L-functions Of Elliptic Curves Research Articles

On a Vernam cipher scheme based on elliptic curve L-functions

In this paper, we present a new Vernam-like cryptosystem based on the L-functions attached to elliptic curves. Actually, we show that for infinitely many elliptic curves, the coefficients of the L-series can be considered as a secure cryptographic cipher key when combined with linear and classical pseudo-random number generators. Hence, fast and widely known classical generators, like linear congruential generators, should be reintroduced into our list of cryptographic design choices.

The Investigation of the Discrete Universality of L-Functions of Elliptic Curves

In the paper, we prove the discrete universality theorem in the sense of the weak convergence of probability measures in the space of analytic functions for the L-functions of elliptic curves. We consider an approximation of analytic functions by translations LE (s+imh) , where h > 0 is a fixed number, the translations of the imaginary part of the complex variable take values from some discrete set such as arithmetical progression. We suppose that the number h > 0 is chosen so that exp{2πk/h } is an rational number for some non-zero integer. The proof of discrete universality of the derivatives of L-functions of elliptic curves is based on a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions.

On the vanishing of twisted L-functions of elliptic curves over rational function fields

We investigate in this paper the vanishing at $$s=1$$ of the twisted L-functions of elliptic curves E defined over the rational function field $${\mathbb {F}}_q(t)$$ (where $${\mathbb {F}}_q$$ is a finite field of q elements and characteristic $$\ge 5$$ ) for twists by Dirichlet characters of prime order $$\ell \ge 3$$ , from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt the techniques of Li (J Number Theory 191:85–103, 2018) and Donepudi and Li (Rocky Mountain J Math 51(5):1615–1628, 2021) who proved vanishing at $$s=1/2$$ for infinitely many Dirichlet L-functions over $${\mathbb {F}}_q(t)$$ based on the existence of one, and we can prove that if there is one $$\chi _0$$ such that $$L(E, \chi _0, 1)=0$$ , then there are infinitely many. Finally, we provide some examples which show that twisted L-functions of constant elliptic curves over $${\mathbb {F}}_q(t)$$ behave differently than the general ones.

Low-lying zeros in families of elliptic curve L-functions over function fields

We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over Fq(T). In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions.

Class field theory, its three main generalisations, and applications

Class Field Theory (CFT) is the main achievement of algebraic number theory of the 20th century. Its reach, beauty and power, stemming from the first steps in algebraic number theory by Gaus, have substantially influenced number theory. Shafarevich wrote: 'Weil was undoubtedly right when he asserted, in the preface to the Russian edition of his book on number theory 1 , that since class field theory pertains to the foundation of mathematics, every mathematician should be as familiar with it as with Galois theory. Moreover, just like Galois theory before it, class field theory was reputed to be very complicated and accessible only to experts ... For class field theory, on the other hand, there is a wide range of essentially different expositions, so that it is not immediately obvious even whether the subject is the same'. 2 Weil's opinion has proved to be quixotic: these days even some number theorists are not familiar with the substance of CFT. This text reviews the enduring process of discovering new branches of CFT and its generalisations. Many of such developments were complicated at their early stages and some were difficult or impossible to understand for their contemporaries. Three main generalisations of CFT and their further extensions will be presented and some of their key fundamental features will be discussed. This text proposes eight fundamental problems. We start with Kummer theory, a purely algebraic exercise, whose highly non-trivial arithmetic analogues over arithmetic fields are supplied by CFT. Kummer theory is an algebraic predecessor of CFT including its existence theorem. Then we discuss the fundamental split of (one-dimensional) CFT into special CFT (SCFT) and general CFT (GCFT). This split has enormously affected many developments in number theory. Section 3 delves into four fundamental parts of CFT including the reciprocity map, existence theorem, explicit formulas for the Hilbert symbol and its generalisations, and interaction with ramification theory. Section 4 briefly touches on higher Kummer theory using Milnor K-groups, i.e. the norm residue isomorphism property. Three generalisations of CFT: Langlands correspondences (LC), higher CFT, and anabelian geometry are discussed in section 5. We note that the split of CFT into SCFT and GCFT is currently somehow reproduced at the level of generalisations of CFT: LC over number fields does not yet have any development parallel to GCFT, while higher CFT is parallel to GCFT and it does not have substantial developments similar to SCFT. In the last section we specialise to elliptic curves over global fields, as an illustration. There we consider two further developments: Mochizuki's inter-universal Teichmuller theory (IUT) which is pivoted on anabelian geometry and two-dimensional adelic analysis and geometry which uses structures of two-dimensional CFT. We also consider the fundamental role of zeta integrals which may unite different generalisations of CFT. Similarly to the situation with LC, the current studies of special values of zeta-and L-functions of elliptic curves over number fields, except two-dimensional adelic analysis and geometry, use special structures and are not of general type. There is no attempt to mention all the main results in CFT and all of its generalisations or all of their parts, and the text does not include all of bibliographical references. 1 [64] 2 in Foreword to [13].

Conjectures for Moments Associated With Cubic Twists of Elliptic Curves

We extend the heuristic introduced by Conrey, Farmer, Keating, Rubinstein, and Snaith in order to formulate conjectures for the -moments of L-functions of elliptic curves twisted by cubic characters. We also apply the work of Keating and Snaith on the -moments of characteristic polynomials of unitary matrices to extend our conjecture to such that and . Our conjectures are then numerically tested for two families. A novelty of our conjectures is that cubic twists naturally lead us to consider the possibility .

A short note on inadmissible coefficients of weight 2 and 2k+1 newforms

Let f(z)=q+sum _{nge 2}a(n)q^n be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for k=2 by ruling out or locating all odd prime values |ell |<100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights kge 1 newforms where the nebentypus is given by a quadratic Dirichlet character.

Mahler Measure of Families of Polynomials Defining Genus 2 and 3 Curves

In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of L-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these families and special values of L-functions of elliptic curves. We also find identities between the Mahler measures of these nonreciprocal families and tempered polynomials defining genus 1 curves. We will explain these relations by considering the pushforward and pullback of certain elements in K 2 of curves defined by these polynomials and applying Beilinson’s conjecture on K 2 of curves. We show that there are two and three explicit linearly independent elements in K 2 of certain families of genus 2 and genus 3 curves, respectively.

On L-functions of modular elliptic curves and certain K3 surfaces

Inspired by Lehmer’s conjecture on the non-vanishing of the Ramanujan tau -function, one may ask whether an odd integer alpha can be equal to tau (n) or any coefficient of a newform f(z). Balakrishnan, Craig, Ono and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight kge 4. We use these methods for weight 2 and 3 newforms and apply our results to L-functions of modular elliptic curves and certain K3 surfaces with Picard number ge 19. In particular, for the complete list of weight 3 newforms f_lambda (z)=sum a_lambda (n)q^n that are eta -products, and for N_lambda the conductor of some elliptic curve E_lambda , we show that if |a_lambda (n)|<100 is odd with n>1 and (n,2N_lambda )=1, then aλ(n)∈{-5,9,±11,25,±41,±43,-45,±47,49,±53,55,±59,±61,±67,-69,±71,±73,75,±79,±81,±83,±89,±93±97,99}.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} a_\\lambda (n) \\in&\\{-5,9,\\pm 11,25, \\pm 41, \\pm 43, -45,\\pm 47,49, \\pm 53,55, \\pm 59, \\pm 61,\\\\&\\pm 67, -69,\\pm 71,\\pm 73,75, \\pm 79,\\pm 81, \\pm 83, \\pm 89,\\pm 93 \\pm 97, 99\\}. \\end{aligned}$$\\end{document}Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving aλ(n)∈{-5,9,±11,25,-45,49,55,-69,75,±81,±93,99}.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} a_\\lambda (n) \\in \\{-5,9,\\pm 11,25,-45,49,55,-69,75,\\pm 81,\\pm 93, 99\\}. \\end{aligned}$$\\end{document}

Elipsinių kreivių L funkcijų išvestinės diskretusis universalumas

Straipsnyje įrodyta elipsinių kreivių L funkcijų išvestinės diskrečiojo universalumo teorema silpnojo tikimybinių matų konvergavimo prasme analizinių funkcijų erdvėje. Nagrinėjamas analizinės funkcijos aproksimavimas postūmiais L‘E (s + imh), čia kompleksinio kintamojo menamosios dalies postūmiai įgyja reikšmes iš diskrečiosios aibės, pavyzdžiui, aritmetinės progresijos. Fiksuotas skaičius h &gt; 0 pasirenkamas taip, kad exp{2πk/h} būtų iracionalusis skaičius visiems k ∈ Z \{0} . Elipsinių kreivių L funkcijų išvestinės diskrečiojo universalumo įrodymas remiasi diskrečiąja ribine teorema tikimybinių matų silpnojo konvergavimo prasme analizinių funkcijų erdvėje.

On a BSD-type formula for L-values of Artin twists of elliptic curves

Abstract This is an investigation into the possible existence and consequences of a Birch–Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted by Artin representations. We translate expected properties of L-functions into purely arithmetic predictions for elliptic curves, and show that these force some peculiar properties of the Tate–Shafarevich group, which do not appear to be tractable by traditional Selmer group techniques. In particular, we exhibit settings where the different p-primary components of the Tate–Shafarevich group do not behave independently of one another. We also give examples of “arithmetically identical” settings for elliptic curves twisted by Artin representations, where the associated L-values can nonetheless differ, in contrast to the classical Birch–Swinnerton-Dyer conjecture.

EFFECTIVE l 2 DECOUPLING FOR THE PARABOLA

We make effective l 2 L p decoupling for the parabola in the range 4 < p < 6 . In an appendix joint with Jean Bourgain, we apply the main theorem to prove the conjectural bound for the sixth-order correlation of the integer solutions of the equation x 2 + y 2 = m in an extremal case. This proves unconditionally a result that was proven in Bombieri and Bourgain [Int. Math. Res. Not. IMRN 2015(11), 3343–3407] under the hypotheses of the Birch and Swinnerton–Dyer conjecture and the Riemann Hypothesis for L-functions of elliptic curves over Q.

Elliptic dilogarithms and parallel lines

We prove Boyd's conjectures relating Mahler's measures and values of L-functions of elliptic curves in the cases when the corresponding elliptic curve has conductor 14.

A Weighted Universality Theorem for L-Functions of Elliptic Curves

In the paper, a short survey on universality results for L-functions of elliptic curves over the field of rational numbers is given and weighted universality theorem is proven. All stated universality theorems are of continuous type. The proof of the universality for L-functions of elliptic curves is based on a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions.

A Discrete Limit Theorem for L-Functions of Elliptic Curves

In the paper, we prove the discrete limit theorem in the sense of the weak convergence of probability measures in the space of analytic on DV = {s ∈ C : 1 &lt; σ &lt; 3/2, |t| &lt; V} functions for L-functions of elliptic curves LE(s). The main statement of the paper is as follows. Let h &gt; 0 be a fixed real number such that exp {2πk/h} is an irrational number for all k∈Z\{0}. Then the probability measure μN(LE(s + imh)∈A), A ∈ B(H(DV)), converges weakly to the measure PLE as N→∞.

Bounds on special values of L-functions of elliptic curves in an Artin\u2013Schreier family

We study a certain Artin–Schreier family of elliptic curves over the function field . We prove an asymptotic estimate on the special values of their L-function in terms of the degree of their conductor; we show that the special values are, in a sense, ‘asymptotically as large as possible’. We also provide an explicit expression for their L-function. The proof of the main result uses this expression and a detailed study of the distribution of character sums related to Kloosterman sums. Via the BSD conjecture, the main result translates into an analogue of the Brauer–Siegel theorem for these elliptic curves.

Exceptional zero formulae for anticyclotomic p-adic L-functions of elliptic curves in the ramified case

Iwasawa theory of modular forms over anticyclotomic Zp-extensions of imaginary quadratic fields K has been studied by several authors, starting from the works of Bertolini–Darmon and Iovita–Spiess, under the crucial assumption that the prime p is unramified in K. We start in this article the systematic study of anticyclotomic p-adic L-functions when p is ramified in K. In particular, when f is a weight 2 modular form attached to an elliptic curve E/Q having multiplicative reduction at p, and p is ramified in K, we show an analogue of the exceptional zeroes phenomenon investigated by Bertolini–Darmon in the setting when p is inert in K. More precisely, we consider situations in which the p-adic L-function Lp(E/K) of E over the anticyclotomic Zp-extension of K does not vanish identically but, by sign reasons, has a zero at certain characters χ of the Hilbert class field of K. In this case we show that the value at χ of the first derivative of Lp(E/K) is equal to the formal group logarithm of the specialization at p of a global point on the elliptic curve (actually, this global point is a twisted sum of Heegner points). This generalizes similar results of Bertolini–Darmon, available when p is inert in K and χ is the trivial character.

Arithmetic properties of coefficients of L-functions of elliptic curves

Let $$\sum _{n\geqslant 1} a_n n^{-s}$$ be the L-series of an elliptic curve E defined over the rationals without complex multiplication. In this paper, we present certain similarities between the arithmetic properties of the coefficients $$\{a_n\}_{n=1}^\infty $$ and Euler’s totient function $$\varphi (n)$$ . Furthermore, we prove that both the set of n such that the regular polygon with $$|a_n|$$ sides is ruler-and-compass constructible, and the set of n such that $$n-a_n+1=\varphi (n)$$ have asymptotic density zero. Finally, we improve a bound of Luca and Shparlinski on the counting function of elliptic pseudoprimes.

Arithmetic statistics of modular symbols

Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.

On the Generalizations of Universality Theorem for L-Functions of Elliptic Curves

In the paper, the continuous type’s universality theorem for L-functions of elliptic curves is discussed and its generalizations in three directions – for positive integer powers and derivatives of L-functions of elliptic curves as well as the weighted universality theorem of L-functions of elliptic curves – are given. The proofs of the universality are based on limit theorems in the sense of weak convergence of probability measures in functional spaces.