Narrow Results

We construct a family of quantum scalar fields over a $p$-adic spacetime which satisfy $p$-adic analogues of the Gårding–Wightman axioms. Most of the axioms can be formulated in the same way for both the Archimedean and non-Archimedean frameworks; however, the axioms depending on the ordering of the background field must be reformulated, reflecting the acausality of $p$-adic spacetime. The $p$-adic scalar fields satisfy certain $p$-adic Klein–Gordon pseudo-differential equations. The second quantization of the solutions of these Klein–Gordon equations corresponds exactly to the scalar fields introduced here. The main conclusion is that there seems to be no obstruction to the existence of a mathematically rigorous quantum field theory (QFT) for free fields in the $p$-adic framework, based on an acausal spacetime.

In this note, we investigate the $p$-degree function of an elliptic curve $E/{\mathbb{Q}}_p$. The $p$-degree measures the least complexity of a non-zero $p$-torsion point on $E$. We prove some properties of this function and compute it explicitly in some special cases.

We present a family of algorithms for computing the Galois group of a polynomial defined over a p-adic field. Apart from the “naive” algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of ${\mathbb{Q}}_2$ of degrees 18, 20 and 22, tables of which are available online.

In this paper, we determine the minimal number of variables ${\mathrm{\Gamma }}^{\ast }(d,K)$ which guarantees a nontrivial solution for every additive form of degree $d=4$ over the four ramified quadratic extensions ${\mathbb{Q}}_2(\sqrt{2}),{\mathbb{Q}}_2(\sqrt{10}),{\mathbb{Q}}_2(\sqrt{-2}),{\mathbb{Q}}_2(\sqrt{-10})$ of ${\mathbb{Q}}_2$. In all four fields, we prove that ${\mathrm{\Gamma }}^{\ast }(4,K)=11$. This is the first example of such a computation for a proper ramified extension of ${\mathbb{Q}}_p$ where the degree is a power of $p$ greater than $p$.

In this paper, we first present the basic ideas of the method of determining reducibility or irreducibility of parabolically induced representations of classical p-adic groups using Jacquet modules. After that we explain the construction of irreducible square integrable representations by considering characteristic examples. We end with a brief presentation of the classification of irreducible square integrable representations of these groups modulo cuspidal data, which was obtained jointly with C. Mœglin.