arXiv subjects

Niven Achenjang

Publications and source records attributed to Niven Achenjang.

4 recordsLinked to original sources

arXiv2025

On Brauer groups of tame stacks

We develop some general tools for computing the Brauer group of a tame algebraic stack $\mathscr X$ by studying the difference between it and the Brauer group of the coarse space $X$ of $\mathscr X$. It is our hope that these tools will be used to simplify future computations of Brauer groups of stacks. Informally, we show that $\operatorname{Br}\mathscr X$ is often built from $\operatorname{Br} X$ and "information about the Picard groups of the fibers of $\mathscr X\to X$''. Along these lines, we compute, for example, the Brauer group of the moduli stack $\mathscr Y(1)_S$ of elliptic curves, over any regular noetherian $\mathbb Z[1/2]$-scheme $S$ as well as the Brauer groups of (many) stacky curves (allowing generic stabilizers) over algebraically closed fields.

math.AG↗

arXiv2024

The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2

Let $K$ be the function field of a smooth curve $B$ over a finite field $k$ of arbitrary characteristic. We prove that the average size of the $2$-Selmer groups of elliptic curves $E/K$ is at most $1+2ζ_B(2)ζ_B(10)$, where $ζ_B$ is the zeta function of the curve $B$. In particular, in the limit as $q=\#k\to\infty$ (with the genus $g(B)$ fixed), we see that the average size of 2-Selmer is bounded above by $3$, even in "bad" characteristics. This completes the proof that the average rank of elliptic curves, over $\textit{any}$ fixed global field, is finite. Handling the case of characteristic $2$ requires us to develop a new theory of integral models of 2-Selmer elements, dubbed "hyper-Weierstrass curves."

math.NT↗

arXiv2023

The Brauer Group of $\mathscr{Y}_0(2)$

We determine the Brauer group of the Deligne-Mumford stack $\mathscr{Y}_0(2)$, the moduli space of elliptic curves with a marked $2$-torsion subgroup over bases of arithmetic interest. Antieau and Meier determine the Brauer group for $\mathscr{M}_{1,1}$, the moduli stack of elliptic curves by exploiting the fact it is covered by the Legendre family and using the Hochschild-Serre spectral sequence. Over an algebraically closed field, Shin uses the coarse space map to determine the Brauer group of $\mathscr{M}_{1,1}$. We combine techniques from both papers to determine the Brauer group of $\mathscr{Y}_0(2)$.

math.AG↗

arXiv2018

On Gaps in the Closures of Images of Divisor Functions

Given a complex number $c$, define the divisor function $σ_c:\mathbb N\to\mathbb C$ by $σ_c(n)=\sum_{d\mid n}d^c$. In this paper, we look at $\overline{σ_{-r}(\mathbb N)}$, the topological closures of the image of $σ_{-r}$, when $r>1$. We exhibit new lower bounds on the number of connected components of $\overline{σ_{-r}(\mathbb N)}$, bringing this bound from linear in $r$ to exponential. Finally, we discuss the general structure of gaps of $\overline{σ_{-r}(\mathbb N)}$ in order to work towards a possible monotonicity result.

math.NT↗