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Let
$E/\mathbb {Q}$
be an elliptic curve. For a prime p of good reduction, let
$r(E,p)$
be the smallest non-negative integer that gives the x-coordinate of a point of maximal order in the group
$E(\mathbb {F}_p)$
. We prove unconditionally that
$r(E,p)> 0.72\log \log p$
for infinitely many p, and
$r(E,p)> 0.36 \log p$
under the assumption of the Generalized Riemann Hypothesis. These can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime.
We correct the proof of the main $\ell$-independence result of the above-mentioned paper by showing that for any smooth and proper variety over an equicharacteristic local field, there exists a globally defined such variety with the same ($p$-adic and $\ell$-adic) cohomology.
Recently E. Bombieri and N. M. Katz (2010) 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.
Given an elliptic curve E in Legendre form $y^2 = x(x - 1)(x - \lambda )$ over the fraction field of a Henselian ring R of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of E over R that depends only on the valuation of $\lambda $. We provide several examples along with an easy corollary concerning $2$-torsion.
Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in terms of $g$ and the size of $S$. In the second part we assume that $L$ is the function field of a hyperelliptic curve $C_{A}:s^{2}=A(t)$, where $A(t)$ is a square-free $k$-polynomial of odd degree. If $\infty$ is the place of $L$ associated to the point at infinity of $C_{A}$, then we prove that the set of separable $\{\infty \}$-points can be bounded solely in terms of $g$ and does not depend on the Mordell–Weil group $E(L)$. This is done by bounding the number of separable integral points over $k(t)$ on elliptic curves of the form $E_{A}:A(t)y^{2}=f(x)$, where $f(x)$ is a polynomial over $k$. Additionally, we show that, under an extra condition on $A(t)$, the existence of a separable integral point of ‘small’ height on the elliptic curve $E_{A}/k(t)$ determines the isomorphism class of the elliptic curve $y^{2}=f(x)$.
We prove an analogue of Belyi’s theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called pseudo-tameness for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a ‘pseudo-tame’ rational function by showing the vanishing of an obstruction class. Finally, we construct a tamely ramified rational function from the ‘pseudo-tame’ rational function.
We generalize the Cohen–Lenstra heuristics over function fields to étale group schemes $G$ (with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg–Venkatesh–Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of $\wedge ^{2}G(1)$, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen–Lenstra heuristics for abelian $\ell$-groups in cases where $\ell \mid q-1$; the nature of this failure was suggested already in the works of Malle, Garton, Ellenberg–Venkatesh–Westerland, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.
We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_{q}$ with $d\leqslant q+1$, then there is an $\mathbb{F}_{q}$-line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ when $q=p^{r}$.
In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let $\ell$ be a prime, $q$ a prime power and consider the ensemble ${\mathcal{H}}_{g,\ell }$ of $\ell$-cyclic covers of $\mathbb{P}_{\mathbb{F}_{q}}^{1}$ of genus $g$. We assume that $q\not \equiv 0,1~\text{mod}~\ell$. If $2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$, then ${\mathcal{H}}_{g,\ell }$ is empty. Otherwise, the number of rational points on a random curve in ${\mathcal{H}}_{g,\ell }$ distributes as $\sum _{i=1}^{q+1}X_{i}$ as $g\rightarrow \infty$, where $X_{1},\ldots ,X_{q+1}$ are independent and identically distributed random variables taking the values $0$ and $\ell$ with probabilities $(\ell -1)/\ell$ and $1/\ell$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$th root of unity, the presence of which was crucial in previous studies.
In this series of papers, we explore moments of derivatives of L-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials P in 𝔽q[T]. When the cardinality q of the ground field is fixed and the degree of P gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of L-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.
In this paper we investigate the moments and the distribution of $L(1,\unicode[STIX]{x1D712}_{D})$, where $\unicode[STIX]{x1D712}_{D}$ varies over quadratic characters associated to square-free polynomials $D$ of degree $n$ over $\mathbb{F}_{q}$, as $n\rightarrow \infty$. Our first result gives asymptotic formulas for the complex moments of $L(1,\unicode[STIX]{x1D712}_{D})$ in a large uniform range. Previously, only the first moment has been computed due to the work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of $L(1,\unicode[STIX]{x1D712}_{D})$ is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of $L(1,\unicode[STIX]{x1D712}_{D})$, which is not present in the number field setting. We also obtain $\unicode[STIX]{x1D6FA}$-results for the extreme values of $L(1,\unicode[STIX]{x1D712}_{D})$, which we conjecture to be the best possible. Specializing $n=2g+1$ and making use of one case of Artin’s class number formula, we obtain similar results for the class number $h_{D}$ associated to $\mathbb{F}_{q}(T)[\sqrt{D}]$. Similarly, specializing to $n=2g+2$ we can appeal to the second case of Artin’s class number formula and deduce analogous results for $h_{D}R_{D}$, where $R_{D}$ is the regulator of $\mathbb{F}_{q}(T)[\sqrt{D}]$.
In this paper, we prove some conjectures of K. Stolarsky concerning the first and third moments of the Beatty sequences with the golden section and its square.
We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an $L$-function away from its zeros and that, when the length of the product tends to infinity, we recover the original $L$-function. We also obtain explicit expressions for the arguments of quadratic Dirichlet $L$-functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet $L$-function over a function field, an auxiliary function based on the approximate functional equation that equals the $L$-function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions and prove that the Riemann hypothesis holds for them and that their zeros are related to those of the associated $L$-function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated $L$-function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple.
In this article we study various forms of $\ell$-independence (including the case $\ell =p$) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of $\ell$-independence for the unipotent fundamental group of smooth and projective varieties over finite fields. By then proving a certain ‘spreading out’ result we are able to deduce a much weaker form of $\ell$-independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduce $\ell$-independence results for the cohomology of smooth and proper varieties over equicharacteristic local fields from the well-known results on $\ell$-independence for smooth and proper varieties over finite fields. As another consequence of this ‘spreading out’ result we are able to deduce the existence of a Clemens–Schmid exact sequence for formal semistable families. Finally, by deforming to characteristic $p$, we show a similar weak version of $\ell$-independence for the unipotent fundamental group of a semistable curve in mixed characteristic.
Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-$p$ extensions of imaginary quadratic number fields for $p$ an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of $\mathbb{F}_{q}(t)$, the Galois groups of the maximal unramified pro-$p$ extensions, as $q\rightarrow \infty$, have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.
For a finite field of odd cardinality $q$, we show that the sequence of iterates of $aX^{2}+c$, starting at $0$, always recurs after $O(q/\text{log}\log q)$ steps. For $X^{2}+1$, the same is true for any starting value. We suggest that the traditional “birthday paradox” model is inappropriate for iterates of $X^{3}+c$, when $q$ is 2 mod 3.
Deninger et Werner ont développé un analogue pour les courbes $p$-adiques de la correspondance classique de Narasimhan et Seshadri entre les fibrés vectoriels stables de degré $0$ et les représentations unitaires du groupe fondamental topologique pour une courbe complexe propre et lisse. Par transport parallèle, ils ont associé fonctoriellement à chaque fibré vectoriel sur une courbe $p$-adique, dont la réduction est fortement semi-stable de degré $0$, une représentation $p$-adique du groupe fondamental de la courbe. Ils se sont posé quelques questions : leur foncteur est-il pleinement fidèle ? La cohomologie des systèmes locaux fournis par celui-ci admet-elle une filtration de Hodge-Tate ? Leur construction est-elle compatible avec la correspondance de Simpson $p$-adique développée par Faltings ? Nous répondons à ces questions dans cet article.
We construct a stable formal model of a Lubin–Tate curve with level three, and study the action of a Weil group and a division algebra on its stable reduction. Further, we study a structure of cohomology of the Lubin–Tate curve. Our study is purely local and includes the case where the characteristic of the residue field of a local field is two.
Consider two ordinary elliptic curves $E,E^{\prime }$ defined over a finite field $\mathbb{F}_{q}$, and suppose that there exists an isogeny $\unicode[STIX]{x1D713}$ between $E$ and $E^{\prime }$. We propose an algorithm that determines $\unicode[STIX]{x1D713}$ from the knowledge of $E$, $E^{\prime }$ and of its degree $r$, by using the structure of the $\ell$-torsion of the curves (where $\ell$ is a prime different from the characteristic $p$ of the base field). Our approach is inspired by a previous algorithm due to Couveignes, which involved computations using the $p$-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of $\tilde{O} (r^{2})p^{O(1)}$ base field operations. On the other hand, the cost of our algorithm is $\tilde{O} (r^{2})\log (q)^{O(1)}$, for a large class of inputs; this makes it an interesting alternative for the medium- and large-characteristic cases.
We present a specialized point-counting algorithm for a class of elliptic curves over $\mathbb{F}_{p^{2}}$ that includes reductions of quadratic $\mathbb{Q}$-curves modulo inert primes and, more generally, any elliptic curve over $\mathbb{F}_{p^{2}}$ with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof–Elkies–Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.