Let $C$ be a smooth projective irreducible curve defined over a finite field $\mathbb{F}_q$ and $K=\mathbb{F}_q(C)$. Let $A\subset K$ be the ring of functions regular outside a fixed place $\infty$ of $K$. Let $ϕ:A\to\text{End}(\mathbb{G}_a)$ be a Drinfeld $A$-module of rank $r$ defined over a finite extension $L$ of $K$ and $\hat{h}_ϕ$ its canonical height. Given a non-torsion point $α$ of $ϕ$ of degree $d$ over $K$, we prove that $\hat{h}_ϕ(α)\ge 1/d$.
A similar statement is proved for the canonical height of a point of infinite order of a non-constant semi-stable elliptic curve defined over $K$, with the absolute constant 1 replaced by a constant depending on the elliptic curve.
Let $R=\mathcal{O}_{\Q(\sqrt{d})}$ for $d<0$, squarefree, $d\neq -1,-3$. We prove Lehmer's conjecture for associated reciprocal polynomials of $R$-matrices; that is, any noncyclotomic $R$-matrix has Mahler measure at least $λ_0=1.176...$.
We solve Lehmer's problem for a class of polynomials arising from Hermitian matrices over the Eisenstein and Gaussian integers, that is, we show that all such polynomials have Mahler measure at least Lehmer's number τ_0 = 1.17628... .
In this paper we prove Lehmer's conjecture on Ramanujan's tau function, namely tau(n) not equal to zero for n >= 1 by investigating the additive group structure attached to tau(n) with the aid of the pigeonhole principle and unique factorization theorem.
In this paper, we obtain analogues of Zsigmondy's theorem and the primitive divisor results for the Lucas and Lehmer sequences in polynomial rings of several variables.
In this paper we attempt to prove Lehmer's conjecture on Ramanujan's tau function, namely tau(n) is never zero, for each n larger than zero by investigating the additive group structure attached to tau(n) with the aid of unique factorization theorem.
Over one year ago, a very long preprint posted on arXiv [arXiv:1709.03771] and HAL announced a proof of Lehmer's Conjecture (and of other related results). Unfortunately, as was remarked by several specialists, this proof contains a (at least one) fatal error. The aim of this very short note is to inform the mathematical community, which could be aware of this.
We generalize a result of Sury and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis) is equivalent to a weak form of Lehmer's conjecture. We include a short survey of related results and conjectures.
A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of the form 1+X+...+X^n for some n > ε*deg(f). We also formulate and prove an analogous statement for elliptic curves.
We state and prove an equivariant version of Lehmer's conjecture on heights, generalizing papers by Zagier (1993) and Dresden (1998) which are special cases of this theorem. We also extend their three cases to a full classification of all groups satisfying the condition that the set of all orbits for which every non-zero element lies on the unit circle is finite and non-empty.
We introduce vector space norms associated to the Mahler measure by using the L^p norm versions of the Weil height recently introduced by Allcock and Vaaler. In order to do this, we determine orthogonal decompositions of the space of algebraic numbers modulo torsion by Galois field and degree. We formulate L^p Lehmer conjectures involving lower bounds on these norms and prove that these new conjectures are equivalent to their classical counterparts, specifically, the classical Lehmer conjecture in the p = 1 case and the Schinzel-Zassenhaus conjecture in the p = infinity case.
We draw connections between the various conjectures which are included in G. Rémond's generalized Lehmer problems. Specifically, we show that the degree one form of his conjecture for the multiplicative group is, in a sense, almost as strong as the strong form. Our insights into the conjectures apply a basic result on what may be called Diophantine approximation in the metric induced by the canonical height. We also present a previously unpublished partial result of Rémond on his conjectures.
In 1947, Lehmer conjectured that the Ramanujan τ-function τ(m) is non-vanishing for all positive integers m, where τ(m) are the Fourier coefficients of the cusp form Δof weight 12. It is known that Lehmer's conjecture can be reformulated in terms of spherical t-design, by the result of Venkov. In this paper, we show that τ(m) = 0 is equivalent to the fact that the homogeneous space of the moonshine vertex operator algebra (V^\natural)_{m+1} is a conformal 12-design. Therefore, Lehmer's conjecture is now reformulated in terms of conformal t-designs.
We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for $n>1$ we prove that $$τ(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 691\}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujan's well-known congruences for $τ(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about ce…
Let $f$ be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer $n$ has a factor common with the $n$-th Fourier coefficient of $f$. The second author \cite{kumar3} showed that this happens very often. In this paper, we give an asymptotic formula for the number of integers $n$ for which $(n, a(n))=1$, where $a(n)$ is the $n$-th Fourier coefficient of a normalized Hecke eigenform $f$ of weight $2$ with rational integer Fourier coefficients and has complex multiplication.
In 1947, Lehmer conjectured that the Ramanujan $τ$-function $τ(m)$ never vanishes for all positive integers $m$, where the $τ(m)$ are the Fourier coefficients of the cusp form $Δ_{24}$ of weight 12. Lehmer verified the conjecture in 1947 for $m<214928639999$. In 1973, Serre verified up to $m<10^{15}$, and in 1999, Jordan and Kelly for $m<22689242781695999$. The theory of spherical $t$-design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's $τ$-function gives the c…
The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{house(α)}(z)$ associated with the dynamical zeta functions $ζ_{house(α)}(z)$ of the Rényi--Parry arithmetical dynamical systems, for $α$ an algebraic integer $α$ of house "$house(α)$" greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{house(α)}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $house(α)$ tends to $1^+$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(house(α))$ of the Mahler…
Motivated by a geometric meaning of Mahler's measure, we introduce two operator analogues of Mahler's measure. This leads to some interesting equalities and inequalities between the two operator-theoretic Mahler measures and the classical Mahler measure. In order to apply these results to the operator version of Lehmer's problem, we introduce and study an important class of operators, the so-called subharmonic operators. It is shown that the operator version of Lehmer's problem fails under some mild condition.
In 1947, Lehmer conjectured that the Ramanujan's tau function $τ(m)$ never vanishes for all positive integers $m$, where $τ(m)$ is the $m$-th Fourier coefficient of the cusp form $Δ_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $τ(m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However,…
We formulate Lehmer's Problem about the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant, and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components then its Lehmer constant vanishes.
Let u(n) be the n-th term of a Lucas sequence or a Lehmer sequence.In this article we shall establish an estimate from below for the greatest prime factor of u(n) which is of the form nexp(logn/104loglogn). In so doing we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdos from 1965.In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.
In the previous paper, we studied the "Toy models for D. H. Lehmer's conjecture". Namely, we showed that the m-th Fourier coefficient of the weighted theta series of the $\mathbb{Z}^2$-lattice and the $A_{2}$-lattice does not vanish, when the shell of norm $m$ of those lattices is not the empty set. In other words, the spherical 4 (resp. 6)-design does not exist among the nonempty shells in the $\mathbb{Z}^2$-lattice (resp. $A_{2}$-lattice). This paper is the sequel to the previous paper. We take 2-dimensional lattices associated to the algebraic integers of imaginary quadratic fields whose cl…
In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in\mathbb{F}_q[x]$ and $φ(q,p(x))$ be the Euler's totient function of $p(x)$ over $\mathbb{F}_q[x],$ where $\mathbb{F}_q$ is a finite field with $q$ elements. We prove that $φ(q,p(x))|(q^{{\rm deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3, \; p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or (iii) $q=2,\; p(x)$ is the product of all irreducibles of degree $1,$ all irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducibles one each…
The Conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{\houseα}(z)$ associated with the dynamical zeta functions $ζ_{\houseα}(z)$ of the Rényi--Parry arithmetical dynamical systems ($β$-shift), for $α$ a reciprocal algebraic integer of house $\houseα$ greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{\houseα}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $\houseα$ tends to $1^+$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(\houseα)$ of …
Let $f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (Γ_0(N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$, define $θ_p\in[0,π]$ to be the angle for which $a(p)=2p^{( k -1)/2}\cos θ_p $. Let $I\subset[0,π]$ be a closed subinterval, and let $dμ_{ST}=\frac{2}π\sin^2θdθ$ be the Sato-Tate measure of $I$. Assuming that the symmetric power $L$-functions of $f$ satisfy certain analytic properties (all of which follow from Langlands functoriality and the Generalized Riemann Hypothesis), we prove that if $x$ is sufficiently large, then \[ \left|…
The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied to deduce the mixed characteristic case of a local analogue of the Grothendieck Conjecture for these fields from its characteristic p case, which was proved earlier by the author.
We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a finite field which have a rational point of order l.
We formulate and study a torsion analogue of the weight-monodromy conjecture for a proper smooth scheme over a non-archimedean local field. We prove it for proper smooth schemes over equal characteristic non-archimedean local fields, abelian varieties, surfaces, varieties uniformized by Drinfeld upper half spaces, and set-theoretic complete intersections in toric varieties. In the equal characteristic case, our methods rely on an ultraproduct variant of Weil II established by Cadoret.
Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the p-adic L-function of a modular abelian variety and showing that it satisfies the appropriate interpolation property. This relies on a careful normalization of the p-adic L-function, which we achieve by a comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, Teitelbaum in dimension 1 and the …
We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of characteristic $p$. We prove the prime-to-$p$ part conditionally on the finiteness of the $p$-primary part of the Tate-Shafarevich group or the equality of the analytic and the algebraic rank. If the base is a product of curves, Abelian varieties and K3 surfaces, we prove the prime-to-$p$ part of the conjecture for constant or isoconstant Abelian schemes, in particular the prime-to-$p$ part for (1) relative elliptic curve…
Let L be a number field and let E be any subgroup of the units O_L^* of L. If rank(E) = 1, Lehmer's conjecture predicts that the height of any non-torsion element of E is bounded below by an absolute positive constant. If rank(E) = rank(O_L^*), Zimmert proved a lower bound on the regulator of E which grows exponentially with [L:Q]. Fernando Rodriguez Villegas made a conjecture in 2002 that "interpolates" between these two extremes of rank. Here we prove a high-rank case of this conjecture. Namely, it holds if L contains a subfield K for which [L:K] >> [K:Q] and E contains the kernel of the nor…
In this paper we show that the map % $$\partial:CH^2(E_1 \times E_2,1)\otimes \Q \longrightarrow PCH^1(\XX_v)$$ % is surjective, where $E_1$ and $E_2$ are two non-isogenous semistable elliptic curves over a local field, $CH^2(E_1 \times E_2,1)$ is one of Bloch's higher Chow groups and $PCH^1(\XX_v)$ is a certain subquotient of a Chow group of the special fibre $\XX_{v}$ of a semi-stable model $\XX$ of $E_1 \times E_2$. On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-$\D$-conjecture of Beilinson - which is known to be true in this case by the work of Chen and Lewis \c…
Lehmer's 1947 conjecture on whether $τ(n)$ vanishes is still unresolved. In this context, it is natural to consider variants of Lehmer's conjecture. We determine many integers that cannot be values of $τ(n)$. For example, among the odd numbers $α$ such that $|α|<99$, we determine that $$τ(n) \notin \{-9, \pm 15, \pm 21, -25, -27, -33, \pm 35, \pm 45, \pm 49, -55, \pm 63, \pm 77, -81, \pm 91 \}.$$ Moreover, under GRH, we have that $τ(n) \neq -|α|$ and that $τ(n) \notin \{9,25,27,39,$ $75,81\}.$ We also consider the level 1 Hecke eigenforms in dimension 1 spaces of cusp forms. For example, for $…
We consider an Archimedean analogue of Tate's conjecture, and verify the conjecture in the examples of isospectral Riemann surfaces constructed by Vigneras and Sunada. We also enunciate a simple lemma in group theory which lies at the heart of T. Sunada's theorem about isospectral manifolds.
In [Ok] Okounkov studies a specific $q$-analogue of multiple zeta values and makes some conjectures on their algebraic structure. In this note we compare Okounkovs $q$-analogues to the generating function for multiple divisor sums defined in [BK1]. We also state a conjecture on their dimensions that complements Okounkovs conjectural formula and present some numerical evidences for it.
We study a class of q-analogues of multiple zeta values given by certain formal q-series with rational coefficients. After introducing a notion of weight and depth for these q-analogues of multiple zeta values we present dimension conjectures for the spaces of their weight- and depth-graded parts, which have a similar shape as the conjectures of Zagier and Broadhurst-Kreimer for multiple zeta values.
In this article we study an abelian analogue of Schanuel's conjecture. This conjecture falls in the realm of the generalised period conjecture of Y. Andr{é}. As shown by C. Bertolin, the generalised period conjecture includes Schanuel's conjecture as a special case. Extending methods of Bertolin, it can be shown that the abelian analogue of Schanuel's conjecture we consider, also follows from Andr{é}'s conjecture. C. Cheng et al. showed that the classical Schanuel's conjecture implies the algebraic independence of the values of the iterated exponential function and the values of the iterated l…
In this paper we formulate two generalizations of Agoh's conjecture. We also formulate conjectures involving congruence modulo primes about hyperbolic secant, hyperbolic tangent, Nörlund numbers, as well as about coefficients of expansions in powers of other analytic functions. We formulate a thesis about combinatorial objects that do not produce fake primes.
The study of Property (B) starts as a special case of Lehmer's conjecture. An algebraic field is said to satisfy Property (B) if there exists a positive constant bounding by below the height of every point of infinite order. In this paper we prove that, under certain uniformity conditions, the compositum of a familly of fields satisfies Property (B).
We examine the polynomial analogues of McMullen's and Zaremba's conjectures on continued fractions with bounded partial quotients. It has already been proved by Blackburn that if the base field is infinite, then the polynomial analogue of Zaremba's conjecture holds; we will prove this again with a different method and examine some known results for finite base fields. Translating to the polynomial setting a result of Mercat, we will prove that the polynomial analogue of McMullen's conjecture holds over infinite algebraic extensions of finite fields and that, over finite fields, it would be a c…
In previous papers we formulated an analogue of the Ichino--Ikeda conjectures for Whittaker--Fourier coefficients of automorphic forms on classical group and the metaplectic group. In the latter case we reduced the conjecture to a local identity. In this paper we will prove the local identity in the $p$-adic case, and hence the global conjecture under simplifying conditions at the archimedean places.
The conjecture of Serre referred in the title is the one about modularity of odd Galois representations into GL(2,F) where F is a finite field of characteristic p. We present an analogous conjecture where GL(2) is replaced by GL(n). We explain the analogue of "oddness." We then present some theoretical and experimental evidence for the conjecture, primarily when n = 3. Our conjecture does not require the Galois representation to be irreducible. Our most interesting examples involve the sum of an irreducible even 2-dimensional representation and a 1-dimensional representation.
In this article we study Semi-abelian analogues of Schanuel conjecture. As showed by the first author, Schanuel Conjecture is equivalent to the Generalized Period Conjecture applied to 1-motives without abelian part. Extending her methods, the second, the third and the fourth authors have introduced the Abelian analogue of Schanuel Conjecture as the Generalized Period Conjecture applied to 1-motives without toric part. As a first result of this paper, we define the Semi-abelian analogue of Schanuel Conjecture as the Generalized Period Conjecture applied to 1-motives. C. Cheng et al. proved tha…
In 1971, Ruzsa conjectured that if $f:\ \mathbb{N}\rightarrow\mathbb{Z}$ with $f(n+k)\equiv f(n)$ mod $k$ for every $n,k\in\mathbb{N}$ and $f(n)=O(θ^n)$ with $θ<e$ then $f$ is a polynomial. In this paper, we investigate the analogous problem for the ring of polynomials over a finite field.
The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews' famous identity for smallest parts function. In the same paper, they also conjectured an inequality between finite analogues of rank and crank moments, analogous to Garvan's conjecture. In the present paper, we give a proof of this conjecture.
Let $L/K$ be a finite Galois extension of complete discrete valued fields of characteristic $p$. Assume that the induced residue field extension $k_L/k_K$ is separable. For an integer $n\geq 0$, let $W_n(\sO_L)$ denote the ring of Witt vectors of length $n$ with coefficients in $\sO_L$. We show that the proabelian group ${H^1(G,W_n(\sO_L))}_{n\in \N}$ is zero. This is an equicharacteristic analogue of Hesselholt's conjecture.
We study the number of ways to decompose a monic polynomial in F_q[t] of degree n as a sum of two monic irreducible polynomials in F_q[t]. Our principal result is an asymptotic formula for the number of such representations in the case when q is large compared to n. In its range of validity, this formula agrees with what is suggested by heuristic arguments from the rational setting. We also present similar results towards an analogue of the twin prime conjecture.
We prove a conjecture of Northshield by determining the maximal order of his analogue of Stern's sequence for $\mathbb{Z}[\sqrt{2}]$. In particular, if $b$ is Northshield's analogue, we prove that $$\limsup_{n\to\infty}\frac{2b(n)}{(2n)^{\log_3 (\sqrt{2}+1)}}=1.$$
