Let C be a smooth irreducible projective curve defined over a finite field $\mathbb{F}_{q}$ of q elements of characteristic p>3 and $K=\mathbb{F}_{q}(C)$ its function field and $ϕ_{\mathcal{E}}:\mathcal{E}\to C$ the minimal regular model of $\mathbf{E}/K$. For each $P\in C$ denote $\mathcal{E}_P=ϕ^{-1}_{\mathcal{E}}(P)$. The elliptic curve $E/K$ has good reduction at $P\in C$ if and only if $\mathcal{E}_P$ is an elliptic curve defined over the residue field $κ_P$ of $P$. This field is a finite extension of $\mathbb{F}_q$ of degree $°(P)$. Let $t(\mathcal{E}_P)=q^{°(P)}+1-#\mathcal{E}_P(κ_P)$ be the trace of Frobenius at P. By Hasse-Weil's theorem (cf. [10, Chapter V, Theorem 2.4]), $t(\mathcal{E}_P)$ is the sum of the inverses of the zeros of the zeta function of $\mathcal{E}_P$. In particular, $|t(\mathcal{E}_P)|\le 2q^{°(P)}$. Let $C_0\subset C$ be the set of points of C at which $E/K$ has good reduction and $C_0(\mathbb{F}_{q^k})$ the subset of $\mathbb{F}_{q^k}$-rational points of $C_0$. We discuss the following question.
Let $k\ge 1$ and t be integers and suppose $|t|\le 2q^{k/2}$. Let $π(k,t)=#\{P\in C_0(\mathbb{F}_{q^k}) | t(\mathcal{E}_P)=t\}$. How big is $π(k,t)$?
Let E and E' be two elliptic curves over a number field. We prove that the reductions of E and E' at a finite place p are geometrically isogenous for infinitely many p, and draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether-Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.
We prove that there are no non-constant global functions and no lifts of Frobenius on the first $p$-jet space on an elliptic curve unless the elliptic curve itself has a lift of Frobenius.
In a previous article, we obtained data on the distribution of traces of Frobenius of non-hyperelliptic genus $3$ curves over small finite fields. In the present one, we give a heuristic explanation of these data, by extrapolating from results on the distribution of traces of Frobenius for plane curves whose degree is small with respect to the cardinality of their finite base field. In particular, our methods shed some new light on the asymmetry of the distribution around its mean value, which is related to the Serre obstruction.
We study the distribution of the Frobenius traces on $K3$ surfaces. We compare experimental data with the predictions made by the Sato--Tate conjecture, i.e. with the theoretical distributions derived from the theory of Lie groups assuming equidistribution. Our sample consists of generic $K3$ surfaces, as well as of such having real and complex multiplication. Each time, the theoretical density and the histogram obtained by counting points match in the range of visible accuracy. Thus, we report evidence for the Sato-Tate conjecture for the surfaces considered.
For an elliptic curve A defined over a global function field K of characteristic p>0, the p-Selmer group of the Frobenius twist of A tends to have larger order than that of A. The aim of this note is to discuss this phenomenon.
Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In 1976, Lang and Trotter conjectured an asymptotic formula for the number $π_{E,r}(X)$ of primes $p \leq X$ of good reduction for which the Frobenius trace at $p$ associated to $E$ is equal to a given fixed integer $r$. We investigate elliptic curves $E$ over $\mathbb{Q}$ that have a missing Frobenius trace, i.e. for which the counting function $π_{E,r}(X)$ remains bounded as $X \rightarrow \infty$, for some $r \in \mathbb{Z}$. In particular, we classify all elliptic curves $E$ over $\mathbb{Q}(t)$ that have a missing Frobenius trace.
We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.
Poor man's transcendence for Frobenius traces of elliptic curves
Stellenbosch University（斯泰伦博斯大学）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）；Department of Computer Science, University of Oxford（牛津大学计算机系）；Department of Mathematics, IMAPP, Radboud University（拉德堡德大学IMAPP数学系）
We present here explicit relations between the traces of Frobenius endomorphisms of certain families of elliptic curves and special values of ${_{2}}F_1$-hypergeometric functions over $\mathbb{F}_q$ for $q \equiv 1 (\text{mod} 6)$ and $q \equiv 1 (\text{mod} 4)$.
In \cite{mccarthy2}, McCarthy defined a function $_{n}G_{n}[\cdots]$ using Teichmüller character of finite fields and quotients of $p$-adic gamma function, and expressed the trace of Frobenius of elliptic curves in terms of special values of $_{2}G_{2}[\cdots]$. We establish two different expressions for the traces of Frobenius of elliptic curves in terms of the function $_{2}G_{2}[\cdots]$. As a result, we obtain two relations between special values of the function $_{2}G_{2}[\cdots]$ with different parameters.
We obtain distribution results for traces of Frobenius for various families of elliptic curves with respect to the Lang-Trotter conjecture, extremal primes, and the central limit theorem. This includes some generalisations and bounds related to the work of Sha-Shparlinski on the average Lang-Trotter conjecture for single-parametric families of elliptic curves and the work of various authors on the trace of Frobenius for primes in congruence classes. Some results are also obtained for modular forms.
In this article we give explicit formulae for a lift of the relative Frobenius morphism between elliptic curves and show how one can compute this lift in the case of ordinary reduction in odd characteristic. Our theory can also be used in the case of supersingular reduction. By means of the explicit formulae that describe a Frobenius lift, we are able to generalize Mestre's 2-adic arithmetic geometric mean (AGM) sequence of elliptic curves to odd characteristic, and prove its convergence. As an application, we give an efficient point counting algorithm for ordinary elliptic curves which is bas…
In the late '60s, B. Dwork studied a Frobenius structure compatible with the classical hypergeometric differential equation with parameters $\left(\frac{1}{2},\frac{1}{2} ; 1 \right)$ by analyzing behavior of solutions of the differential equation under Frobenius transformation. Recently, P. Scholze conjectured the existence of $q$-de Rham cohomology groups for any $\mathbb{Z}$-scheme. In this paper, we give a Frobenius structure compatible with the $q$-hypergeometric differential equation with parameters $(q^{\frac12},q^{\frac12};q)$ by showing a $q$-analogue of some results of Dwork. This co…
We present here a formula for expressing the trace of the Frobenius endomorphism of an elliptic curve $E$ over $\mathbb{F}_q$ satisfying $j(E)\neq 0,1728$ and $q\equiv 1 \pmod{12}$ in terms of special values of Gaussian hypergeometric series. This paper uses methods introduced in Fuselier's work for one parameter families of curves to express the trace of Frobenius of $E$ as a function of its $j$-invariant and discriminant instead of a parameter, which are more intrinsic characteristics of the curve.
Recently E. Bombieri and N. M. Katz (2010) have demonstrated that several well-known results about the distribution of values of linear recurrence sequences lead to interesting statements for Frobenius traces of algebraic curves. Here we continue this line of study and establish the Möbius randomness law quantitatively for the normalised form of Frobenius traces.
Let $C$ be a smooth projective curve over $\mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $ϕ:\mathcal{E}\to C$ its minimal regular model. For each $P\in C$ such that $E$ has good reduction at $P$, i.e., the fiber $\mathcal{E}_P=ϕ^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $\mathcal{E}_P$ over the residue field $κ_P$ of $P$ are of the form $q_P^{1/2}e^{iθ_P},q_{P}e^{-iθ_P}$, where $q_P=q^{°(P)}$ and $0\leθ_P\leπ$. The goal of this note is to determine given an integer $B\ge 1$, $α,β\in[0,π]$ the number of $P\in C$ where the reduction of $E$ is goo…
We define a function in terms of quotients of the $p$-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the $p$-adic setting. We prove, for primes $p > 3$, that the trace of Frobenius of any elliptic curve over $\mathbb{F}_p$, whose $j$-invariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over $\mathbb{F}_p$ when $p \equiv 1 \pmod {12}$.
Let $\E/\Q$ be a fixed elliptic curve over $\Q$ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W. Duke have obtained an asymptotic formula for the number of primes $p\le x$ such that the reduction of $\E$ modulo p has a trivial Tate-Shafarevich group. Recent results of A. C. Cojocaru and C. David lead to a better error term. We introduce a new argument in the scheme of the proof which gives further improvement.
We study the averages of products of traces of high powers of the Frobenius class of hyperelliptic curves of genus g over a fixed finite field. We show that for increasing genus g, the limiting expectation of these products equals to the expectation when the curve varies over the unitary symplectic group USp(2g). We also consider the scaling limit of linear statistics for eigenphases of the Frobenius class of hyperelliptic curves, and show that their first few moments are Gaussian.
Let $K$ be a fixed number field, assumed to be Galois over $\mathbb Q$. Let $r$ and $f$ be fixed integers with $f$ positive. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree $f$ prime ideals of $K$ with trace of Frobenius equal to $r$. Except in the case $f=2$, we show that "on average," the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and extends the work of several authors.
Let $q$ be an odd prime power, $a \in \mathbb{F}_q[T]$ and $u \in \mathbb{F}_q^*$. Provided $q \geq 17$, we compute the average number of primes $p$ for which the characteristic polynomial of the Frobenius at $p$ is $X^2 - aX + up$ over a family of rank 2 Drinfeld $ \mathbb{F}_q[T]$-modules. Our results give asymptotic formulas in the $x$-limit.
Frobenius Traces for Rank-2 Drinfeld Modules, Higher-Dimensional Galois Representations, and a Strong Multiplicity One Theorem in Positive Characteristic
We study the moments of $\mbox{Tr}(Θ_χ)$ as $χ$ runs over Dirichlet characters defined over $\mathbb{F}_q[T]$ of fixed order $r$. In particular, we show that after an appropriate normalization, the $q$-limit of the power sum moments behave like the power sum moments of the group of unitary matrices multiplied by a weight function.
Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D<-4 of an imaginary quadratic field K . If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal P of K dividing p and there exist positive integers u and v such that 4p=u^2-Dv^2. It is well known that square of the trace a_P of Frobenius endomorphism of the reduction of E modulo P is equal to u^2. We determine whether a_P=u or a_P=-u in the case the class number of R is 2 or 3 and D is divided by 3,4 or 5. This article is a…
Denote $Θ_C$ as the Frobenius class of a curve $C$ over the finite field $\mathbb{F}_q$. In this paper we determine the expected value of Tr$(Θ_C^n)$ where $C$ runs over all biquadratic curves when $q$ is fixed and $g$ tends to infinity. This extends work done by Rudnick and Chinis who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda who looked at $\ell$-cyclic curves, for $\ell$ a prime, as well as cubic non-Galois curves.
We show that a two dimensional $\ell $-adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a $GL(2)$-$\ell$-adic representation $ρ$ at a set of places of $K$ of positive upper density is potentially equivalent to $ρ$. For an elliptic curver \( E \) defined over a number field \( K \) and a finite place \( v \) of \( K \) of good reduction for \( E \), let \( F(E,v) \) denote the Frobenius field of \( E \) at \( v \), given by the splitting field of the characteristic polynomial of the Frobenius automorphism at \( v \) acting on the Tate…
In this paper we propose conjectures that assert that, the sequence of Frobenius angles of a given elliptic curve over $\mathbf{Q}$ without complex multiplication is pseudorandom, in other words that the Frobenius angles are statistically independently distributed with respect to the Sato-Tate measure. Numerical evidences are presented to support the conjectures.
We consider the expected value of $\mbox{Tr}(Θ_C^n)$ where $C$ runs over a thin family of $r$-cyclic covers of $\mathbb{P}^1_{\mathbb{F}_q}$ for any $r$. We obtain many lower order terms dependent on the divisors of $r$. We use these results to calculate the one-level density of the family and hypothesize a refined one-level density result.
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be a prime of good reduction for $E$. Then, for a prime $p \neq \ell$, the Frobenius automorphism associated to $p$ (unique up to conjugation) acts on the $\ell$-adic Tate module of $E$. The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is independent of $\ell$. Its splitting field is called the Frobenius field of $E$ at $p$. Let $E_1$ and $E_2$ be two elliptic curves defined over $\mathbb{Q}$ that are non-isogenous over $\overline{\mathbb{Q}}$ and also without complex multiplication over $\…
The zeta function of a curve over a finite field may be expressed in terms of the characteristic polynomial of a unitary symplectic matrix, called the Frobenius class of the curve. We compute the expected value of the trace of the n-th power of the Frobenius class for an ensemble of hyperelliptic curves of genus g over a fixed finite field in the limit of large genus, and compare the results to the corresponding averages over the unitary symplectic group USp(2g). We are able to compute the averages for powers n almost up to 4g, finding agreement with the Random Matrix results except for small …
The Zeta function of a curve $C$ over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix $Θ_C$. Following the work of Rudnick, we compute the expected value of $\mbox{tr}(Θ_C^n)$ over the moduli space of hyperelliptic curves of genus $g$, over a fixed finite field $\mathbb{F}_q$, in the limit of large genus. As an application, we compute the expected value of the number of points on $C$ in $\mathbb{F}_{q^n}$ as the genus tends to infinity. We also look at biases in both expected values for small values of $n$.
Let $A$ be an abelian variety over $\mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $ρ_A$ is open in $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. We investigate the arithmetic of the traces $a_{1, p}$ of the Frobenius at $p$ in $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ under $ρ_A$, modulo varying primes $p$. In particular, we obtain upper bounds for the counting function $\#\{p \leq x: a_{1, p} = t\}$ and we prove an Erdös-Kac type theorem for the number of prime factors of $a_{1, p}$. We also formulate a conjecture about the asymp…
Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.
We obtain new results concerning Lang-Trotter conjecture on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A.C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterized families of elliptic curves when the parameter runs through a set of rational numbers of bounded height. Some of the families we consider are much thinner than the ones previously studied.
Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves
Temple University（天普大学）；University of Connecticut（康涅狄格大学）；Korea Institute for Advanced Study（韩国高等研究院）；University of California, Berkeley（加州大学伯克利分校）；University of Westminster（威斯敏斯特大学）；Princeton University（普林斯顿大学）
For primes p congruent to 1 mod 12, we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728/t and values of a particular 2F1-hypergeometric function over F_p. Additionally, we determine a formula for traces of Hecke operators T_k(p) on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's tau-function in terms of hypergeometric functions.
We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Basis in the second family, the so-called normal elliptic basis are normal basis and allow fast (quasi linear) arithmetic. We prove that all extensions admit models of this kind.
Let $S$ be a smooth cubic surface over a finite field $\mathbb F_q$. It is known that $\#S(\mathbb F_q) = 1 + aq + q^2$ for some $a \in \{-2,-1,0,1,2,3,4,5,7\}$. Serre has asked which values of a can arise for a given $q$. Building on special cases treated by Swinnerton-Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton-Dyer's tables on cubic surfaces over finite fields.
We express the Frobenius-Hecke traces on the compactly supported cohomology of a Shimura variety of abelian type in terms of elliptic parts of stable Arthur-Selberg trace formulas for the endoscopic groups. This confirms predictions of Langlands and Kottwitz at primes where the level is hyperspecial.
We show that if $G$ is a linear algebraic group over a number field and if $G$ is not a torus then for all but finitely many primes $p$ the $p$-adic completion of $G$ does not possess a Frobenius lift that is "Lie invariant mod $p$" (in the sense of \cite{alie1}). This is in contrast with the situation of elliptic curves studied in \cite{alie1}.
Let $g \geq 1$ be an integer and let $A/\mathbb{Q}$ be an abelian variety that is isogenous over $\mathbb{Q}$ to %the product $E_1 \times \ldots \times E_g$ of elliptic curves $E_1/\mathbb{Q}$, $\ldots$, $E_g/\mathbb{Q}$, without complex multiplication and pairwise non-isogenous over $\overline{\mathbb{Q}}$. a product of $g$ elliptic curves defined over $\mathbb{Q}$, pairwise non-isogenous over $\overline{\mathbb{Q}}$ and each without complex multiplication. %pairwise non-isogenous over $\overline{\mathbb{Q}}$. For an integer $t$ and a positive real number $x$, denote by $π_A(x, t)$ the number…
We obtain new results concerning the Sato--Tate conjecture on the distribution of Frobenius angles over parametric families of elliptic curves with a rational parameter of bounded height.
Let F be a finite field and let b and N be integers. We prove explicit estimates for the probability that the number of rational points on a randomly chosen elliptic curve E over F equals b modulo N. The underlying tool is an equidistribution result on the action of Frobenius on the N-torsion subgroup of E. Our results subsume and extend previous work by Achter and Gekeler.
A standard observation in algebraic geometry and number theory is that a ramified cover of an algebraic variety $\widetilde{X}\rightarrow X$ over a finite field $F_q$ furnishes the rational points $x\in X(F_q)$ with additional arithmetic structure: the Frobenius action on the fiber over $x$. For example, in the case of the Vieta cover of polynomials over $F_q$ this structure describes a polynomial's irreducible decomposition type. Furthermore, the distribution of these Frobenius actions is encoded in the cohomology of $\widetilde{X}$ via the Grothendieck-Lefschetz trace formula. This note pres…
We show that the distribution of elliptic curves in isogeny classes of curves with a given value of the Frobenius trace $t$ becomes close to uniform even when $t$ is averaged over very short intervals inside the Hasse-Weil interval.
Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Omega of the order. If the prime p splits completely in Omega, then we can reduce E modulo one the factors of p and get a curve Ep defined over GF(p). The trace of the Frobenius of Ep is known up to sign and we need a fast way to find this sign. For this, we propose to use the action of the Frobenius on torsion points of small order built with class invariants a la Weber, in a manner reminiscent of the Schoof-Elkies-Atkin al…
Suppose we have a elliptic curve over a number field whose mod $l$ representation has image isomorphic to $SL_2(\mathbb{F}_l)$. We present a method to determine Frobenius elements of the associated Galois group which incorporates the linear structure available. We are able to distinguish $SL_n(\mathbb{F}_l)$-conjugacy from $GL_n(\mathbb{F}_l)$-conjugacy; this can be thought of as being analogous to a result which distinguishes $A_n$-conjugacy from $S_n$-conjugacy when the Galois group is considered as a permutation group.
