arXiv subjects

Andrew Fiori

Publications and source records attributed to Andrew Fiori.

28 records · Page 2Linked to original sources

arXiv2020

Arthur packets for $G_2$ and perverse sheaves on cubics

This paper begins the project of defining Arthur packets of all unipotent representations for the $p$-adic exceptional group $G_2$. Here we treat the most interesting case by defining and computing Arthur packets with component group $S_3$. We also show that the distributions attached to these packets are stable, subject to a hypothesis. This is done using a self-contained microlocal analysis of simple equivariant perverse sheaves on the moduli space of homogeneous cubics in two variables. In forthcoming work we will treat the remaining unipotent representations and their endoscopic classification and strengthen our result on stability.

math.RT↗

arXiv2020

Average liar count for degree-2 Frobenius pseudoprimes

In this paper we obtain lower and upper bounds on the average number of liars for the Quadratic Frobenius Pseudoprime Test of Grantham, generalizing arguments of Erdős and Pomerance, and Monier. These bounds are provided for both Jacobi symbol plus and minus cases, providing evidence for the existence of several challenge pseudoprimes.

math.NT↗

arXiv2019

Lower Bounds for the Least Prime in Chebotarev

In this paper we show there exists an infinite family of number fields $L$, Galois over $\mathbb{Q}$, for which the smallest prime $p$ of $\mathbb{Q}$ which splits completely in $L$ has size at least $( \log(|D_L|) )^{2+o(1)}$. This gives a converse to various upper bounds, which shows that they are best possible.

math.NT↗

arXiv2019

Embeddings of maximal tori in groups of type $F_4$

We classify maximal tori in groups of type $F_4$ over a local or global field of characteristic different from $2$ and $3$. We prove a local-global principle for embeddings of maximal tori in groups of type $F_4$.

math.AG↗

arXiv2017

Rational Conjugacy Classes of Maximal Tori In Groups of Type $D_n$

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type $D_n$, with specific emphasis on the case of number fields and p-adic fields. This includes the forms associated to quadratic spaces, all of their inner and outer forms as well as the Spin groups, their simply connected covers. In particular, in this work we handle all (simply connected) outer forms of $D_4$.

math.GR↗

arXiv2017

A Riemann-Hurwitz Theorem for the Algebraic Euler Characteristic

We prove an analogue of the Riemann-Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

math.AG↗

arXiv2016

Toroidal Compactifications and Dimension Formulas for Spaces of Modular Forms for Orthogonal Shimura Varieties

In this paper we describe the general theory of constructing toroidal compactifications of locally symmetric spaces and using these to compute dimension formulas for spaces of modular forms. We focus explicitly on the case of the orthogonal locally symmetric spaces arising from quadratic forms of signature $(2,n)$, giving explicit details of the constructions, structures and results in these cases. This article does not give explicit cone decompositions, compute explicit intersection pairings, or count cusps and thus does not give any complete formulas for the dimensions. This article is still `in preparation'.

math.NT↗

arXiv2016

Sub-Shimura Varieties for type O(2,n)

We give a classification, up to consideration of component groups, of sub-Shimura varieties of those Shimura Varieties attached to orthogonal groups of signature (2,n) over Q.

math.NT↗

arXiv2016

On The $j$-Invariants of CM-Elliptic Curves Defined Over $\mathbb{Z}_p$

We characterize the possible reductions of $j$-invariants of elliptic curves which admit complex multiplication by an order $\mathcal{O}$ where the curve itself is defined over $\mathbb{Z}_p$. In particular, we show that the distribution of these $j$-invariants depends on which primes divide the discriminant and conductor of the order.

math.NT↗

arXiv2015

Classification of Certain Subgroups of G2

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type G2, focusing on the case of number fields and p-adic fields. In the same context we characterize the rational conjugacy classes of A2 subgroups of G2. Having obtained the concrete characterization, we then relate it to the more abstract characterization which can be given in terms of Galois cohomology. We note that these results on A2 subgroups were simultaneously and independently developed in the work of Hooda whereas the results on tori were simultaneously and independently developed in the work of Beli-Gille-Lee.

math.GR↗