arXiv subjects

Joe Kramer-Miller

Publications and source records attributed to Joe Kramer-Miller.

14 recordsLinked to original sources

arXiv2025

Ekedahl-Oort types of $\mathbb{Z}/2\mathbb{Z}$-covers in characteristic $2$

In this article we study the Ekedahl-Oort types of $\mathbb{Z}/2\mathbb{Z}$-Galois covers $\pi:Y \to X$ in characteristic two. When the base curve $X$ is ordinary, we show that the Ekedahl-Oort type of $Y$ is completely determined by the genus of $X$ and the ramification of $\pi$. For a general base curve $X$, we prove bounds on the Ekedahl-Oort depending on the Ekedahl-Oort type of $X$ and the ramification of $\pi$. Along the way, we develop a theory of \emph{enhanced differentials of the second kind}. This theory allows us to study algebraic de Rham cohomology in any characteristic by working directly with differentials, in contrast to the standard \v{C}ech resolution.

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arXiv2025

The diagonal and Hadamard grade of hypergeometric functions

Diagonals of rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. In this paper we study the diagonal grade of a function $f$, which is defined to be the smallest $n$ such that $f$ is the diagonal of a rational function in variables $x_0,\dots, x_n$. We relate the diagonal grade of a function to the nilpotence of the associated differential equation. This allows us to determine the diagonal grade of many hypergeometric functions and answer affirmatively the outstanding question on the existence of functions with diagonal grade greater than $2$. In particular, we show that $\prescript{}{n}F_{n-1}(\frac{1}{2},\dots, \frac{1}{2};1\dots,1 \mid x)$ has diagonal grade $n$ for each $n\geq 1$. Our method also applies to the generating function of the Ap\'ery sequence, which we find to have diagonal grade $3$. We also answer related questions on Hadamard grades posed by Allouche and Mend\`es France. For example, we show that $\prescript{}{n}F_{n-1}(\frac{1}{2},\dots, \frac{1}{2};1\dots,1 \mid x)$ has Hadamard grade $n$ for all $n\geq 1$.

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arXiv2024

Iwasawa theory of Frobenius-torsion class group schemes

We establish a new Iwasawa theory for the kernel of Frobenius on Jacobians of curves in geometric $\mathbf{Z}_p$-towers over the projective line in characteristic $p$, thereby proving several of the main conjectures of [arXiv:2107.12555].

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arXiv2024

Transcendence properties of the Artin-Hasse exponential modulo $p$

Let $E_p(x)$ denote the Artin-Hasse exponential and let $\overline{E}_p(x)$ denote its reduction modulo $p$ in $\mathbb{F}_p[[x]]$. In this article we study transcendence properties of $\overline{E}_p(x)$ over $\mathbb{F}_p[x]$. We give two proofs that $\overline{E}_p(x)$ is transcendental, affirmatively answering a question of Thakur. We also prove algebraic independence results: i) for $f_1,\dots,f_r \in x\mathbb{F}_p[x]$ satisfying certain linear independence properties, we show that the $\overline{E}_p(f_1), \dots, \overline{E}_p(f_r)$ are algebraically independent over $\mathbb{F}_p[x]$ and ii) we determine the algebraic relations between $\overline{E}_p(cx)$, where $c \in \mathbb{F}_p^\times$. Our proof studies the higher derivatives of $\overline{E}_p(x)$ and makes use of iterative differential Galois theory.

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arXiv2023

Zeros of the Goss zeta function

Let $X$ be a smooth proper curve over a finite field and let $\infty \in X$ be a closed point. Let $A$ be the ring of functions on $X - \infty$. The Goss zeta function $\zeta_A$ of $A$ is an equicharacteristic analogue of the Riemann zeta function. In this article we study the zeros of $\zeta_A$ under the generic condition that $X$ is ordinary. We prove an analogue of the Riemann hypothesis, which verifies a corrected version of a conjecture of Goss. We also show that the zeros of $\zeta_A$ at negative even integers are `simple' and that $\zeta_A$ is nonzero at negative odd integers. This answers questions posed by Goss and Thakur. Both of these results were previously only known under the restrictive hypothesis that $A$ has class number one. Finally, we prove versions of these results for $v$-adic interpolations of the Goss zeta function.

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arXiv2021

Newton Polygons of Sums on Curves I: Local-to-Global Theorems

The purpose of this article is to study Newton polygons of certain abelian $L$-functions on curves. Let $X$ be a smooth affine curve over a finite field $\mathbb{F}_q$ and let $\rho:\pi_1(X) \to \mathbb{C}_p^\times$ be a finite character of order $p^n$. By previous work of the first author, the Newton polygon $\mathrm{NP}(\rho)$ lies above a `Hodge polygon' $\mathrm{HP}(\rho)$, which is defined using local ramification invariants of $\rho$. In this article we study the touching between these two polygons. We prove that $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$ share a vertex if and only if a corresponding vertex is shared between the Newton and Hodge polygons of `local' $L$-functions associated to each ramified point of $\rho$. As a consequence, we determine a necessary and sufficient condition for the coincidence of $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$.

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arXiv2021

Newton Polygons of Sums on Curves II: Variation in $p$-adic Families

In this article we study the behavior of Newton polygons along $\mathbb{Z}_p$-towers of curves. Fix an ordinary curve $X$ over a finite field $\mathbb{F}_q$ of characteristic $p$. By a $\mathbb{Z}_p$-tower $X_\infty/X$ we mean a tower of covers $ \dots \to X_2 \to X_1 \to X$ with $\mathrm{Gal}(X_n/X) \cong \mathbb{Z}/p^n\mathbb{Z}$. We show that if the ramification along the tower is sufficiently moderate, then the slopes of the Newton polygon of $X_n$ are equidistributed in the interval $[0,1]$ as $n$ tends to $\infty$. Under a stronger congruence assumption on the ramification invariants, we completely determine the slopes of the Newton polygon of each curve. This is the first result towards `regularity' in Newton polygon behavior for $\mathbb{Z}_p$-towers over higher genus curves. We also obtain similar results for $\mathbb{Z}_p$-towers twisted by a generic tame character.

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arXiv2020

Some unlikely intersections between the Torelli locus and Newton strata in $\mathcal{A}_g$

Let $p$ be an odd prime. What are the possible Newton polygons for a curve in characteristic $p$? Equivalently, which Newton strata intersect the Torelli locus in $\mathcal{A}_g$? In this note, we study the Newton polygons of certain curves with $\mathbb{Z}/p\mathbb{Z}$-actions. Many of these curves exhibit unlikely intersections between the Torelli locus and the Newton stratification in $\mathcal{A}_g$. Here is one example of particular interest: fix a genus $g$. We show that for any $k$ with $\frac{2g}{3}-\frac{2p(p-1)}{3}\geq 2k(p-1)$, there exists a curve of genus $g$ whose Newton polygon has slopes $\{0,1\}^{g-k(p-1)} \sqcup \{\frac{1}{2}\}^{2k(p-1)}$. This provides evidence for Oort's conjecture that the amalgamation of the Newton polygons of two curves is again the Newton polygon of a curve. We also construct families of curves $\{C_g\}_{g \geq 1}$, where $C_g$ is a curve of genus $g$, whose Newton polygons have interesting asymptotic properties. For example, we construct a family of curves whose Newton polygons are asymptotically bounded below by the graph $y=\frac{x^2}{4g}$. The proof uses a Newton-over-Hodge result for $\mathbb{Z}/p\mathbb{Z}$-covers of curves due to the author, in addition to recent work of Booher-Pries on the realization of this Hodge bound.

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arXiv2020

$p$-adic estimates of abelian Artin $L$-functions on curves

The purpose of this article is to prove a "Newton over Hodge" result for finite characters on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $\rho:\pi_1^{et}(V) \to \mathbb{C}^\times$. In this article, we prove a lower bound on the Newton polygon of the $L$-function $L(\rho,s)$. The estimate depends on monodromy invariants of $\rho$: the Swan conductor and the local exponents. Under certain nondegeneracy assumptions this lower bound agrees with the irregular Hodge filtration introduced by Deligne. In particular, our result further demonstrates Deligne's prediction that the irregular Hodge filtration would force $p$-adic bounds on $L$-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.

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arXiv2020

$p$-adic estimates of exponential sums on curves

The purpose of this article is to prove a ``Newton over Hodge'' result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. For a regular function $\overline{f}$ on $V$, we may form the $L$-function $L(\overline{f},V,s)$ associated to the exponential sums of $\overline{f}$. In this article, we prove a lower estimate on the Newton polygon of $L(\overline{f},V,s)$. The estimate depends on the local monodromy of $f$ around each point $x \in X-V$. This confirms a hope of Deligne that the irregular Hodge filtration forces bounds on $p$-adic valuations of Frobenius eigenvalues. As a corollary, we obtain a lower estimate on the Newton polygon of a curve with an action of $\mathbb{Z}/p\mathbb{Z}$ in terms of local monodromy invariants.

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arXiv2019

Log-decay $F$-isocrystals on higher dimensional varieties

Let $k$ be a perfect field of positive characteristic and let $X$ be a smooth irreducible quasi-compact scheme over $k$. The Drinfeld-Kedlaya theorem states that for an irreducible $F$-isocrystal on $X$, the gap between consecutive generic slopes is bounded by one. In this note we provide a new proof of this theorem. Our proof utilizes the theory of $F$-isocrystals with $r$-log decay. We first show that a rank one $F$-isocrystal with $r$-log decay is overconvergent if $r<1$. Next, we establish a connection between slope gaps and the rate of log-decay of the slope filtration. The Drinfeld-Kedlaya theorem then follows from a simple patching argument.

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arXiv2018

The monodromy of unit-root $F$-isocrystals with geometric origin

Let $C$ be a smooth curve over a finite field in characteristic $p$ and let $M$ be an overconvergent $F$-isocrystal over $C$. After replacing $C$ with a dense open subset $M$ obtains a slope filtration, whose steps interpolate the Frobenius eigenvalues of $M$ with bounded slope. This is a purely $p$-adic phenomenon; there is no counterpart in the theory of lisse $\ell$-adic sheaves. The graded pieces of this slope filtration correspond to lisse $p$-adic sheaves, which we call geometric. Geometric lisse $p$-adic sheaves are mysterious. While they fit together to build an overconvergent $F$-isocrystal, which should have motivic origin, individually they are not motivic. In this article we study the monodromy of geometric lisse $p$-adic sheaves with rank one. We prove that the ramification breaks grow exponentially. In the case where $M$ is ordinary we prove that the ramification breaks are predicted by polynomials in $p^n$, which implies a variant of Wan's genus stability conjecture. The crux of the proof is the theory of $F$-isocrystals with log-decay. We prove a monodromy theorem for these $F$-isocrystals, as well as a theorem relating the slopes of $M$ to the rate of log-decay of the slope filtration. As a consequence of these methods, we provide a new proof of the Drinfeld-Kedlaya theorem for irreducible $F$-isocrystals on curves.

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arXiv2016

The monodromy of $F$-isocrystals with log-decay

Let U be a smooth geometrically connected affine curve over $\mathbb{F}_p$ with compactification X. Following Dwork and Katz, a $p$-adic representation $ρ$ of $π_1(U)$ corresponds to an étale $F$-isocrystal. By work of Tsuzuki and Crew an $F$-isocrystal is overconvergent precisely when $ρ$ has finite monodromy at each $x \in X-U$. However, in practice most F-isocrystals arising geometrically are not overconvergent and have logarithmic growth at singularities (e.g. characters of the Igusa tower over a modular curve). We give a Galois-theoretic interpretation of these log growth $F$-isocrystals in terms of asymptotic properties of higher ramification groups.

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arXiv2015

$p$-adic $L$-functions on Hida Families

A major theme in the theory of $p$-adic deformations of automorphic forms is how $p$-adic $L$-functions over eigenvarieties relate to the geometry of these eigenvarieties. In this article we prove results in this vein for the ordinary part of the eigencurve (i.e. Hida families). We address how Taylor expansions of one variable $p$-adic $L$-functions varying over families can detect "bad" geometric phenomena: crossing components of a certain intersection multiplicity and ramification over the weight space. Our methods involve proving a converse to a result of Vatsal relating congruences between eigenforms to their algebraic special $L$-values and then $p$-adically interpolating congruences using formal models.

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