arXiv subjects

Douglas Ulmer

Publications and source records attributed to Douglas Ulmer.

At least 19 recordsLinked to original sources

arXiv2026

Minimal $a$-numbers of Artin--Schreier covers of ordinary curves

Let $k$ be a perfect field of characteristic $p>0$, and let $d$ be a positive integer not divisible by $p$. We define a non-empty Zariski open subset $U$ of the space of polynomials of degree $d$, and for $f(x)\in U(k)$, we compute the $a$-number of the curve defined by $y^p-y=f(x)$. This $a$-number realizes a lower bound given by Booher and Cais, so the latter is tight. Our result also implies that the bound of Booher and Cais for minimal $a$-numbers of Artin-Schreier covers of ordinary curves is tight.

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arXiv2025

New unlikely intersections on elliptic surfaces

Consider a Jacobian elliptic surface $E \to C$ with a section $P$ of infinite order. Previous work of the first author and Urz\'ua over the complex numbers gives a bound on the number of tangencies between $P$ and a torsion section of $E$ (an ``unlikely intersection''), and more precisely, an exact formula for the weighted number of tangencies between $P$ and elements of the ``Betti foliation''. This work used analytic techniques that apparently do not generalize to positive characteristic. In this paper, we extend their work to characteristic $p$, and we develop a second approach to tangency properties of algebraic curves on a complex elliptic surface, yielding a new family of unlikely intersections with a strong connection to a famous homomorphism of Manin. We also correct inaccuracies in the literature about this homomorphism.

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arXiv2023

$p$-torsion for unramified Artin--Schreier covers of curves

Let $Y\to X$ be an unramified Galois cover of curves over a perfect field $k$ of characteristic $p>0$ with $\mathrm{Gal}(Y/X)\cong\mathbb{Z}/p\mathbb{Z}$, and let $J_X$ and $J_Y$ be the Jacobians of $X$ and $Y$ respectively. We consider the $p$-torsion subgroup schemes $J_X[p]$ and $J_Y[p]$, analyze the Galois-module structure of $J_Y[p]$, and find restrictions this structure imposes on $J_Y[p]$ (for example, as manifested in its Ekedahl--Oort type) taking $J_X[p]$ as given.

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arXiv2021

On $BT_1$ group schemes and Fermat Jacobians

Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. We compare three classifications of $BT_1$ group schemes, due in large part to Kraft, Ekedahl, and Oort, and defined using words, canonical filtrations, and permutations. Using this comparison, we determine the Ekedahl--Oort types of Fermat quotient curves and we compute four invariants of the $p$-torsion group schemes of these curves.

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arXiv2021

Every $BT_1$ group scheme appears in a Jacobian

Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. Our main result is that every $BT_1$ scheme group over $k$ occurs as a direct factor of the $p$-torsion group scheme of the Jacobian of an explicit curve defined over $\mathbb{F}_p$. We also treat a variant with polarizations. Our main tools are the Kraft classification of $BT_1$ group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.

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arXiv2020

Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with zero section $O$ and another section $P$ of infinite order. If $k$ has characteristic zero, we show there are only finitely many points where $O$ is tangent to a multiple of $P$. Equivalently, there is a finite list of integers such that if $n$ is not divisible by any of them, then $nP$ is not tangent to $O$. Such tangencies can be interpreted as unlikely intersections. If $k$ has characteristic zero or $p>3$ and $\mathcal{E}$ is very general, then we show there are no tangencies between $O$ and $nP$. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with $K$ ample and $K^2$ unbounded.

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arXiv2020

Bounding tangencies of sections on elliptic surfaces

Given an elliptic surface $\mathcal{E}\to\mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where $O$ is tangent to a multiple of $P$.

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arXiv2019

On the arithmetic of a family of twisted constant elliptic curves

Let $\mathbb{F}_r$ be a finite field of characteristic $p>3$. For any power $q$ of $p$, consider the elliptic curve $E=E_{q,r}$ defined by $y^2=x^3 + t^q -t$ over $K=\mathbb{F}_r(t)$. We describe several arithmetic invariants of $E$ such as the rank of its Mordell--Weil group $E(K)$, the size of its Néron--Tate regulator $\text{Reg}(E)$, and the order of its Tate--Shafarevich group $III(E)$ (which we prove is finite). These invariants have radically different behaviors depending on the congruence class of $p$ modulo 6. For instance $III(E)$ either has trivial $p$-part or is a $p$-group. On the other hand, we show that the product $|III(E)|\text{Reg}(E)$ has size comparable to $r^{q/6}$ as $q\to\infty$, regardless of $p\pmod{6}$. Our approach relies on the BSD conjecture, an explicit expression for the $L$-function of $E$, and a geometric analysis of the Néron model of $E$.

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arXiv2019

On the Brauer-Siegel ratio for abelian varieties over function fields

Hindry has proposed an analogue of the classical Brauer-Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell-Weil group and the order of the Tate-Shafarevich group should have size similar to the exponential differential height. Hindry-Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate-Shafarevich group and the regulator. We recover the results of Hindry-Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.

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arXiv2017

Explicit points on the Legendre curve III

We continue our study of the Legendre elliptic curve $y^2=x(x+1)(x+t)$ over function fields $K_d=\mathbf{F}_p(μ_d,t^{1/d})$. When $d=p^f+1$, we have previously exhibited explicit points generating a subgroup $V_d$ of $E(K_d)$ of rank $d-2$ and of finite, $p$-power index. We also proved the finiteness of $III(E/K_d)$ and a class number formula: $[E(K_d):V_d]^2=|III(E/K_d)|$. In this paper, we compute $E(K_d)/V_d$ and $III(E/K_d)$ explicitly as modules over $\mathbf{Z}_p[\mathrm{Gal}(K_d/F_p(t))]$.

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arXiv2017

Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields

We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $\mathbb{F}_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, we compute the $L$-function of $J$ over $\mathbb{F}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb{F}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^ν+1$, and $K_d := \mathbb{F}_p(μ_d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the Tate-Shafarevich group of $J$ over $K_d$ is $[J(K_d):V]^2$. When $r>2$, we prove that the "new" part of $J$ is isogenous over $\overline{\mathbb{F}_p(t)}$ to the square of a simple abelian variety of dimension $ϕ(r)/2$ with endomorphism algebra $\mathbb{Z}[μ_r]^+$. For a prime $\ell$ with $\ell \nmid pr$, we prove that $J[\ell](L)=\{0\}$ for any abelian extension $L$ of $\overline{\mathbb{F}}_p(t)$.

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arXiv2016

On the number of rational points on special families of curves over function fields

We construct families of curves which provide counterexamples for a uniform boundedness question. These families generalize those studied previously by several authors. We show, in detail, what fails in the argument of Caporaso, Harris, Mazur that uniform boundedness follows from the Lang conjecture. We also give a direct proof that these curves have finitely many rational points and give explicit bounds for the heights and number of such points.

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arXiv2015

Conductors of l-adic representations

We give a new formula for the Artin conductor of an $\ell$-adic representation of the Weil group of a local field of residue characteristic $p\neq\ell$.

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arXiv2015

Arithmetic of abelian varieties in Artin-Schreier extensions

We study abelian varieties defined over function fields of curves in positive characteristic $p$, focusing on their arithmetic within the system of Artin-Schreier extensions. First, we prove that the $L$-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer conjecture holds. Third, we give a formula for the rank of the Mordell-Weil groups of these Jacobians in terms of the geometry of their fibers of bad reduction and homomorphisms between Jacobians of auxiliary Artin-Schreier curves. We illustrate these theorems by computing the rank for explicit examples of Jacobians of arbitrary dimension $g$, exhibiting Jacobians with bounded rank and others with unbounded rank in the tower of Artin-Schreier extensions. Finally, we compute the Mordell-Weil lattices of an isotrivial elliptic curve and a family of non-isotrivial elliptic curves. The latter exhibits an exotic phenomenon whereby the angles between lattice vectors are related to point counts on elliptic curves over finite fields. Our methods also yield new results about supersingular factors of Jacobians of Artin-Schreier curves.

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arXiv2014

Low-dimensional factors of superelliptic Jacobians

Given a polynomial $f\in\mathbb{C}[x]$, we consider the family of superelliptic curves $y^d=f(x)$ and their Jacobians $J_d$ for varying integers $d$. We show that for any integer $g$ the number of abelian varieties up to isogeny of dimension $\le g$ which appear in any $J_d$ is finite and their multiplicities are bounded.

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arXiv2014

Rational curves on elliptic surfaces

We prove that a very general elliptic surface $\mathcal{E}\to\mathbb{P}^1$ over the complex numbers with a section and with geometric genus $p_g\ge2$ contains no rational curves other than the section and components of singular fibers. Equivalently, if $E/\mathbb{C}(t)$ is a very general elliptic curve of height $d\ge3$ and if $L$ is a finite extension of $\mathbb{C}(t)$ with $L\cong\mathbb{C}(u)$, then the Mordell-Weil group $E(L)=0$.

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arXiv2013

Explicit points on the Legendre curve II

Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\Fp(t)$ where $p$ is an odd prime. We study the arithmetic of $E$ over extensions $\Fq(t^{1/d})$ where $q$ is a power of $p$ and $d$ is an integer prime to $p$. The rank of $E$ is given in terms of an elementary property of the subgroup of $(\Z/d\Z)^\times$ generated by $p$. We show that for many values of $d$ the rank is large. For example, if $d$ divides $2(p^f-1)$ and $2(p^f-1)/d$ is odd, then the rank is at least $d/2$. When $d=2(p^f-1)$, we exhibit explicit points generating a subgroup of $E(\Fq(t^{1/d}))$ of finite index in the "2-new" part, and we bound the index as well as the order of the "2-new" part of the Tate-Shafarevich group.

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arXiv2013

Explicit points on the Legendre curve

We study the elliptic curve E given by y^2=x(x+1)(x+t) over the rational function field k(t) and its extensions K_d=k(μ_d,t^{1/d}). When k is finite of characteristic p and d=p^f+1, we write down explicit points on E and show by elementary arguments that they generate a subgroup V_d of rank d-2 and of finite index in E(K_d). Using more sophisticated methods, we then show that the Birch and Swinnerton-Dyer conjecture holds for E over K_d, and we relate the index of V_d in E(K_d) to the order of the Tate-Shafarevich group \sha(E/K_d). When k has characteristic 0, we show that E has rank 0 over K_d for all d.

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