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Tristan Phillips

Publications and source records attributed to Tristan Phillips.

14 recordsLinked to original sources
arXiv2026

Lattice point visibility along powers of quadratic polynomials

We study the growth of the number of invisible lattice points along powers of quadratic polynomials. Let $f(x)=Ax^2+Bx+C\in\mathbb{Z}[x]$ have a positive leading coefficient and nonzero discriminant, and let $F(x)=f(x)^m$ with $m\geq 2$. For $m\geq 3$ we prove that the number of invisible lattice points in $[1,N]^2$ has order $N\log N$, and when $m=2$ the number of invisible lattice points satisfies $N\log N \ll_F\#\mathrm{Invisible}_F(N)\ll_F N(\log N)^4$. These estimates refine a previous result of the authors.

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arXiv2026

On the largest strongly connected component of randomly oriented divisor graphs

We introduce the study of \textit{randomly oriented divisor graphs}. For each $\rho \in [0,1]$, the randomly oriented divisor graph $\mathcal{D}_\rho(N)$ is obtained from the divisor graph on $\{1, 2, \ldots, N\}$ by directing each edge according to divisibility and independently reversing the direction of each edge with probability $\rho$. We study the expected size of the largest strongly connected component, $\textbf{E}[\#\Phi(\mathcal{D}_\rho(N))]$. Our main result gives a lower bound for this quantity in terms of the distribution of values of the divisor function $\tau(n)$. As a consequence, we show that for any fixed $\rho \in (0,1)$, the largest strongly connected component has expected size asymptotic to $N$. To obtain explicit bounds, we prove an effective version of a theorem of Hardy and Ramanujan on the normal order of $\log \tau(n)$, which may be of independent interest.

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arXiv2025

Unbounded average Selmer ranks of elliptic curves in torsion families

Let $M$ and $N$ be positive integers for which the modular curve $X_1(M,MN)$ has genus $0$, and let $p$ be a prime divisor of $MN$. This article gives asymptotic lower bounds for the average size of the $p$-Selmer group of elliptic curves over a number field, with torsion subgroup $\mathbb{Z}/M\mathbb{Z} \oplus \mathbb{Z}/MN\mathbb{Z}$. In many cases, it is shown that this average is unbounded.

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arXiv2025

100% of elliptic curves with a marked point have positive rank

As a consequence of their work on average Selmer ranks of elliptic curves with marked points, Bhargava and Ho proved that $100\%$ of elliptic curves over $\mathbb{Q}$ with an additional marked point have positive rank. In this note we provide an alternate proof which extends the result to global fields of characteristic not two or three.

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arXiv2022

Counting Drinfeld modules with prescribed local conditions

We give asymptotics for the number of Drinfeld $\mathbb{F}_q[T]$-modules over $\mathbb{F}_q(T)$ of a given height, which satisfy prescribed sets of local conditions. This is done by relating our problem to a problem about counting points on weighted projective spaces. Our results for counting points of bounded height on weighted projective spaces over global function fields may be of independent interest.

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arXiv2022

Most Elliptic Curves over Global Function Fields are Torsion Free

Given an elliptic curve $E$ over a global function field $K$, the Galois action on the $n$-torsion points of $E$ gives rise to a mod-n Galois representation $ρ_{E,n}$. For $K$ satisfying some mild conditions, we show that the set of $E$ for which $ρ_{E,n}$ is as large as possible for all $n$, has density $1$.

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arXiv2022

Average analytic ranks of elliptic curves over number fields

A conditional bound is given for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which satisfy the Generalized Riemann Hypothesis, it is shown that the average analytic rank of isomorphism classes of elliptic curves over $K$ is bounded above by $(9\ \text{deg}(K)+1)/2$, when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.

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arXiv2022

Points of bounded height in images of morphisms of weighted projective stacks with applications to counting elliptic curves

Asymptotics are given for the number of rational points in the domain of a morphism of weighted projective stacks whose images have bounded height and satisfy a (possibly infinite) set of local conditions. As a consequence we obtain results for counting elliptic curves over number fields with prescribed level structures, including the cases of $\Gamma(N)$ for $N\in\{1,2,3,4,5\}$, $\Gamma_1(N)$ for $N\in\{1,2,\dots,10,12\}$, and $\Gamma_0(N)$ for $N\in\{1,2,4,6,8,9,12,16,18\}$. In all cases we give an asymptotic with an expression for the leading coefficient, and in many cases we also give a power-saving error term.

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arXiv2021

Local Monodromy of 1-Dimensional p-Divisible Groups

Let $G$ be a $p$-divisible group over a complete discrete valuation ring $R$ of characteristic $p$. The generic fiber of $G$ determines a Galois representation $\rho$. The image of $\rho$ admits a ramification filtration and a Lie filtration. We relate these filtrations in the case $G$ is one dimensional, giving an equicharacteristic version of Sen's theorem in this setting. This result generalizes a result of Gross. Additionally, we prove that the representation associated to the \'etale part of $G$ is irreducible, generalizing a result of Chai.

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