Some Breuil-Mezard identities in moduli spaces of Breuil-Kisin modules
The Breuil--Mezard conjecture predicts certain identities between cycles in moduli spaces of mod p Galois representations in terms of the Fp-representation theory of GLn(Fq).
In this talk I will discuss work in progress which considers the situtation arising from (the reduction modulo p of) two dimensional crystalline Galois representation with suitably small* Hodge--Tate weights. We will discuss how the predected identities can also seen in ``resolutions`` of these spaces of Galois representations described in terms of semilinar algebra.
*small will be precisely the bound which ensures that the Fp-representation theory of GL2(Fq) appearing behaves precisely as it would with char 0 coefficients.