Felipe Gambardella (Orsay), 2-3pm
Transfer principles and the Kato–Kuzumaki conjecture.
Seeking a Diophantine characterisation of cohomological dimension of fields, Kato and Kuzumaki introduced the C_i^q property for every pair of non-negative integers i and q. These properties are inspired by Lang’s notion of a C_i field and Bloch-Kato’s conjecture relating Milnor K-theory with Galois cohomology. An interesting property of C_i fields is that they satisfy a transfer principle with respect to equicharacteristic complete discrete valuations – i.e. if a field k is C_i, then the field of Laurent series k((t)) is C_{i+1}. However, for the C_i^q property, no such transfer result is known to hold and the situation is more subtle than the one of C_i fields. For instance, the optimistic statement “k((t)) satisfies C_{i+1}^q whenever k satisfies C_i^q ” is false. This talk reports on joint work with Konstantinos Kartas. I will present a transfer principle for C_i fields with respect to non-discrete valuations. This principle has as a corollary a transfer principle for a variant of the Kato-Kuzumaki properties. Moreover, this is enough to establish the Kato-Kuzumaki conjecture for several fields of arithmetic interest such as the perfection of Fp((t_1)) . . . ((t_n)) or C(x_1, . . . , x_n)((t_1)) . . . ((t_m)) and to establish a strengthening of the C_1^1 property for Q_p introduced by Wittenberg.

Quentin Gazda (IMJ-PRG), 3.30-4.30pm
Régulateur cyclosyntomique des corps de nombres
Dans cet exposé, nous présenterons un travail en commun avec Tess Bouis où nous construisons une version q-déformée du régulateur de Dirichlet p-adique des corps de nombres. De manière surprenante, notre construction ne nécessite ni la complétion en p, ni que p soit premier. Nous interpréterons ce régulateur à travers la notion de q-Gauges, objets globaux ayant vocation à interpoler les structures de Hodge p-adiques en chaque p.