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Everett W. Howe

Publications and source records attributed to Everett W. Howe.

At least 19 recordsLinked to original sources
arXiv2026

Doubly isogenous curves of genus two with a rational action of $D_6$

Let $C$ and $C'$ be curves over a finite field $K$, provided with embeddings $ι$ and $ι'$ into their Jacobian varieties. Let $D\to C$ and $D'\to C'$ be the pullbacks (via these embeddings) of the multiplication-by-$2$ maps on the Jacobians. We say that $(C,ι)$ and $(C',ι')$ are \emph{doubly isogenous} if $\mathrm{Jac}(C)$ and $\mathrm{Jac}(C')$ are isogenous over $K$ and $\mathrm{Jac}(D)$ and $\mathrm{Jac}(D')$ are isogenous over~$K$. When we restrict attention to the case where $C$ and $C'$ are curves of genus $2$ whose groups of $K$-rational automorphisms are isomorphic to the dihedral group $D_6$ of order $12$, we find many more doubly isogenous pairs than one would expect from reasonable heuristics. Our analysis of this overabundance of doubly isogenous curves over finite fields leads to the construction of a pair of doubly isogenous curves over a number field. That such a global example exists seems extremely surprising. We show that the Zilber--Pink conjecture implies that there can only be finitely many such examples. When we exclude reductions of this pair of global curves in our counts, we find that the data for the remaining curves is consistent with our original heuristic. Computationally, we find that doubly isogenous curves in our family of $D_6$ curves can be distinguished from one another by considering the isogeny classes of the Prym varieties of certain unramified covers of exponent $3$ and $4$. We discuss how our family of curves can be potentially be used to obtain a deterministic polynomial-time algorithm to factor univariate polynomials over finite fields via an argument of Kayal and Poonen.

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arXiv2026

Curves of genus two with maps of every degree to a fixed elliptic curve

We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that for every genus-$2$ curve $C$, there is an integer $n$ with $1 < n \le 59$ such that there is no minimal degree-$n$ map from $C$ to an elliptic curve.

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arXiv2025

Enumerating places of $\mathbf P^1$ up to automorphisms of $\mathbf P^1$ in quasilinear time

We present an algorithm that, for every fixed degree $n\ge 3$, will enumerate all degree-$n$ places of the projective line over a finite field $k$ up to the natural action of $\operatorname{PGL}_2(k)$ using $O(\log q)$ space and $\widetilde{O}(q^{n-3})$ time, where $q=\#k$. Since there are $Θ(q^{n-3})$orbits of $\operatorname{PGL}_2(k)$ acting on the set of degree-$n$ places, the algorithm is quasilinear in the size of its output. The algorithm is probabilistic unless we assume the extended Riemann hypothesis. We also present an algorithm for enumerating orbit representatives for the action of $\operatorname{PGL}_2(k)$ on the degree-$n$ effective divisors of $\mathbf{P}^1$ over finite fields $k$. The two algorithms depend on one another; our method of enumerating orbits of places of odd degree $n$ depends on enumerating orbits of effective divisors of degree $(n+1)/2$. As an application of the second algorithm, for $g=2$, $3$, and $4$ we implement an algorithm in Magma that computes all hyperelliptic curves of genus $g$ over finite fields $k$ using $O(q^{g-1})$ space and $\widetilde{O}(q^{2g-1})$ time, where $q=\#k$. Our implementation runs $60$--$80$ times faster than existing algorithms for computing genus-$2$ hyperelliptic curves, and about $280$ times faster than existing algorithms for computing genus-$3$ hyperelliptic curves. We know of no other implementations of algorithms to compute genus-$4$ hyperelliptic curves.

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arXiv2025

Purely inseparable Richelot isogenies

We show that if $C$ is a supersingular genus-$2$ curve over an algebraically-closed field of characteristic $2$, then there are infinitely many Richelot isogenies starting from $C$. This is in contrast to what happens with non-supersingular curves in characteristic $2$, or to arbitrary curves in characteristic not $2$: In these situations, there are at most fifteen Richelot isogenies starting from a given genus-$2$ curve. More specifically, we show that if $C_1$ and $C_2$ are two arbitrary supersingular genus-$2$ curves over an algebraically-closed field of characteristic $2$, then there are exactly sixty Richelot isogenies from $C_1$ to $C_2$, unless either $C_1$ or $C_2$ is isomorphic to the curve $y^2 + y = x^5$. In that case, there are either twelve or four Richelot isogenies from $C_1$ to $C_2$, depending on whether $C_1$ is isomorphic to $C_2$. (Here we count Richelot isogenies up to isomorphism.) We give explicit constructions that produce all of the Richelot isogenies between two supersingular curves.

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arXiv2024

Enumerating hyperelliptic curves over finite fields in quasilinear time

We present an algorithm that, for every fixed genus $g$, will enumerate all hyperelliptic curves of genus $g$ over a finite field $k$ of odd characteristic in quasilinear time; that is, the time required for the algorithm is $\widetilde{O}(q^{2g-1})$, where $q=\#k$. Such an algorithm already exists in the case $g=2$, thanks to work of Mestre and Cardona and Quer, and in the case $g=3$, thanks to work of Lercier and Ritzenthaler. Experimentally, it appears that our new algorithm is about two orders of magnitude faster in practice than ones based on their work.

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arXiv2024

On the Maximum Gonality of a Curve over a Finite Field

The gonality of a smooth geometrically connected curve over a field $k$ is the smallest degree of a nonconstant $k$-morphism from the curve to the projective line. In general, the gonality of a curve of genus $g \ge 2$ is at most $2g - 2$. Over finite fields, a result of F.K. Schmidt from the 1930s can be used to prove that the gonality is at most $g+1$. Via a mixture of geometry and computation, we improve this bound: for a curve of genus $g \ge 5$ over a finite field, the gonality is at most $g$. For genus $g = 3$ and $g = 4$, the same result holds with exactly $217$ exceptions: There are two curves of genus $4$ and gonality $5$, and $215$ curves of genus $3$ and gonality $4$. The genus-$4$ examples were found in other papers, and we reproduce their equations here; in supplementary material, we provide equations for the genus-$3$ examples.

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arXiv2023

Refinements of Katz-Sarnak theory for the number of points on curves over finite fields

This paper goes beyond Katz-Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus $g \geq 3$. The experiments point to a stronger notion of convergence than the one provided by the Katz-Sarnak framework for all curves of genus $\geq 3$. However, for elliptic curves and for hyperelliptic curves of every genus we prove that this stronger convergence cannot occur.

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arXiv2023

Powers of 3 with few nonzero bits and a conjecture of Erdős

Using completely elementary methods, we find all powers of 3 that can be written as the sum of at most twenty-two distinct powers of 2, as well as all powers of 2 that can be written as the sum of at most twenty-five distinct powers of 3. The latter result is connected to a conjecture of Erdős, namely, that 1, 4, and 256 are the only powers of 2 that can be written as a sum of distinct powers of 3. We present this work partly as a reminder that for certain exponential Diophantine equations, elementary techniques based on congruences can yield results that would be difficult or impossible to obtain with more advanced techniques involving, for example, linear forms in logarithms.

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arXiv2023

Doubly isogenous genus-2 curves with $D_4$-action

We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C' are curves over a finite field K, with K-rational base points P and P', and let D and D' be the pullbacks (via the Abel-Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C,P) and (C',P') are *doubly isogenous* if Jac(C) and Jac(C') are isogenous over K and Jac(D) and Jac(D') are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naive heuristics predict, and we provide an explanation for this phenomenon.

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arXiv2022

Isogeny classes of abelian varieties with no principal polarizations

We provide a simple method of constructing isogeny classes of abelian varieties over certain fields $k$ such that no variety in the isogeny class has a principal polarization. In particular, given a field $k$, a Galois extension $\ell$ of $k$ of odd prime degree $p$, and an elliptic curve $E$ over $k$ that has no complex multiplication over $k$ and that has no $k$-defined $p$-isogenies to another elliptic curve, we construct a simple $(p-1)$-dimensional abelian variety $X$ over $k$ such that every polarization of every abelian variety isogenous to $X$ has degree divisible by $p^2$. We note that for every odd prime $p$ and every number field $k$, there exist $\ell$ and $E$ as above. We also provide a general framework for determining which finite group schemes occur as kernels of polarizations of abelian varieties in a given isogeny class. Our construction was inspired by a similar construction of Silverberg and Zarhin; their construction requires that the base field $k$ have positive characteristic and that there be a Galois extension of $k$ with a certain non-abelian Galois group. Note: Theorem 3.2 in this paper is incorrect. The current version of the paper includes comments explaining the mistake.

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arXiv2022

Deducing information about curves over finite fields from their Weil polynomials

We discuss methods for using the Weil polynomial of an isogeny class of abelian varieties over a finite field to determine properties of the curves (if any) whose Jacobians lie in the isogeny class. Some methods are strong enough to show that there are no curves with the given Weil polynomial, while other methods can sometimes be used to show that a curve with the given Weil polynomial must have nontrivial automorphisms, or must come provided with a map of known degree to an elliptic curve with known trace. Such properties can sometimes lead to efficient methods for searching for curves with the given Weil polynomial. Many of the techniques we discuss were inspired by methods that Serre used in his 1985 Harvard class on rational points on curves over finite fields. The recent publication of the notes for this course gives an incentive for reviewing the developments in the field that have occurred over the intervening years.

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arXiv2022

Lower bounds on the maximal number of rational points on curves over finite fields

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all $q$ large enough, the existence of a curve of genus $g$ over ${\mathbb F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result, with a bound of the form $1+q+4 \sqrt{q} -32$ valid for all $g\ge 2$ and for all $q$.

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arXiv2020

The maximum number of points on a curve of genus eight over the field of four elements

The Oesterlé bound shows that a curve of genus 8 over the finite field $\mathbb{F}_4$ can have at most 24 rational points, and Niederreiter and Xing used class field theory to show that there exists such a curve with 21 points. We improve both of these results: We show that a genus-8 curve over $\mathbb{F}_4$ can have at most 23 rational points, and we provide an example of such a curve with 22 points, namely the curve defined by the two equations $y^2 + (x^3 + x + 1)y = x^6 + x^5 + x^4 + x^2$ and $z^3 = (x+1)y + x^2.$

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arXiv2020

Variations in the distribution of principally polarized abelian varieties among isogeny classes

We show that for a large class of rings $R$, the number of principally polarized abelian varieties over a finite field in a given simple ordinary isogeny class and with endomorphism ring $R$ is equal either to 0, or to a ratio of class numbers associated to $R$, up to some small computable factors. This class of rings includes the maximal order of the CM field $K$ associated to the isogeny class (for which the result was already known), as well as the order $R$ generated over $\mathbf{Z}$ by Frobenius and Verschiebung. For this latter order, we can use results of Louboutin to estimate the appropriate ratio of class numbers in terms of the size of the base field and the Frobenius angles of the isogeny class. The error terms in our estimates are quite large, but the trigonometric terms in the estimate are suggestive: Combined with a result of Vladut on the distribution of Frobenius angles of isogeny classes, they give a heuristic argument in support of the theorem of Katz and Sarnak on the limiting distribution of the multiset of Frobenius angles for principally polarized abelian varieties of a fixed dimension over finite fields.

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arXiv2020

Algorithm to enumerate superspecial Howe curves of genus $4$

A Howe curve is a curve of genus $4$ obtained as the fiber product over $\mathbf{P}^1$ of two elliptic curves. Any Howe curve is canonical. This paper provides an efficient algorithm to find superspecial Howe curves and that to enumerate their isomorphism classes. We discuss not only an algorithm to test the superspeciality but also an algorithm to test isomorphisms for Howe curves. Our algorithms are much more efficient than conventional ones proposed by the authors so far for general canonical curves. We show the existence of a superspecial Howe curve in characteristic $7<p\le 331$ and enumerate the isomorphism classes of superspecial Howe curves in characteristic $p\le 53$, by executing our algorithms over the computer algebra system Magma.

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arXiv2020

Hasse-Witt and Cartier-Manin matrices: A warning and a request

Let X be a curve in positive characteristic. A Hasse--Witt matrix for X is a matrix that represents the action of the Frobenius operator on the cohomology group H^1(X,O_X) with respect to some basis. A Cartier--Manin matrix for X is a matrix that represents the action of the Cartier operator on the space of holomorphic differentials of X with respect to some basis. The operators that these matrices represent are adjoint to one another, so Hasse--Witt matrices and the Cartier--Manin matrices are related to one another, but there seems to be a fair amount of confusion in the literature about the exact nature of this relationship. This confusion arises from differences in terminology, from differing conventions about whether matrices act on the left or on the right, and from misunderstandings about the proper formulae for iterating semilinear operators. Unfortunately, this confusion has led to the publication of incorrect results. In this paper we present the issues involved as clearly as we can, and we look through the literature to see where there may be problems. We encourage future authors to clearly distinguish between Hasse--Witt and Cartier--Manin matrices, in the hope that further errors can be avoided.

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arXiv2020

Kernels of polarizations of abelian varieties over finite fields

Suppose $C$ is an isogeny class of abelian varieties over a finite field $k$. In this paper we give a partial answer to the question of which finite group schemes over $k$ occur as kernels of polarizations of varieties in $C$. We show that there is an element $I_C$ of a finite two-torsion group that determines which Jordan-Hölder isomorphism classes of finite commutative group schemes over $k$ contain kernels of polarizations. We indicate how the two-torsion group can be computed from the characteristic polynomial of the Frobenius endomorphism of the varieties in $C$, and we give some relatively weak sufficient conditions for the element $I_C$ to be zero. Using these conditions, we show that every isogeny class of simple odd-dimensional abelian varieties over a finite field contains a principally polarized variety. As a step in the proofs of these theorems, we prove that if $K$ is a CM-field and $A$ is a central simple $K$-algebra with an involution of the second kind, then every totally positive real element of $K$ is the reduced norm of a positive symmetric element of $A$.

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