Weights of mod pp automorphic forms and partial Hasse invariants

For a connected, reductive group $G$ over a finite field endowed with a cocharacter $\mu$, we define the zip cone of $(G,\mu)$ as the cone of all possible weights of mod $p$ automorphic forms on the stack of $G$-zips. This cone is conjectured to coincide with the cone of weights of characteristic $p$ automorphic forms for Hodge-type Shimura varieties of good reduction. We prove in full generality that the cone of weights of characteristic $0$ automorphic forms is contained in the zip cone, which gives further evidence to this conjecture. Furthermore, we determine exactly when the zip cone is generated by the weights of partial Hasse invariants, which is a group-theoretical generalization of a result of Diamond--Kassaei and Goldring--Koskivirta.Comment: 47 pages, appendix by Wushi Goldrin

Dessins, their delta-matroids and partial duals

Given a map $\mathcal M$ on a connected and closed orientable surface, the delta-matroid of $\mathcal M$ is a combinatorial object associated to $\mathcal M$ which captures some topological information of the embedding. We explore how delta-matroids associated to dessins d'enfants behave under the action of the absolute Galois group. Twists of delta-matroids are considered as well; they correspond to the recently introduced operation of partial duality of maps. Furthermore, we prove that every map has a partial dual defined over its field of moduli. A relationship between dessins, partial duals and tropical curves arising from the cartography groups of dessins is observed as well.Comment: 34 pages, 20 figures. Accepted for publication in the SIGMAP14 Conference Proceeding

Dynamics of the w function and primes

AbstractWe begin by defining a function w on the setA3={n=p1e1⋯pses∈Z>1|∑i=1sei=3,ei>0,s>1}, where pi is prime and pi≠pj for i≠j. If n∈A3 then was can write n=pqr where p, q, r are primes and possibly two, but not all three of them are equal. For any positive integer m, let P(m) be its largest prime factor. Define the function w on A3 byw(n)=w(pqr)=P(p+q)P(p+r)P(q+r). Our goal is to study the dynamics of w. One of our main results is that every element of A3 is periodic with period a cyclic permutation of the period of 20

The μ-ordinary Hasse invariant of\break unitary Shimura varieties

We construct a generalization of the Hasse invariant for any Shimura variety of PEL-typ

The μ-ordinary Hasse invariant of\break unitary Shimura varieties

We construct a generalization of the Hasse invariant for any Shimura variety of PEL-type A over a prime of good reduction, whose non-vanishing locus is the open and dense μ-ordinary locus