Related papers: On The Inertia Conjecture for Alternating group Co…

This paper verifies Abhyankar's Inertia Conjecture for certain sporadic groups $G$ in particular characteristics by showing that all possible inertia groups occur for $G$-Galois covers of the affine line. For a larger set of sporadic…

We prove Abhyankar's Inertia Conjecture for the alternating group A_{p+2} on p+2 letters when p = 2 mod 3, by showing that every possible inertia group occurs for a (wildly ramified) A_{p+2}-Galois cover of the projective k-line branched…

We obtain new evidence for the Purely Wild Inertia Conjecture posed by Abhyankar and for its generalization. We show that this generalized conjecture is true for any product of simple Alternating groups in odd characteristics, and for any…

Studying two point branched Galois covers of the projective line we prove the Inertia Conjecture for the Alternating groups $A_{p+1}$, $A_{p+3}$, $A_{p+4}$ for any odd prime $p \equiv 2 \pmod{3}$ and for the group $A_{p+5}$ when…

For \ell \neq p odd primes, we examine PSL_2(\ell)-covers of the projective line branched at one point over an algebraically closed field of characteristic p, where PSL_2(\ell) has order divisible by p. We show that such covers can be…

In the present paper, we study a purely inseparable counterpart of Abhyankar's conjecture for the affine line in positive characteristic, and prove its validity for all the finite local non-abelian simple group schemes in characteristic…

We prove that for any prime $p>2$, $q=p^\nu$ a power of $p$, $n\ge p$ and $G=S_n$ or $G=A_n$ (symmetric or alternating group) there exists a Galois extension $K/\mathbb F_q(T)$ ramified only over $\infty$ with $\mathrm{Gal}(K/\mathbb…

Let $U$ be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar Conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck's \'etale…

In this paper, we prove that a refinement of the Alperin-McKay Conjecture for $p$-blocks of finite groups, formulated by I. M. Isaacs and G. Navarro in 2002, holds for all covering groups of the symmetric and alternating groups, whenever…

We prove special cases of a general conjecture: If an invertible field theory admits a projectively topological boundary theory, then it has finite order in the abelian group of invertible field theories. One can substitute `gapped' for…

The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications of these incidence bounds, building on the recent work of…

In the note we construct a family of \'etale coverings of the affine line. More specifically, let $F$ be a finite field of characteristic $p$ and suppose that the cardinality of $F$ is at least 4. Let $A = F[T]$ be the polynomial ring in…

Sp\"ath showed that the Alperin-McKay conjecture in the representation theory of finite groups holds if the so-called inductive Alperin-McKay condition holds for all finite simple groups. In a previous article, we showed that the…

We prove the multiple cover formula conjecture for abelian surfaces for a large class of insertions, including all stationary invariants. The proof uses the reduced degeneration formula expressing the invariants in terms of the correlated…

We describe the additive subgroups of fields which are closed with respect to taking inverses. In particular, in characteristic different from two any such subgroup is either a subfield or the kernel of the trace map of a quadratic…

Given an abelian variety over a field with a discrete valuation, Grothendieck defined a certain open normal subgroup of the absolute inertia group. This subgroup encodes information on the extensions over which the abelian variety acquires…

Given asymptotic counts in number theory, a question of Venkatesh asks what is the topological nature of lower order terms. We consider the arithmetic aspect of the inertia stack of an algebraic stack over finite fields to partially answer…

We prove a nontrivial energy bound for a finite set of affine transformations over a general field and discuss a number of implications. These include new bounds on growth in the affine group, a quantitative version of a theorem by Elekes…

Given a class of objects, a pattern theorem is a powerful result describing their structure. We show that alternating knots exhibit a pattern theorem, and use this result to prove a long-standing conjecture that alternating knots grow rare.…

A cover of normal varieties is exceptional over a finite field if the map on points over infinitely many extensions of the field is one-one. A cover over a number field is exceptional if it is exceptional over infinitely many residue class…