5 June 2025 – 10.30 Aula 4
Tess Bouis (Regensburg)

Beilinson–Lichtenbaum phenomenon for motivic cohomology

Recently, a lot of progress has been made in the development of a theory of non-A^1-invariant motives, which interpolates between the categories of coefficients for A^1-invariant motivic cohomology (given by Bloch’s cycle complexes in the smooth case) and for p-adic cohomology theories (such as prismatic cohomology). In this talk, I want to explain how the Beilinson–Lichtenbaum phenomenon for non-A^1-invariant motivic cohomology can be used to compare the A^1-invariant and non-A^1-invariant theories of motivic cohomology in the smooth case, and to prove motivic refinements of certain results appearing in p-adic Hodge theory. This is based on a joint work with Arnab Kundu, where we develop a version in families of Gabber’s presentation lemma to prove such a Beilinson–Lichtenbaum phenomenon over general valuation rings.

23 May 2025 – University of Genova
Giuseppe Ancona (Strasbourg)
Antonio Cauchi (Dublin)
Veronika Ertl-Bleimhofer (Caen)
Stefano Morra (Sorbonne Paris Nord)

Arithmetica Transalpina

More information on this page.

25 March 2025 – 11.00-12.00 Aula Dottorato
Adrian Ioviță (Concordia – Padova)

BGG decomposition for de Rham sheaves

I will report on joint work with F. Andreatta and M. Baracchini. The typical example we’d like to analyze is the following: given a prime $p>0$ and a $p$-adic weight $k$, we have constructed a sheaf of Banach modules of weight $k$, with an integrable connection, on a strict neighborhood of the ordinary locus in a modular curve of level prime to $p$, which interpolates the integral symmetric powers of the relative de Rham cohomology of the generalized universal elliptic curve over the modular curve. The question is: how do we compute the finite slope de Rham cohomology of this sheaf with connection?
We show that one can use Lie-algebra techniques à la Bernstein-Gelfand-Gelfand (BGG) to find a much simpler complex of coherent sheaves, which computes the desired cohomology groups. Although this is a somewhat classical topic discussed in the 90’s by Faltings-Chai, we found some new sides of the story that we think are interesting and can be applied in many other situations, for example for Hilbert and Siegel modular varieties for which such sheaves with connections have been constructed.

24 February 2025 – 14.30 Aula 9
Hélène Esnault (Uni Freie Berlin – Harvard – Copenhagen)

Colloquium: Seminari Enriques – Algebra e Geometria
Vanishing at the generic point in cohomology

If X is a smooth projective variety defined over the field of complex num- bers, its i-th Betti cohomology Hi(X,C) is said to have coniveau one if there is a Zariski dense open U ⊂ X such that the restriction map Hi(X, C) → Hi(U, C) dies. Equivalently, the restriction map to the generic point of X in i-th coho- mology vanishes. Grothendieck’s generalized Hodge conjecture is in general difficult to express as one needs the notion of Hodge sub-structure, but one particular instance has a purely algebraic formulation. It predicts that if X has no non-trivial global differential forms of degree i, then Hi(X,C) should have coniveau one. The converse is easily seen to be true. Aside of i = 1, 2, for which complex Hodge theory gives a positive answer, we know nothing. On the other hand, the philosophy behind is very useful to draw analogies, e.g. it helps to find rational points over finite fields of rationally connected varieties (Lang-Manin conjecture). So it is worth trying to understand whether more modern p-adic methods yield some non-trivial information.
With Mark Kisin and Alexander Petrov, in work in progress, we formulate and prove a vanishing result in the separate quotient of p-completed de Rham cohomology, and a weaker version in the separate quotient of prismatic cohomology. I’ll present the ‘program’ and a few questions which at present are not understood.

15 November 2024 – 11.30-18.00 Aula 3
Giada Grossi (LAGA – Sorbonne Paris Nord)
Ben Heuer (Frankfurt)
Guido Kings (Regensburg)
Pol Van Hoften (VU Amsterdam)

Arithmetica Transalpina

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24 October 2024 – 11.00 Aula 10
Jarod Alper (Washington)

Colloquium: Seminari Enriques – Algebra e Geometria
Evolution of Moduli

In the rich landscape of algebraic varieties, moduli spaces stand out as some of the most enchanting varieties, capturing the imagination of algebraic geometers with their profound elegance and deep connections to other branches of mathematics.   Moduli, the plural of modulus, is a term coined by Riemann to describe a space whose points afford an alternative description as certain classes of geometric objects.  We will trace the origins of moduli spaces through the discoveries of Riemann, Hilbert, Grothendieck, Mumford, and Deligne, as a means to explain many of the fundamental concepts and results.  We will then survey how the foundations of moduli theory have further evolved over the last 50 years.

“Pizza seminars”, funded by the ALGANT Consortium, offer a laid-back atmosphere in which students and young researchers can interact with the speaker and with each other…eating pizza!

14 November 2024 – 12.30 Aula Dottorato
Giada Grossi (LAGA – Sorbonne Paris Nord)

Iwasawa theory: from class groups to elliptic curves