Characteristic p geometry in a broad sense

Location: All talks will be in Albano House 1, Floor 2, Classroom 4

Speakers:

• Amina Abdurrahman (IHES): Square roots of L-functions of symplectic local systems on curves

Abstract: I will present some of the consequences (including work in progress) of a cohomological formula for the central value of symplectic L-functions on curves established in previous joint work with A. Venkatesh. I will sketch ideas of the proofs and crucial topological inputs.

• Giuseppe Ancona (Strasbourg): Ramified periods and field of definition

Abstract: In a joint work with Dragos Fratila and Alberto Vezzani, we construct hyperelliptic curves of large genus, defined over quadratic fields that are isomorphic to their Galois conjugates but do not descend to Q. The obstruction to descent is new and we call it “ramified periods”. These are p-adic numbers that arise from the comparison between de Rham cohomology and crystalline cohomology (hence the term periods). These numbers can reveal interesting information if p ramifies in the quadratic field.

• Hélène Esnault (Berlin): On the vanishing of the restriction map to the  generic point  in cohomology  

Abstract: If $X$ is a smooth projective variety defined over the field of complex numbers, its $i$-th Betti cohomology $H^i(X, \mathbb C)$ is said to have {\it coniveau one}  if there is a Zariski dense open $U\subset X$ such that the restriction map $H^i(X,\mathbb C) \to H^i(U,\mathbb C)$ dies.  Equivalently, the restriction map to the generic point  of $X$ in $i$-th cohomology 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 $H^i(X, \mathbb C)$ 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 to try to understand whether more modern $p$-adic  methods yield some non-trivial  information.

With  Mark Kisin and Alexander Petrov, in  work in progress, we

1) compute that $X$ has no non-trivial global differential forms of degree $i$ over $\mathbb C$ if and only if  mod $p$ for almost all $p$ de Rham cohomology $H^i$ has coniveau one;

2)   formulate and prove a vanishing result in a separated quotient of $p$-completed de Rham cohomology of $X$ restricted to a (unramified) $p$-adic field for $p$ large, and a weaker version in the separate quotient of prismatic cohomology;

3) lift 2) to prismatic cohomology;

4) find an example for which a small open (in the sens of Faltings) does not suffice. 

• Marion Jeannin (SU): Semistability of G-torsors and parabolic subgroups in positive characteristic

Abstract: Let X be a curve over a field k. Let also G be a reductive group scheme over X. Semistability

for G-torsors can be defined in several ways that depend on assumptions on k and G. These

approaches are both well-defined and equivalent when k is of characteristic zero. In this talk I

will explain in which generality it is possible to extend some of these to the positive characteristic framework and compare them. This requires to investigate whether some well-known results in representation theory in characteristic zero still hold to be true in characteristic p > 0. More specifically, an analogous statement of a theorem of Morozov (which classifies, in characteristic 0, parabolic subalgebras of the Lie algebra of a reductive group by means of their nilradical) is a cornerstone of all this unification attempt.

In the first part of the talk, I will provide an overview of the geometric context and emphasize the role played by parabolic subgroups. The second part of the talk will be dedicated to the extension of Morozov’s theorem to positive characteristics, and the way it allows one to get a more uniform vision of the different historical definitions of semistability of G-torsors.

Ben Moonen (Nijmegen): The addition map on subvarieties of abelian varieties

Abstract: I'll report on joint work with Olivier Debarre. The main result is that if A is an absolutely simple abelian variety over some field and X_1,..., X_r are subvarieties of A, then the dimension of their sum X_1 + ... + X_r equals the minimum of dim(A) and \sum dim(X_i). In characteristic 0, there is a simple geometric proof for this, but that argument breaks down in characteristic p. Instead, we prove this result as a consequence of a theorem on perverse sheaves, building upon work of Krämer and Weissauer.

• Richard Pink (ETH Zurich):  A Cohen-Lenstra Heuristic for Schur $\sigma$-Groups

Abstract: For any odd prime $p$ and any imaginary quadratic field~$K$, the $p$-tower group $G_K$  associated to~$K$ is the Galois group over $K$ of the maximal unramified pro-$p$-extension of~$K$. This group comes with an action of a finite group $\{1,\sigma\}$  of order~$2$ induced by complex conjugation and is known to possess a number of other properties, making it a so-called Schur $\sigma$-group. Its maximal abelian quotient is naturally isomorphic to the $p$-primary part of the narrow ideal class group of~${\mathcal O}_K$, and the Cohen-Lenstra heuristic gives a probabilistic explanation for how often this group is isomorphic to a given finite abelian $p$-group.

I will present the content of two recent preprints https://arxiv.org/abs/2505.05569 and https://arxiv.org/abs/2505.05580. The first is joint with Luca Rubio and develops an analogue of this heuristic for the full group~$G_K$. It is based on a detailed analysis of general pro-$p$-groups with an action of $\{1,\sigma\}$, which we call $\sigma$-pro-$p$-groups. We construct a probability space whose underlying set consists of $\sigma$-isomorphism classes of weak Schur $\sigma$-groups and whose measure is constructed from the principle that the relations defining $G_K$ should be randomly distributed according to the Haar measure. We also compute the measures of certain basic subsets, the result being inversely proportional to the order of the $\sigma$-automorphism group of a certain finite $\sigma$-$p$-group, as has often been observed before. Finally, we show that the $\sigma$-isomorphism classes of weak Schur $\sigma$-groups for which each open subgroup has finite abelianization form a subset of measure~$1$.

In the second preprint we prove that if $p>3$, any infinite Schur $\sigma$-group of Zassenhaus type $(3,3)$, for which every open subgroup has finite abelianization, is isomorphic to an open subgroup of a form of ${\rm PGL}_2$ over~${\mathbb Q}_p$. Combined with the results of the first preprint, or with the Fontaine-Mazur conjecture, this lends credence to the ``if'' part of a conjecture of McLeman.

• Sjoerd de Vries (SU): Arithmetic aspects of Drinfeld modules

Abstract: There is a strong interplay between characteristic p geometry and the theory of modular forms: for instance, Deligne famously showed that the Weil conjectures imply the Ramanujan--Petersson bound on the coefficients of cusp forms. In this talk I will discuss a function field analogue of this picture. The Eichler--Shimura theory developed by Böckle, using the formalism of crystals due to Böckle--Pink, allows one to express traces of Hecke operators on Drinfeld modular forms in terms of Frobenius elements of Drinfeld modules over finite fields. This leads to some quite explicit consequences in the theory of Drinfeld modular forms. I will also highlight historical developments and current open problems.

Organizers:  Eric Ahlqvist, Jonas Bergström, Wushi Goldring

Funding:

We thank the following for their support: 

•  The Knut & Alice Wallenbeg Foundation

• The Swedish Research Council