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Journal articles on the topic 'Curves and Jacobians over finite fields'

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Published: 4 June 2021
Last updated: 20 February 2023

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1

Voloch, José Felipe. "Jacobians of Curves over Finite Fields." Rocky Mountain Journal of Mathematics 30, no. 2 (June 2000): 755–59. http://dx.doi.org/10.1216/rmjm/1022009294.

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2

FORD, KEVIN, and IGOR SHPARLINSKI. "ON CURVES OVER FINITE FIELDS WITH JACOBIANS OF SMALL EXPONENT." International Journal of Number Theory 04, no. 05 (October 2008): 819–26. http://dx.doi.org/10.1142/s1793042108001687.

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We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of Duke in the case of g = 1. We also show that when g = 1 or g = 2, our lower bounds on the exponent, valid for almost all finite fields 𝔽q and all curves over 𝔽q, are best possible.
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3

Xiong, Maosheng, and Alexandru Zaharescu. "Statistics of the Jacobians of hyperelliptic curves over finite fields." Mathematical Research Letters 19, no. 2 (2012): 255–72. http://dx.doi.org/10.4310/mrl.2012.v19.n2.a1.

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4

Shparlinski, Igor. "On the size of the Jacobians of curves over finite fields." Bulletin of the Brazilian Mathematical Society, New Series 39, no. 4 (December 2008): 587–95. http://dx.doi.org/10.1007/s00574-008-0006-4.

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5

Overkamp, Otto. "Jumps and Motivic Invariants of Semiabelian Jacobians." International Mathematics Research Notices 2019, no. 20 (January 29, 2018): 6437–79. http://dx.doi.org/10.1093/imrn/rnx303.

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Abstract We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields and their behavior under tame base change. For a semiabelian variety, this behavior is governed by a finite sequence of (a priori) real numbers between 0 and 1, called jumps. The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions and generalize Raynaud’s description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence that decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.
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6

Ahmadi, Omran, Gary McGuire, and Antonio Rojas-León. "Decomposing Jacobians of curves over finite fields in the absence of algebraic structure." Journal of Number Theory 156 (November 2015): 414–31. http://dx.doi.org/10.1016/j.jnt.2015.04.014.

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7

Lauder, Alan G. B., and Daqing Wan. "Computing Zeta Functions of Artin–schreier Curves over Finite Fields." LMS Journal of Computation and Mathematics 5 (2002): 34–55. http://dx.doi.org/10.1112/s1461157000000681.

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AbstractThe authors present a practical polynomial-time algorithm for computing the zeta function of certain Artin–Schreier curves over finite fields. This yields a method for computing the order of the Jacobian of an elliptic curve in characteristic 2, and more generally, any hyperelliptic curve in characteristic 2 whose affine equation is of a particular form. The algorithm is based upon an efficient reduction method for the Dwork cohomology of one-variable exponential sums.
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8

Bruin, Peter, and Filip Najman. "Hyperelliptic modular curves and isogenies of elliptic curves over quadratic fields." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 578–602. http://dx.doi.org/10.1112/s1461157015000157.

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We study elliptic curves over quadratic fields with isogenies of certain degrees. Let $n$ be a positive integer such that the modular curve $X_{0}(n)$ is hyperelliptic of genus ${\geqslant}2$ and such that its Jacobian has rank $0$ over $\mathbb{Q}$. We determine all points of $X_{0}(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, every elliptic curve over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by a quadratic extension $L$ of $K$. We determine $d$ and $L$ explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, all elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb{Q}$-curves.
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9

Cornelissen, G. "Two-torsion in the Jacobian of hyperelliptic curves over finite fields." Archiv der Mathematik 77, no. 3 (September 2001): 241–46. http://dx.doi.org/10.1007/pl00000487.

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10

Garra, Ricard, Josep M. Miret, Jordi Pujolàs, and Nicolas Thériault. "The 2-adic valuation of the cardinality of Jacobians of genus 2 curves over quadratic towers of finite fields." Journal of Algebra and Its Applications 18, no. 07 (July 2019): 1950135. http://dx.doi.org/10.1142/s0219498819501354.

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Given a genus 2 curve [Formula: see text] defined over a finite field [Formula: see text] of odd characteristic such that [Formula: see text], we study the growth of the 2-adic valuation of the cardinality of the Jacobian over a tower of quadratic extensions of [Formula: see text]. In the cases of simpler regularity, we determine the exponents of the 2-Sylow subgroup of [Formula: see text].
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11

Khuri-Makdisi, Kamal. "Upper bounds for some Brill–Noether loci over a finite field." International Journal of Number Theory 14, no. 03 (March 25, 2018): 739–49. http://dx.doi.org/10.1142/s1793042118500471.

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Let [Formula: see text] be a smooth projective algebraic curve of genus [Formula: see text], over the finite field [Formula: see text]. A classical result of H. Martens states that the Brill–Noether locus of line bundles [Formula: see text] in [Formula: see text] with [Formula: see text] and [Formula: see text] is of dimension at most [Formula: see text], under conditions that hold when such an [Formula: see text] is both effective and special. We show that the number of such [Formula: see text] that are rational over [Formula: see text] is bounded above by [Formula: see text], with an explicit constant [Formula: see text] that grows exponentially with [Formula: see text]. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form [Formula: see text] for the probability that a line bundle in [Formula: see text] is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree [Formula: see text] is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.
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12

Velichka, M. D., M. J. Jacobson, and A. Stein. "Computing discrete logarithms in the Jacobian of high-genus hyperelliptic curves over even characteristic finite fields." Mathematics of Computation 83, no. 286 (July 23, 2013): 935–63. http://dx.doi.org/10.1090/s0025-5718-2013-02748-2.

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13

Ulmer, Douglas. "On Mordell–Weil groups of Jacobians over function fields." Journal of the Institute of Mathematics of Jussieu 12, no. 1 (May 15, 2012): 1–29. http://dx.doi.org/10.1017/s1474748012000618.

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AbstractWe study the arithmetic of abelian varieties over $K= k(t)$ where $k$ is an arbitrary field. The main result relates Mordell–Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield completely explicit points on elliptic curves with unbounded rank over ${ \overline{ \mathbb{F} } }_{p} (t)$ and a new construction of elliptic curves with moderately high rank over $ \mathbb{C} (t)$.
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14

Maisner, Daniel, Enric Nart, and Everett W. Howe. "Abelian Surfaces over Finite Fields as Jacobians." Experimental Mathematics 11, no. 3 (January 2002): 321–37. http://dx.doi.org/10.1080/10586458.2002.10504478.

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15

González, Josep. "Fermat Jacobians of Prime Degree over Finite Fields." Canadian Mathematical Bulletin 42, no. 1 (March 1, 1999): 78–86. http://dx.doi.org/10.4153/cmb-1999-009-7.

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AbstractWe study the splitting of Fermat Jacobians of prime degree l over an algebraic closure of a finite field of characteristic p not equal to l. We prove that their decomposition is determined by the residue degree of p in the cyclotomic field of the l-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.
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16

van der Geer, Gerard. "Counting curves over finite fields." Finite Fields and Their Applications 32 (March 2015): 207–32. http://dx.doi.org/10.1016/j.ffa.2014.09.008.

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17

Garcia, A., and J. F. Voloch. "Fermat curves over finite fields." Journal of Number Theory 30, no. 3 (November 1988): 345–56. http://dx.doi.org/10.1016/0022-314x(88)90007-8.

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18

Howe, Everett, Enric Nart, and Christophe Ritzenthaler. "Jacobians in isogeny classes of abelian surfaces over finite fields." Annales de l’institut Fourier 59, no. 1 (2009): 239–89. http://dx.doi.org/10.5802/aif.2430.

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19

Auer, R. "Legendre Elliptic Curves over Finite Fields." Journal of Number Theory 95, no. 2 (August 2002): 303–12. http://dx.doi.org/10.1016/s0022-314x(01)92760-x.

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20

Katz, Nicholas M. "Space filling curves over finite fields." Mathematical Research Letters 6, no. 6 (1999): 613–24. http://dx.doi.org/10.4310/mrl.1999.v6.n6.a2.

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21

Auer, Roland, and Jaap Top. "Legendre Elliptic Curves over Finite Fields." Journal of Number Theory 95, no. 2 (August 2002): 303–12. http://dx.doi.org/10.1006/jnth.2001.2760.

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22

Lebacque, Philippe, and Alexey Zykin. "On the number of rational points of Jacobians over finite fields." Acta Arithmetica 169, no. 4 (2015): 373–84. http://dx.doi.org/10.4064/aa169-4-5.

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23

Bruin, Nils, and Kevin Doerksen. "The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians." Canadian Journal of Mathematics 63, no. 5 (October 18, 2011): 992–1024. http://dx.doi.org/10.4153/cjm-2011-039-3.

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Abstract In this paper we study genus 2 curves whose Jacobians admit a polarized (4, 4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion, and we derive the relation their absolute invariants satisfy.As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered.Our main tool is a Galois theoretic characterization of genus 2 curves admitting multiple Richelot isogenies.
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24

Berger, Lisa, Chris Hall, René Pannekoek, Jennifer Park, Rachel Pries, Shahed Sharif, Alice Silverberg, and Douglas Ulmer. "Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields." Memoirs of the American Mathematical Society 266, no. 1295 (July 2020): 0. http://dx.doi.org/10.1090/memo/1295.

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25

Balakrishnan, Jennifer S., Sorina Ionica, Kristin Lauter, and Christelle Vincent. "Constructing genus-3 hyperelliptic Jacobians with CM." LMS Journal of Computation and Mathematics 19, A (2016): 283–300. http://dx.doi.org/10.1112/s1461157016000322.

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Given a sextic CM field $K$, we give an explicit method for finding all genus-$3$ hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field $K$, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field $\mathbb{F}_{p}$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo $p$.
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26

Smith, Benjamin. "Families of explicitly isogenous Jacobians of variable-separated curves." LMS Journal of Computation and Mathematics 14 (August 1, 2011): 179–99. http://dx.doi.org/10.1112/s1461157010000410.

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AbstractWe construct six infinite series of families of pairs of curves (X,Y ) of arbitrarily high genus, defined over number fields, together with an explicit isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by 2, 3 or 4. For each family, we compute the isomorphism type of the isogeny kernel and the dimension of the image of the family in the appropriate moduli space. The families are derived from Cassou-Noguès and Couveignes’ explicit classification of pairs (f,g) of polynomials such that f(x1)−g(x2) is reducible.Supplementary materials are available with this article.
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27

Bridy, Andrew. "Automatic sequences and curves over finite fields." Algebra & Number Theory 11, no. 3 (May 6, 2017): 685–712. http://dx.doi.org/10.2140/ant.2017.11.685.

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28

Skałba, M. "Points on elliptic curves over finite fields." Acta Arithmetica 117, no. 3 (2005): 293–301. http://dx.doi.org/10.4064/aa117-3-7.

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29

Shparlinski, Igor E., and José Felipe Voloch. "Visible Points on Curves over Finite Fields." Bulletin of the Polish Academy of Sciences Mathematics 55, no. 3 (2007): 193–99. http://dx.doi.org/10.4064/ba55-3-1.

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30

Hirschfeld, J. W. P. "Book Review: Algebraic curves over finite fields." Bulletin of the American Mathematical Society 27, no. 2 (October 1, 1992): 327–33. http://dx.doi.org/10.1090/s0273-0979-1992-00321-x.

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31

Garcia, Arnaldo, and Luciane Quoos. "A construction of curves over finite fields." Acta Arithmetica 98, no. 2 (2001): 181–95. http://dx.doi.org/10.4064/aa98-2-8.

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32

JEON, DAEYEOL. "POINTS ON MODULAR CURVES OVER FINITE FIELDS." Journal of the Chungcheong Mathematical Society 28, no. 3 (August 15, 2015): 443–49. http://dx.doi.org/10.14403/jcms.2015.28.3.443.

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33

Stöhr, Karl-Otto, and José Felipe Voloch. "Weierstrass Points and Curves Over Finite Fields." Proceedings of the London Mathematical Society s3-52, no. 1 (January 1986): 1–19. http://dx.doi.org/10.1112/plms/s3-52.1.1.

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34

von zur Gathen, Joachim, Igor Shparlinski, and Alistair Sinclair. "Finding Points on Curves over Finite Fields." SIAM Journal on Computing 32, no. 6 (January 2003): 1436–48. http://dx.doi.org/10.1137/s0097539799351018.

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35

Montanucci, Maria, and Giovanni Zini. "Generalized Artin–Mumford curves over finite fields." Journal of Algebra 485 (September 2017): 310–31. http://dx.doi.org/10.1016/j.jalgebra.2017.05.020.

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36

Aubry, Yves, and Marc Perret. "Coverings of singular curves over finite fields." Manuscripta Mathematica 88, no. 1 (December 1995): 467–78. http://dx.doi.org/10.1007/bf02567835.

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37

Fukshansky, Lenny, and Hiren Maharaj. "Lattices from elliptic curves over finite fields." Finite Fields and Their Applications 28 (July 2014): 67–78. http://dx.doi.org/10.1016/j.ffa.2014.01.007.

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38

Padmanabhan, R., and Alok Shukla. "Orchards in elliptic curves over finite fields." Finite Fields and Their Applications 68 (December 2020): 101756. http://dx.doi.org/10.1016/j.ffa.2020.101756.

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39

Schoof, René. "Nonsingular plane cubic curves over finite fields." Journal of Combinatorial Theory, Series A 46, no. 2 (November 1987): 183–211. http://dx.doi.org/10.1016/0097-3165(87)90003-3.

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40

ACHTER, JEFFREY D., and SIMAN WONG. "QUOTIENTS OF ELLIPTIC CURVES OVER FINITE FIELDS." International Journal of Number Theory 09, no. 06 (September 2013): 1395–412. http://dx.doi.org/10.1142/s1793042113500334.

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Fix a prime ℓ, and let 𝔽q be a finite field with q ≡ 1 (mod ℓ) elements. If ℓ > 2 and q ≫ℓ 1, we show that asymptotically (ℓ - 1)2/2ℓ2 of the elliptic curves E/𝔽q with complete rational ℓ-torsion are such that E/〈P〉 does not have complete rational ℓ-torsion for any point P ∈ E(𝔽q) of order ℓ. For ℓ = 2 the asymptotic density is 0 or 1/4, depending whether q ≡ 1 (mod 4) or 3 (mod 4). We also show that for any ℓ, if E/𝔽q has an 𝔽q-rational point R of order ℓ2, then E/〈ℓR〉 always has complete rational ℓ-torsion.
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41

Nestler, Andrew. "SK1 of Affine Curves over Finite Fields." Journal of Algebra 225, no. 2 (March 2000): 943–46. http://dx.doi.org/10.1006/jabr.1999.8187.

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42

Huang, Ming-Deh, and Doug Ierardi. "Counting Points on Curves over Finite Fields." Journal of Symbolic Computation 25, no. 1 (January 1998): 1–21. http://dx.doi.org/10.1006/jsco.1997.0164.

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43

Maisner, Daniel, and Enric Nart. "Zeta Functions of Supersingular Curves of Genus 2." Canadian Journal of Mathematics 59, no. 2 (April 1, 2007): 372–92. http://dx.doi.org/10.4153/cjm-2007-016-6.

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AbstractWe determine which isogeny classes of supersingular abelian surfaces over a finite field k of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to k-isomorphism, leading to the same zeta function.
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44

Bröker, Reinier, Everett W. Howe, Kristin E. Lauter, and Peter Stevenhagen. "Genus-2 curves and Jacobians with a given number of points." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 170–97. http://dx.doi.org/10.1112/s1461157014000461.

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AbstractWe study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-$2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus-$2$ curves each having exactly $10^{2013}$ points.In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus-$2$ curves over $k$ that have $k$-rational degree-$3$ maps to elliptic curves, including formulas for the genus-$2$ curves, the associated elliptic curves, and the degree-$3$ maps.Supplementary materials are available with this article.
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45

Koike, Masao. "Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields." Hiroshima Mathematical Journal 25, no. 1 (1995): 43–52. http://dx.doi.org/10.32917/hmj/1206127824.

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46

Hu, Yong. "Weak approximation over function fields of curves over large or finite fields." Mathematische Annalen 348, no. 2 (January 22, 2010): 357–77. http://dx.doi.org/10.1007/s00208-010-0481-y.

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47

Mânzăţeanu, Adelina. "RATIONAL CURVES ON CUBIC HYPERSURFACES OVER FINITE FIELDS." Mathematika 67, no. 2 (February 13, 2021): 366–87. http://dx.doi.org/10.1112/mtk.12073.

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48

Cheon, J., and S. Hahn. "Division polynomials of elliptic curves over finite fields." Proceedings of the Japan Academy, Series A, Mathematical Sciences 72, no. 10 (1996): 226–27. http://dx.doi.org/10.3792/pjaa.72.226.

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49

Voloch, J. F. "A note on elliptic curves over finite fields." Bulletin de la Société mathématique de France 116, no. 4 (1988): 455–58. http://dx.doi.org/10.24033/bsmf.2107.

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50

Schoof, René. "Counting points on elliptic curves over finite fields." Journal de Théorie des Nombres de Bordeaux 7, no. 1 (1995): 219–54. http://dx.doi.org/10.5802/jtnb.142.

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