over an algebraically closed field of characteristic p>0 and its Hasse–Witt matrix, as well as a positive integer n, computes all étale Galois covers of X with group Z/p^nZ. We provide a complete implementation of this algorithm in SageMath. We then apply this algorithm to the computation of the cohomology complex of a locally constant sheaf of Z/p^nZ-modules on such a curve.
6. Using de Rham–Witt cohomology in Kedlaya's algorithm (draft), SageMath code

   We explain how to replace Monsky–Washnitzer cohomology with overconvergent de Rham–Witt cohomology in Kedlaya's algorithm in the case of hyperelliptic curves over a finite field of odd characteristic. This method yields a simpler formula for the action of the Frobenius. We describe how to construct cohomological reduction formulae allowing us to compute the zeta function of the curve. Finally, we give an implementation in SageMath of the algorithm.

Lecture notes

1. p-adic cohomology theories and point couting, draft updated on June 26, 2023, comments welcome!
2. Introduction à la théorie des groupes, online version updated on September 06, 2024.

How to spell my name

I have noticed that my name is often misspelled. This is because of its two diacritical signs, but mostly because of the two dashes in my family name. It is indeed a double dash, and not a simple one, nor a long one, and certainly not a typo!

If you use LaTeX2e with a version released after 2018, you simply have to use that code:

Rubén Muñoz-\relax-Bertrand

In Plain TeX, or for obsolete versions of LaTeX2e, you can use the following code:

Rub\'en Mu\~noz-\relax-Bertrand

I would be very grateful if you use the proper spelling of my name.