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Jeffrey Lin Thunder

Publications and source records attributed to Jeffrey Lin Thunder.

6 recordsLinked to original sources
arXiv2024

The distribution of $a$-numbers of hyperelliptic curves in characteristic three

In this paper we present a new approach to counting the proportion of hyperelliptic curves of genus $g$ defined over a finite field $\mathbb{F}_q$ with a given $a$-number. In characteristic three this method gives exact probabilities for curves of the form $Y^2=f(X)$ with $f(X)\in\mathbb{F}_q[X]$ monic and cubefree, probabilities that match the data presented by Cais et al. in previous work. These results are sufficient to derive precise estimates (in terms of $q$) for these probabilities when restricting to squarefree $f$. As a consequence, for positive integers $a$ and $g$ we show that the nonempty strata of the moduli space of hyperelliptic curves of genus $g$ consisting of those curves with $a$-number $a$ are of codimension $2a-1$. This contrasts with the analogous result for the moduli space of abelian varieties in which the codimensions of the strata are $a(a+1)/2$. Finally, our results allow for an alternative heuristic conjecture to that of Cais et al.; one that matches the available data.

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arXiv2015

Thue Equations and Lattices

We consider Diophantine equations of the kind $|F(x,y)|= m$, where $F(X,Y )\in \bz [X,Y]$ is a homogeneous polynomial of degree $d\ge 3$ that has non-zero discriminant and $m$ is a positive integer. We prove results that simplify those of Stewart and provide heuristics for a conjecture of Stewart.

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arXiv2013

On sums involving products and quotients of L-functions over function fields

We estimate the sum of products or quotients of $L$-functions, where the sum is taken over all quadratic extensions of given genus over a fixed global function field. Our estimate for the sum of the quotient of two $L$-functions is analogous to a result of Schmidt where he estimates the sum of the quotient of two $L$-series, where the sum is over quadratic extensions of $\bq$ with absolute value of the discriminant less than a given bound.

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arXiv2011

Counting points of fixed degree and given height over function fields

Let $k$ be a finite field extension of the function field $\bfF_p(T)$ and $\bar{k}$ its algebraic closure. We count points in projective space $\Bbb P ^{n-1}(\bar{k})$ with given height and of fixed degree $d$ over the field $k$. If $n>2d+3$ we derive an asymptotic estimate for their number as the height tends to infinity. As an application we also deduce asymptotic estimates for certain decomposable forms.

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arXiv2001

Decomposable form inequalities

We consider Diophantine inequalities of the kind |f(x)| \le m, where F(X) \in Z[X] is a homogeneous polynomial which can be expressed as a product of d homogeneous linear forms in n variables with complex coefficients and m\ge 1. We say such a form is of finite type if the total volume of all real solutions to this inequality is finite and if, for every n'-dimensional subspace S\subseteq R^n defined over Q, the corresponding n'-dimensional volume for F restricted to S is also finite. We show that the number of integral solutions x \in Z^n to our inequality above is finite for all m if and only if the form F is of finite type. When F is of finite type, we show that the number of integral solutions is estimated asymptotically as m\to \infty by the total number of integral solutions is estimated asymptotically as m\to \infty by the total volume of all real solutions. This generalizes a previous result due to Mahler for the case n=2. Further, we prove a conjecture of W. M. Schmidt, showing that for F of finite type the number of integral solutions is bounded above by c(n,d)m^(n/d), where c(n,d) is an effectively computable constant depending only on n and d.

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