We determine what 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 to count the number of curves, up to k-isomorphism, leading to the same zeta function.
We prove that there exists a supersingular nonsingular curve of genus $4$ in arbitrary characteristic $p$. For $p>3$ we shall prove that the desingularization of a certain fiber product over $\mathbb{P}^1$ of two supersingular elliptic curves is supersingular.
One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus $g$? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on genus 4 hyperelliptic supersingular curves over finite fields of order $2^n$, $n$ odd.
We describe the intersection of the Torelli locus $j(\mathcal{M}_4^{ct}) = \mathcal{J}_4$ with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic two. We show that the locus of supersingular Jacobians $\mathcal{S}_4\cap\mathcal{J}_4$ in characteristic two is pure of dimension three. One way to obtain that result uses an analysis of the data of smooth genus four curves and principally polarized abelian fourfolds defined over $\mathbb{F}_2$, and another involves studying the Ekedahl-Oort types and using the indecomposability of some $\mathrm{BT}_1$ group schemes.
We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of such fibrations without purely inseparable multisections. Finally, we discuss the consequences for the claimed proof of the Artin conjecture on unirationality of supersingular K3 surfaces.
Let C be a supersingular genus-2 curve over an algebraically closed field of characteristic 3. We show that if C is not isomorphic to the curve y^2 = x^5 + 1 then up to isomorphism there are exactly 20 degree-3 maps phi from C to the elliptic curve E with j-invariant 0. We study the coarse moduli space of triples (C,E,phi), paying particular attention to questions of rationality. The results we obtain allow us to determine, for every finite field k of characteristic 3, the polynomials that occur as Weil polynomials of supersingular genus-2 curves over k.
This paper determines the normal forms of hyperelliptic supersingular curves of genus g over an algebraically closed field F of characteristic 2 for 0 < g< 9. We also show that every hyperelliptic supersingular curve of genus 9 over F has an equation y^2-y=x^19+c^8x^9+c^3x for some c in F_2bar. The paper determines the dimensions of the open locus of hyperelliptic supersingular curves of genus g < 10 over F_2bar.
We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. We display these computations in an appendix where we select a family of representatives of all these curves up to geometric isomorphism and we exhibit equations and the zeta function of all their twists. As an application we obtain a direct computation of the cryptographic exponent of the Jacobians of these curves.
We exhibit the isogeny classes of supersingular abelian threefolds over F_{2^n} containing the Jacobian of a genus 3 curve. In particular, we prove that for even n>6 there always exist a maximal and a minimal curve over F_{2^n}. All the curves can be obtained explicitly.
We study the behaviour near s=1/2 of zeta functions of varieties over finite fields F_q with q a square. The main result is an Euler-characteristic formula for the square of the special value at s=1/2. The Euler-characteristic is constructed from the Weil-etale cohomology of a certain supersingular elliptic curve.
We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus $3$ in c…
In late 2012 and early 2013 the discrete logarithm problem (DLP) in finite fields of small characteristic underwent a dramatic series of breakthroughs, culminating in a heuristic quasi-polynomial time algorithm, due to Barbulescu, Gaudry, Joux and Thomé. Using these developments, Adj, Menezes, Oliveira and Rodríguez-Henríquez analysed the concrete security of the DLP, as it arises from pairings on (the Jacobians of) various genus one and two supersingular curves in the literature, which were originally thought to be $128$-bit secure. In particular, they suggested that the new algorithms have n…
Let $A/\overline{\mathbb{F}}\_p$ and $A'/\overline{\mathbb{F}}\_p$ be supersingular principally polarized abelian varieties of dimension $g>1$. For any prime $\ell \ne p$, we give an algorithm that finds a path $ϕ\colon A \rightarrow A'$ in the $(\ell, \dots , \ell)$-isogeny graph in $\widetilde{O}(p^{g-1})$ group operations on a classical computer, and $\widetilde{O}(\sqrt{p^{g-1}})$ calls to the Grover oracle on a quantum computer. The idea is to find paths from $A$ and $A'$ to nodes that correspond to products of lower dimensional abelian varieties, and to recurse down in dimension until an…
We study when Hurwitz curves are supersingular. Specifically, we show that the curve $H_{n,\ell}: X^nY^\ell + Y^nZ^\ell + Z^nX^\ell = 0$, with $n$ and $\ell$ relatively prime, is supersingular over the finite field $\mathbb{F}_{p}$ if and only if there exists an integer $i$ such that $p^i \equiv -1 \bmod (n^2 - n\ell + \ell^2)$. If this holds, we prove that it is also true that the curve is maximal over $\mathbb{F}_{p^{2i}}$. Further, we provide a complete table of supersingular Hurwitz curves of genus less than 5 for characteristic less than 37.
In this paper, we establish a Voronoi formula for the spinor zeta function of a Siegel cusp form of genus 2. We deduce from this formula quantitative results on the number of its positive (resp. negative) coefficients in some short intervals.
We study genus 2 curves over finite fields of small characteristic. The $p$-rank $f$ of a curve induces a stratification of the coarse moduli space $\mathcal{M}_2$ of genus 2 curves up to isomorphism. We are interested in the size of those strata for all $f \in \{0,1,2\}$. In characteristic 2 and 3, previous results show that the supersingular $f=0$ stratum has size $q$. We show that for $q=3^r$, over $\mathbb{F}_q$ the non-ordinary $f=1$ and ordinary $f=2$ strata are of size $q(q-1)$ and $q^2(q-1)$, respectively. We give results found from computer calculations which suggest that these formul…
We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann Hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval I will contain 2g|I| angles as the genus grows. We show that for the variance of number of angles in I is asymptotically a constant multiple of log(2g|I|) and prove a central limit theorem: The normalized fluctuations are Gaussian. These results continue to hold fo…
Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated since the full torsion subgroup has rank 2g. In this paper we prove that distortion maps always exist for supersingular curves of genus g>1 and we construct distortion maps in genus 2 (for embedding degrees 4,5,6 and 12).
We prove or conjecture several relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or conjecturally equivalently algebraic). These are the first known relations between multizetas, which are not with prime field coefficients. We seem to have one universal family. We also find that interestingly the mechanism with which the relations work is quite different from the rational function field case, raising interesting questions about the expected…
The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier's work on elliptic curves. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the denominators of the discriminants of the Igusa class polynomials all have the property 1) that they are bounded by d, the absolute value of the discriminant of K, and 2) that they divide d-x^2, for some integer x whose square is less than d. A slightly stronger condition is given in Section 3. Such primes are primes of bad reduction for the genus 2 curve and primes of …
It is known from work of du Sautoy and Grunewald in \cite{duSG1} that the zeta functions counting subgroups of finite index in infinite nilpotent groups depend upon the behaviour of some associated system of algebraic varieties on reduction $\modp.$ Further to this, in \cite{duS3, duS4} du Sautoy constructed a group whose local zeta function was determined by the number of points on the elliptic curve $E:Y^2=X^3-X.$ In this work we generalise du Sautoy\rq s construction to define a class of groups whose local zeta functions are dependent upon the number of points on the reduction of a given el…
We determine the zeta functions of trinomial curves in terms of Gauss sums and Jacobi sums, and we obtain an explicit formula of the genus of a trinomial curve over a finite field, then we study the conditions for a trinomial curve to be a maximal curve over a finite field.
In this PhD thesis we study holomorphic and non-holomorphic elliptic analogues of multiple zeta values, namely elliptic multiple zeta values and modular graph functions. Both classes of functions have been discovered very recently, and are involved in the computation of genus one superstring amplitudes. In particular, we obtain new results on the asymptotic expansion of these functions that allow us to perform explicit computations and point out analogies between genus zero and genus one amplitudes.
In this paper we show that in perturbative string theory the genus-one contribution to formal 2-point amplitudes can be related to the genus-zero contribution to 4-point amplitudes. This is achieved by studying special linear combinations of multiple zeta values that appear as coefficients of the amplitudes. We also exploit our results to relate closed strings to open strings at genus one using Brown's single-valued projection, proving a conjecture of Broedel, Schlotterer and the second author.
Deriving motivic coactions and single-valued maps at genus zero from zeta generators
Institute for Advanced Study（高等研究院）；ETH Zürich（苏黎世联邦理工学院）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；Uppsala University（乌普萨拉大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学数学研究所）
本文通过 zeta 生成器证明了零 genus 多重 polylogarithms 的动机共作用和单值映射猜想，结合了非交换变量的操作和生成函数。
In any positive genus case, we show an explicit formula of the Mumford forms expressed by infinite products like Selberg type zeta values for Schottky groups. This result is considered as an extension of the formula in terms of Ramanujan's Delta function in the genus 1 case, and is especially applied to studying the rationality of Ruelle zeta values for Schottky groups.
We study the modular graph functions introduced by Green, Russo, Vanhove in the context of type II superstring scattering amplitudes of 4 gravitons on a torus. In particular we describe a method to algorithmically compute the coefficients in their expansion at the cusp in terms of conical sums. We perform explicit computations for 3-graviton functions, which naturally suggest to conjecture that only single-valued multiple zeta values appear.
A curve over a perfect field $K$ of characteristic $p > 0$ is said to be superspecial if its Jacobian is isomorphic to a product of supersingular elliptic curves over the algebraic closure $\overline{K}$. In recent years, isomorphism classes of superspecial nonhyperelliptic curves of genus $4$ over finite fields in small characteristic have been enumerated. In particular, the non-existence of superspecial curves of genus $4$ in characteristic $p = 7$ was proved. In this note, we give an elementary proof of the existence of superspecial nonhyperelliptic curves of genus $4$ for infinitely many p…
Let $p$ be a prime, and let $Γ=\Sp_g(\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic $L$-functions attached to $\Sp_g$, and their analytic properties are given. Critical values for the spinor $L$-functions and $p$-adic constructions are discussed. Rankin's lemma of higher genus is established. A general conjecture on a lifting from $ GSp_{2m} \times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modu…
We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\ell-monodromy of every irreducible component of the stratum H_g^f of hyperelliptic curves of genus g and p-rank f is the symplectic group Sp_{2g}(Z/\ell) if g > 2, f > 0 and \ell is an odd prime distinct from p. These results yield applications about the generic behavior of hyperelliptic curves of given genus and p-rank. The first application is that a generic hyperelliptic curve of genus g > 2 and p-rank 0 is not su…
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 nu…
We study the distribution of the zeroes of the zeta functions of the family of Artin-Schreier covers of the projective line over $\mathbb{F}_q$ when $q$ is fixed and the genus goes to infinity. We consider both the global and the mesoscopic regimes, proving that when the genus goes to infinity, the number of zeroes with angles in a prescribed non-trivial subinterval of $[-π,π)$ has a standard Gaussian distribution (when properly normalized).
We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in Arizona winter school 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactification. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at infinity and definition of a special tangential base point on the moduli space.
Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over $\mathbb{F}_p$ in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve $\mathcal{C}$ of genus $g\geq 2$ has level $2$. We provide a good number of examples and raise a conjecture.
In this paper we prove that there are no hyperelliptic supersingular curves over F_2bar of genus 2^n-1 for any integer n>1. Let g be a natural number, and h=floor(log_2(g+1)+1). Let X be a hyperelliptic curve over F_2bar of genus g>2 and 2-rank zero, given by an affine equation y^2-y=c_{2g+1} x^{2g+1} +...+ c_1 x. We prove that the first slope of the Newton polygon of X is bigger than or equal to 1/h. We also prove that the equality holds if (I) g<2^h-2, c_{2^h-1} is nonzero; or (II) g=2^h-2, c_{2^h-1} or c_{3(2^{h-1})-1} is nonzero. We prove that genus-4 hyperelliptic curve over F_2bar are pr…
In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and C_{ab} curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over F_{p^n}, the expected running time is O(n^3g^6 + n^2g^{6.5}), whereas the space compl…
For a compact Riemann surface $M$ of genus $g\ge 2$, we study the functional equations of the Selberg zeta functions attached with the Tate motives $f$. We prove that certain functional equations hold if and only if $f$ has the absolute automorphy.
We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field $\mathbb F_q$, with explicit real multiplication by an order $\mathbb Z[η]$ in a totally real cubic field. Our main result states that this algorithm requires an expected number of $\widetilde O((\log q)^6)$ bit-operations, where the constant in the $\widetilde O()$ depends on the ring $\mathbb Z[η]$ and on the degrees of polynomials representing the endomorphism $η$. As a proof-of-concept, we compute the zeta function of a curve defined over a 64-bit prime f…
We relate the low-energy expansions of world-sheet integrals in genus-one amplitudes of open- and closed-string states. The respective expansion coefficients are elliptic multiple zeta values in the open-string case and non-holomorphic modular forms dubbed "modular graph forms" for closed strings. By inspecting the differential equations and degeneration limits of suitable generating series of genus-one integrals, we identify formal substitution rules mapping the elliptic multiple zeta values of open strings to the modular graph forms of closed strings. Based on the properties of these rules, …
We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the sequence $D_1$, $D_2, \ldots$ of denominators of multiples $P$, $2P,\ldots$ of $P$ is a strong divisibility sequence in the sense that $\gcd(D_m, D_n) = D_{\gcd(m,n)}$. This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences. A number $N$ is called a Zsigmondy bound of the sequence if each term $D_{n}$ with $n>N$ presents a new p…
In this paper, we show that half of non-zero coefficients of the spinor zeta function of a Siegel cusp form of genus 2 are positive and half are negative.
