arXiv subjects

Eric Larson

Publications and source records attributed to Eric Larson.

40 records · Page 3Linked to original sources

arXiv2011

Integrality Properties of the CM-values of Certain Weak Maass Forms

In a recent paper, Bruinier and Ono prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. In particular, for the partition function $p(n)$, they prove that \[p(n)=\frac{1}{24n-1} \sum P(α_Q),\] where $P$ is a weak Maass form and $α_Q$ ranges over a finite set of discriminant $-24n+1$ CM points. Moreover, they show that $6 (24n-1) P(α_Q)$ is always an algebraic integer, and they conjecture that $(24n-1) P(α_Q)$ is always an algebraic integer. Here we prove a general theorem which implies this conjecture as a corollary.

math.NT↗

arXiv2011

Upper Bounds for the Number of Number Fields with Alternating Galois Group

We study the number $N(n, A_n, X)$ of number fields of degree $n$ whose Galois closure has Galois group $A_n$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect that $N(n, A_n, X) \sim C_n X^{1/2} (\log X)^{b_n}$, for constants $b_n$ and $C_n$. For $5 < n < 84394$, the best known upper bound is $N(n, A_n, X) \ll X^{\frac{n + 2}{4}}$; this bound follows from Schmidt's Theorem, which implies there are $\ll X^{\frac{n + 2}{4}}$ number fields of degree $n$. (For $n > 84393$, there are better bounds due to Ellenberg and Venkatesh.) We show, using the important work of Pila on counting integral points on curves, that $N(n, A_n, X) \ll X^{\frac{n^2 - 2}{4(n - 1)}+ε}$, thereby improving the best previous exponent by approximately 1/4 for $5 < n < 84394$.

math.NT↗

arXiv2009

The DNA Inequality in Non-Convex Regions

A simple plane closed curve $Γ$ satisfies the DNA Inequality if the average curvature of any closed curve contained inside $Γ$ exceeds the average curvature of $Γ$. In 1997 Lagarias and Richardson proved that all convex curves satisfy the DNA Inequality and asked whether this is true for any non-convex curve. They conjectured that the DNA Inequality holds for certain L-shaped curves. In this paper, we disprove this conjecture for all L-Shapes and construct a large class of non-convex curves for which the DNA Inequality holds. We also give a polynomial-time procedure for determining whether any specific curve in a much larger class satisfies the DNA Inequality.

math.MG↗

arXiv2009

On the classification of certain fusion categories

We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories -- and consequently, semi-simple Hopf algebras -- of dimension $pq^2$, where $p$ and $q$ are distinct primes. This case is especially interesting because it is the simplest class of dimensions where not all integral fusion categories are group-theoretical. Secondly, we classify a certain family of $\ZZ/3\ZZ$-graded fusion categories, which are generalizations of the $\ZZ/2\ZZ$-graded Tambara-Yamagami categories. Our proofs are based on the recently developed theory of extensions of fusion categories.

math.QA↗