Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(\bar{K})$. More precisely, we prove that there are constants $C_1,C_2>0$ such that the normalized Bernoulli-part of the canonical height is bounded below by $$ \hat{h}_A^{\mathbb{B}}(P) \ge C_1\bigl[K(P):K\bigr]^{-2} $$ for all points $P\in{A(\bar{K})}$ whose height satisfies $0<\hat{h}_A(P)\le{C_2}$.
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.
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.
Let $A/K$ be an abelian variety over a number field $K$. We prove in this article that a good lower bound (in terms of the degree $[K(P):K]$) for the Néron-Tate height of the points $P$ of infinite order modulo every strict abelian subvarieties of $A$ implies a good lower bound for the height of all the non-torsion points of $A$. In particular when $A$ is of C.M. type, a theorem of David and Hindry enables us to deduce, up to ``log'' factors, an optimal lower bound for the height of the non-torsion points of $A$. In the C.M. type case, this improves the previous result of Masser \cite{lettre}.…
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…
We propose a lower bound estimate in Dobrowolski's style of the canonical height on a certain family of Drinfeld modules of characteristic 0, including under some hypothesis on their degree and their base field, the complex multiplication case, extending so a previous result of L. Denis on Carlitz modules. Our study is focused on the highest possible level of precision on the parameters involved with rapport to the main values which characterize the Drinfeld module (height, base field degree and rank) and it provides an estimate in function of both separable and inseparable degree of the algeb…
We obtain a lower bound for the normalised height of a non-torsion hypersurface $V$ of a C.M. abelian variety $A$ which is a refinement of a precedent result. This lower bound is optimal in terms of the geometric degree of $V$, up to an absolute power of a ``log'' (independant of the dimension of $A$). We thus extend the results of F. Amoroso and S. David on the same problem on a multiplicative group $\mathbb{G}_m^n$. When $A$ is an elliptic curve and $V=\bar{P}$ is the set of conjugates of a non torsion $\bar{k}$-point, we reobtain the result of M. Laurent on the elliptic Lehmer's problem.
We seek to understand how the technical definition of Lehmer pair can be related to more analytic properties of the Riemann zeta function, particularly the location of the zeros of $ζ^\prime(s)$. Because we are interested in the connection between Lehmer pairs and the de Bruijn-Newman constant $Λ$, we assume the Riemann Hypothesis throughout. We define strong Lehmer pairs via an inequality on the derivative of the pre-Schwarzian of Riemann's function $Ξ(t)$, evaluated at consecutive zeros. Theorem 1 shows that strong Lehmer pairs are Lehmer pairs. Theorem 2 describes the derivative of the pre-…
In this note, we propose the modular height of an abelian variety defined over a field of finite type over Q. Moreover, we prove its finiteness property.
We prove that for abelian varieties with semistable ordinary reduction the p-adic Mazur-Tate height pairing is induced by the unit root splitting of the Hodge filtration on the first deRham cohomology.
Let A be an abelian variety defined over a number field K, and consider the canonical height function attached to a symmetric ample line bundle L on A. We prove that there is a positive lower bound C (depending on A, K, and L) for the canonical height of non-torsion points on A defined over the maximal abelian extension K^ab of K.
This paper is the sequel of our paper "Arithmetic height functions over finitely generated fields" (cf. math.NT/9809016). In this paper, we define the canonical height of subvarieties of an abelian variety over a finitely generated field over Q. We also prove that the canonical height of a subvariety is zero if and only if it is a translation of an abelian subvariety by a torsion point.
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…
We prove a special case of the following conjecture of Zilber-Pink generalising the Manin-Mumford conjecture : let $X$ be a curve inside an Abelian variety $A$ over $\bar{\Q}$, provided $X$ is not contained in a torsion subvariety, the intersection of $X$ with the union of all subgroup schemes of codimension at least 2 is finite ; we settle the case where $A$ is a power of a simple Abelian variety of C.M. type. This generalises the previous known result, due to Viada and Rémond-Viada (who was able to prove the conjecture for power of an elliptic curve with complex multiplication). The proof is…
In 1983 Silverman and Tate showed that the set of points in a 1-dimensional family of abelian varieties where a section of infinite order has `small height' is finite. We conjecture a generalisation to higher-dimensional families, where we replace `finite' by `not Zariski dense'. We show that this conjecture would imply the Uniform Boundedness Conjecture for torsion points on abelian varieties. We then prove a few special cases of this new conjecture.
We consider a family of heights defined by the $L_p$ norms of polynomials with respect to the equilibrium measure of a lemniscate for $0 \le p \le \infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler measure) and $p=\infty$ corresponds to the standard supremum norm. This special choice of the measure allows to find an explicit form for the geometric mean of a polynomial, and estimate it via certain resultant. For lemniscates satisfying appropriate hypotheses, we establish explicit polynomials of lowest height, and also show their uniqueness. We discuss relations betwe…
Canonical heights and Arakelov geometry on semi-abelian varieties. In this paper, we propose a construction of the canonical heights on an extension of an abelian variety by the multiplicative group, in the framework of Arakelov geometry. These canonical heights are the sum of some height coming from the abelian variety and something we call a relative height. We finally give some complements about the points whose relative height is zero.
Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of non-torsion points on E defined over the maximal abelian extension K^ab of K. This is analogous to results of Amoroso-Dvornicich and Amoroso-Zannier for the multiplicative group. We also show that if E has non-integral j-invariant (so that in particular E does not have complex multiplication), then there exists C > 0 such that there are only finitely many points P in E(K^ab) of canonical height less than C. This strengthens a result…
We obtain a lower bound for the normalised height of a non-torsion subvariety $V$ of a C.M. abelian variety. This lower bound is optimal in terms of the geometric degree of $V$, up to a power of a ``log''. We thus extend the results of F. Amoroso and S. David on the same problem on a multiplicative group $\mathbb{G}_m^n$. We prove furthermore that the optimal lower bound (conjectured by S. David and P. Philippon) is a corollary of the conjecture of S. David and M. Hindry on the abelian Lehmer's problem. We deduce these results from a density theorem on the non-torsion points of $V$.
Let f : X --> X be an endomorphism of a normal projective variety defined over a global field K, and let D_0,D_1,D_2,... be divisor classes that form a Jordan block with eigenvalue b for the action of f^* on Pic(X) tensored with C. We construct appropriately normalized canonical heights h_0,h_1,h_2,... associated to D_0,D_1,D_2,... and satisfying Jordan transformation formulas h_k(f(x)) = b h_k(x) + h_{k-1}(x). As an application, we prove that for every x in X, the arithmetic degree a_f(x) exists, is an algebraic integer, and takes on only finitely many values as x varies over X. Further, if X…
We study the problem of describing local components of height functions on abelian varieties over characteristic $0$ local fields as functions on spaces of torsors under various realisations of a $2$-step unipotent motivic fundamental group naturally associated to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the $\mathbb Q_\ell$- and $\mathbb Q_p$-pro-unipotent étale realisations when the base field is $p$-adic, and in terms of the $\mathbb R$-pro-unipotent Betti--de Rham realisation when the base field is archimedean. In the cours…
On an abelian scheme over a smooth curve over $\overline{\mathbb Q}$ a symmetric relatively ample line bundle defines a fiberwise Néon-Tate height. If the base curve is inside a projective space, we also have a height on its $\overline{\mathbb Q}$-points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the abelian scheme. As an application we prove the Geometric Bogomolov Conjecture for the …
In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties over the complex numbers of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are moreover able to use the technique to develop new Colmez-type formulas for the Faltings height.
Given a family of abelian varieties over a quasiprojective smooth curve $T^0$ over a global field and a point $P$ on the generic fiber, we show that the Néron-Tate canonical height $h_{X_t}(P_t)$ of $P_t$ along each fiber is exactly equal to a Weil height $h_{\overline M}(t)$ given by an adelic metrized line bundle $\overline M$ on the unique smooth projective curve $T$ containing $T^0$. As a consequence, we show that a conjecture of Zhang on the finiteness of small-height specializations of $P$ is equivalent to $\overline M$ being big.
We prove that assuming the Colmez conjecture and the ``no Siegel zeros" conjecture, the stable Faltings height of a CM abelian variety over a number field is less than or equal to the logarithm of the root discriminant of the field of definition of the abelian variety times an effective constant depending only on the dimension of the abelian variety. In view of the fact that the Colmez conjecture for abelian CM fields, the averaged Colmez conjecture, and the ``no Siegel zeros" conjecture for CM fields with no complex quadratic subfields are already proved, we prove unconditional analogues of t…
Let M be the Shimura variety associated with the group of spinor similitudes of a rational quadratic space over of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain $L$-functions. As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.
We lower bound the Faltings height of an abelian variety over a number field by the sum of its injectivity diameter and the norm of its bad reduction primes. It leads to an unconditional bound on the rank of Mordell-Weil groups. Assuming the height conjecture of Lang and Silverman, we then obtain a Northcott property for the regulator on the set of simple abelian varieties defined over a fixed number field, of fixed dimension $g$, bounded rank and with dense rational points over a number field. We remove the simplicity assumption in the principally polarized case by giving a refined version of…
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).
Let E be an elliptic curve over the rationals. Let L be an infinite Galois extension of the rationals with uniformly bounded local degrees at almost all primes. We will consider the infinite extension L(E_tor) of the rationals where we adjoin all coordinates of torsion points of E. In this paper we will prove an effective lower bound for the height of non-zero elements in L(E_tor) that are not a root of unity.
Let E/K be an ellptic curve defined over a number field, let h be the canonical height on E, and let K^ab be the maximal abelian extension of K. Extending work of M. Baker, we prove that there is a positive constant C(E/K) so that every nontorsion point P in E(K^ab) satisfies h(P) > C(E/K).
Let $E$ be an elliptic curve defined over the rationals without complex multiplication. The field $F$ generated by all torsion points of $E$ is an infinite, non-abelian Galois extension of the rationals which has unbounded, wild ramification above all primes. We prove that the absolute logarithmic Weil height of an element of $F$ is either zero or bounded from below by a positive constant depending only on $E$. We also show that the Néron-Tate height has a similar gap on $E(F)$ and use this to determine the structure of the group $E(F)$.
Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer's conjecture for algebraic vectors …
Let $E$ be an elliptic curve over the rationals. We will consider the infinite extension $\mathbb{Q}(E_{\text{tor}})$ of the rationals where we adjoin all coordinates of torsion points of $E$. In this paper we will prove an explicit lower bound for the height of non-zero elements in $\mathbb{Q}(E_{\text{tor}})$ that are not a root of unity, only depending on the conductor of the elliptic curve. As a side result we will give an explicit bound for a small supersingular prime for an elliptic curve.
Let $B$ be a simple CM abelian variety over a CM field $E$, $p$ a rational prime. Suppose that $B$ has potentially ordinary reduction above $p$ and is self-dual with root number $-1$. Under some further conditions, we prove the generic non-vanishing of (cyclotomic) $p$-adic heights on $B$ along anticyclotomic $\Z_{p}$-extensions of $E$. This provides evidence towards Schneider's conjecture on the non-vanishing of $p$-adic heights. For CM elliptic curves over $\Q$, the result was previously known as a consequence of work of Bertrand, Gross--Zagier and Rohrlich in the 1980s. Our proof is based o…
This work is the third part of a series of papers. In the first two we consider curves and varieties in a power of an elliptic curve. Here we deal with subvarieties of an abelian variety in general. Let V be an irreducible variety of dimension d embedded in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. For V transverse, we prove that the algebraic…
In this note, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's Specialization Theorem for families of Abelian varieties over $K$, where $K$ is any field finitely generated over ${\Q}$.
Let $F$ be an algebraic extension of the rational numbers and $E$ an elliptic curve defined over some number field contained in $F$. The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on $F^*$ modulo torsion, respectively on $E(F)$ modulo torsion. The groups $F^*$ and $E(F)$ are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. In this paper we prove the failure of the converse to this statement by explicitly constructing counterexamples.
We compare general inequalities between invariants of number fields and invariants of abelian varieties over number fields. On the number field side, we remark that there is only a finite number of non-CM number fields with bounded regulator. On the abelian side, assuming the height conjecture of Lang and Silverman, we obtain a Northcott property for the regulator on the set of abelian varieties with dense rational points over a number field. This amounts to say that the arithmetic of CM fields is similar, with respect to the invariants considered here, to the arithmetic of abelian varieties o…
We prove a formula, which, given a principally polarized abelian variety $(A,λ)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron--Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,λ)$.
We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties satisfy this property, when counting by height relative to a fixed finite map to projective space.
We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$. In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\Z$ there will be $\Q$-linear relations among the values of a canonical height-pairing evaluated at a basis modulo torsion of $A(k)$.
We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number-theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces.
