Search ResultsShowing 1-13 of 13
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arXiv:2410.20262 (Published 2024-10-26)
Supersingular curves of genus five for primes congruent to three modulo four
For every prime number $p$ congruent to three modulo four, we prove that there exists a smooth curve of genus five in characteristic $p$ that is supersingular. This extends earlier work of Ekedahl and of Harashita, Kudo, and Senda.
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arXiv:2401.09396 (Published 2024-01-17)
Curves with prescribed rational points
Categories: math.NTGiven a smooth curve $C/\mathbb{Q}$ with genus $\geq 2$, we know by Faltings' Theorem that $C(\mathbb{Q})$ is finite. Here we ask the reverse question: given a finite set of rational points $S\subseteq \mathbb{P}^n(\mathbb{Q})$, does there exist a smooth curve $C/\mathbb{Q}$ contained in $\mathbb{P}^n$ such that $C(\mathbb{Q})=S$? We answer this question in the affirmative by providing an effective algorithm for constructing such a curve.
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arXiv:2310.08493 (Published 2023-10-12)
The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2
Comments: 71 pages; comments welcome!Subjects: 11G05Let $K$ be the function field of a smooth curve $B$ over a finite field $k$ of arbitrary characteristic. We prove that the average size of the $2$-Selmer groups of elliptic curves $E/K$ is at most $1+2\zeta_B(2)\zeta_B(10)$, where $\zeta_B$ is the zeta function of the curve $B/k$. In particular, in the limit as $q=\#k\to\infty$ (with the genus $g(B)$ fixed), we see that the average size of 2-Selmer is bounded above by $3$, even in ``bad'' characteristics. Along the way, we also produce new bounds on 2-Selmer groups of elliptic curves over characteristic $2$ global function fields.
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arXiv:2302.03986 (Published 2023-02-08)
Rational Points of some genus $3$ curves from the rank $0$ quotient strategy
Comments: 9 pagesCategories: math.NTIn 1922, Mordell conjectured that the set of rational points on a smooth curve $C$ over $\mathbb{Q}$ with genus $g \ge 2$ is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of $\#C(\mathbb{Q})$ by following Chabauty's approach which considers the special case when the Jacobian variety of $C$ has Mordell-Weil rank $< g$. In 2006, Stoll improved the Coleman's bound. Balakrishnan with her co-authors in [1] implemented the Chabauty-Coleman method to compute the rational points of genus $3$ hyperelliptic curves. Then, Hashimoto and Morrison [8] did the same work for Picard curves. But it happens that this work has not yet been done for all genus 3 curves. In this paper, we describe an algorithm to compute the complete set of rational points $C(\mathbb{Q})$ for any genus $3$ curve $C/\mathbb{Q}$ that is a degree-$2$ cover of a genus $1$ curve whose Jacobian has rank $0$. We implemented this algorithm in Magma, and we ran it on approximately $40, 000$ curves selected from databases of plane quartics and genus $3$ hyperellitic curves. We discuss some interesting examples, and we exhibit curves for which the number of rational points meets the Stoll's bound
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Fields of moduli and the arithmetic of tame quotient singularities
Given a perfect field $k$ with algebraic closure $\bar{k}$ and a variety $X$ over $\bar{k}$, the field of moduli of $X$ is the subfield of $\bar{k}$ of elements fixed by field automorphisms $\gamma\in\operatorname{Gal}(\bar{k}/k)$ such that the twist $X_{\gamma}$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset\bar{k}$ such that $X$ descends to $k'$. In this paper we extend the formalism, and define the field of moduli when $k$ is not perfect. D\`ebes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and proved that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As an application we give conditions on the automorphism group of a variety $X$ with a smooth marked point $p$ that ensure that the pair $(X,p)$ is defined over its field of moduli.
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arXiv:2004.08989 (Published 2020-04-19)
Big fields that are not large
Categories: math.NTA subfield $K$ of $\bar{\mathbb{Q}}$ is $large$ if every smooth curve $C$ over $K$ with a rational point has infinitely many rational points. A subfield $K$ of $\bar{\mathbb{Q}}$ is $big$ if for every positive integer $n$, $K$ contains a number field $F$ with $[F:\mathbb{Q}]$ divisible by $n$. The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large.
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arXiv:1811.00604 (Published 2018-11-01)
Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties
Categories: math.NTWe study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain PEL-type Shimura varieties. We develop a clutching method to show that the intersection of the Torelli locus with some Newton polygon strata is non-empty and has the expected codimension. This yields results about the Newton polygon stratification of Hurwitz spaces of cyclic covers of the projective line. The clutching method allows us to guarantee the existence of a smooth curve whose Newton polygon is the amalgamate sum of two other Newton polygons, under certain compatibility conditions. As an application, we produce infinitely many new examples of Newton polygons that occur for Jacobians of smooth curves in characteristic $p$. Most of these arise in inductive systems which demonstrate unlikely intersections of the Torelli locus with the Newton polygon stratification. As another application, for the PEL-type Shimura varieties associated to the twenty special families of cyclic covers of the projective line found by Moonen, we prove that all Newton polygon strata intersect the open Torelli locus (assuming $p$ sufficiently large for certain supersingular cases).
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arXiv:1708.03652 (Published 2017-08-11)
Non-ordinary curves with a Prym variety of low $p$-rank
If $\pi: Y \to X$ is an unramified double cover of a smooth curve of genus $g$, then the Prym variety $P_\pi$ is a principally polarized abelian variety of dimension $g-1$. When $X$ is defined over an algebraically closed field $k$ of characteristic $p$, it is not known in general which $p$-ranks can occur for $P_\pi$ under restrictions on the $p$-rank of $X$. In this paper, when $X$ is a non-hyperelliptic curve of genus $g=3$, we analyze the relationship between the Hasse-Witt matrices of $X$ and $P_\pi$. As an application, when $p \equiv 5 \bmod 6$, we prove that there exists a curve $X$ of genus $3$ and $p$-rank $f=3$ having an unramified double cover $\pi:Y \to X$ for which $P_\pi$ has $p$-rank $0$ (and is thus supersingular); for $3 \leq p \leq 19$, we verify the same for each $0 \leq f \leq 3$. Using theoretical results about $p$-rank stratifications of moduli spaces, we prove, for small $p$ and arbitrary $g \geq 3$, that there exists an unramified double cover $\pi: Y \to X$ such that both $X$ and $P_\pi$ have small $p$-rank.
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arXiv:1705.00901 (Published 2017-05-02)
Plane model-fields of definition, fields of definition, the field of moduli of smooth plane curves
Given a smooth plane curve $\overline{C}$ of genus $g\geq 3$ over an algebraically closed field $\overline{k}$, a field $L\subseteq\overline{k}$ is said to be a \emph{plane model-field of definition for $\overline{C}$} if $L$ is a field of definition for $\overline{C}$, i.e. $\exists$ a smooth curve $C'$ defined over $L$ where $C'\times_L\overline{k}\cong \overline{C}$, and such that $C'$ is $L$-isomorphic to a non-singular plane model $F(X,Y,Z)=0$ in $\mathbb{P}^2_{L}$. {In this short note, we construct a smooth plane curve $\overline{C}$ over $\overline{\mathbb{Q}}$, such that the field of moduli of $\overline{C}$ is not a field of definition for $\overline{C}$, and also fields of definition do not coincide with plane model-fields of definition for $\overline{C}$.} As far as we know, this is the first example in the literature with the above property, since this phenomenon does not occur for hyperelliptic curves, replacing plane model-fields of definition with the so-called hyperelliptic model-fields of definition.
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Generic Newton polygons for curves of given p-rank
Comments: 16 pages, to appear in Algebraic Curves and Finite Fields: Codes, Cryptography, and other emergent applications, edited by H. Niederreiter, A. Ostafe, D. Panario, and A. WinterhofWe survey results and open questions about the $p$-ranks and Newton polygons of Jacobians of curves in positive characteristic $p$. We prove some geometric results about the $p$-rank stratification of the moduli space of (hyperelliptic) curves. For example, if $0 \leq f \leq g-1$, we prove that every component of the $p$-rank $f+1$ stratum of ${\mathcal M}_g$ contains a component of the $p$-rank $f$ stratum in its closure. We prove that the $p$-rank $f$ stratum of $\overline{\mathcal M}_g$ is connected. For all primes $p$ and all $g \geq 4$, we demonstrate the existence of a Jacobian of a smooth curve, defined over $\overline{\mathbb F}_p$, whose Newton polygon has slopes $\{0, \frac{1}{4}, \frac{3}{4}, 1\}$. We include partial results about the generic Newton polygons of curves of given genus $g$ and $p$-rank $f$.
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arXiv:1306.5102 (Published 2013-06-21)
Frobenius lifts and point counting for smooth curves
Comments: 29 pages, 2 figuresWe describe an algorithm to compute the zeta-function of a proper, smooth curve over a finite field, when the curve is given together with some auxiliary data. Our method is based on computing the matrix of the action of a semi-linear Frobenius on the first cohomology group of the curve by means of Serre duality. The cup product involved can be computed locally, after first computing local expansions of a globally defined lift of Frobenius. The resulting algorithm's complexity is softly cubic in the field degree, which is also the case with Kedlaya's algorithm in the hyperelliptic case.
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On the canonical representation of curves in positive characteristic
Comments: 13 pages, 0 figuresJournal: New York J. Math. 18 (2012) 911-924Keywords: positive characteristic, automorphism group, global holomorphic differential, smooth curve, rham hypercohomologyTags: journal articleGiven a smooth curve, the canonical representation of its automorphism group is the space of global holomorphic differential 1-forms as a representation of the automorphism group of the curve. In this paper, we study an explicit set of curves in positive characteristic with irreducible canonical representation whose genus is unbounded. Additionally, we study the implications this has for the de Rham hypercohomology as a representation of the automorphism group.
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Independence of l in Lafforgue's theorem
Comments: 19 pages, AMSTeX; revised version 2 to appear in Advances in MathematicsJournal: Adv. Math. 180 (2003), no. 1, 64--86Keywords: lafforgues theorem, prime number, independence, irreducible lisse l-adic sheaf, smooth curveTags: journal articleLet X be a smooth curve over a finite field of characteristic p, let l be a prime number different from p, and let L be an irreducible lisse l-adic sheaf on X whose determinant is of finite order. By a theorem of Lafforgue, for each prime number l' different from p, there exists an irreducible lisse l'-adic sheaf L' on X which is compatible with L, in the sense that at every closed point x of X, the characteristic polynomials of Frobenius at x for L and L' are equal. We prove an "independence of l" assertion on the fields of definition of these irreducible l'-adic sheaves L' : namely, that there exists a number field F such that for any prime number l' different from p, the l'-adic sheaf L' above is defined over the completion of F at one of its l'-adic places.