arXiv subjects
Publications and source records attributed to David Roe.
We present a new method to propagate $p$-adic precision in computations, which also applies to other ultrametric fields. We illustrate it with many examples and give a toy application to the stable computation of the SOMOS 4 sequence.
We generalize the work of DeBacker and Reeder to the case of unitary groups split by a tame extension. The approach is broadly similar and the restrictions on the parameter the same, but many of the details of the arguments differ. Let $G$ be a unitary group defined over a local field $K$ and splitting over a tame extension $E/K$. Given a Langlands parameter $φ: \mathcal{W}_K \rightarrow {^L G}$ that is tame, discrete and regular, we give a natural construction of an $L$-packet $Π_φ$ associated to $φ$, consisting of representations of pure inner forms of $G(K)$ and parametrized by the characters of the finite abelian group $A_φ= \operatorname{Z}_{\hat{G}}(φ)$.
This paper generalizes work of Buzzard and Kilford to the case $p=3$, giving an explicit bound for the overconvergence of the quotient $E_κ/ V(E_κ)$ and using this bound to prove that the eigencurve is a union of countably many annuli over the boundary of weight space.
We generalize the rectifier of Bushnell and Henniart, which occurs in the local Langlands correspondence for $GL_n(K)$, to certain Langlands parameters for unramified connected reductive groups.
Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing the Frobenius action on p-adic cohomology to a small degree of p-adic accuracy. We have implemented this procedure in Magma; using this implementation, we exhibit several examples, such as smooth quartics over F_2 and F_3 with arithmetic Picard number 1, and a smooth quintic over F_2 with geometric Picard number 1. We also produce some examples of smooth quartics with geometric Picard number 2, which by a construction of van Luijk also have trivial geometric automorphism group.