We solve an energy minimization problem for local fields. As an application of these results, we improve on lower bounds set by Bombieri and Zannier for the limit infimum of the Weil height in fields of totally p-adic numbers and generalizations thereof. In the case of fields with mixed archimedean and non-archimedean splitting conditions, we are able to combine our bounds with similar bounds at the archimedean places for totally real fields.
In the paper we obtain the asymtotic number of integral quadratic polynomials with bounded heights and discriminants as the upper bound of heights tends to infinity.
We study the asymptotic distribution of S-integral points of bounded height on partial bi-equivariant compactifications of semi-simple groups of adjoint type.
This paper discusses examples of integral factorial ratios of height 2 or more. It classifies (apart from finitely many examples) such factorial ratios with height 2 and norm at most 1/3, and describes a general result which exhibits more than 50 two parameter families of integral factorial ratios with height 2. These examples are shown to be irreducible in the sense that they do not arise from multiplying two factorial ratios of height 1.
In the paper we partially solved the problem of the distribution of the discriminants of integral polynomials in the cubic case. We proved the asymptotic formula for the number of integral cubic polynomials having bounded height and bounded discriminant. The main term of the obtained formula is the best possible for the considered range of polynomial discriminant values.
We study small points for the Arakelov height on the projective line. First, we identify the smallest positive value taken by the Arakelov height, and we characterize all cases of equality. Next we solve several archimedean energy minimization problems with respect to the chordal metric on the projective line, and as an application, we obtain lower bounds on the Arakelov height in fields of totally real and totally p-adic numbers.
We establish new upper bounds for the height of the S-integral points of an elliptic curve. This bound is explicitly given in terms of the set S of places of the number field K involved, but also in terms of the degree of K, as well as the rank, the regulator and the height of a basis of the Mordell-Weil group of the curve. The proof uses the elliptic analogue of Baker's method, based on lower bounds for linear forms in elliptic logarithms.
By Northcott's Theorem there are only finitely many algebraic points in affine $n$-space of fixed degree over a given number field and of height at most $X$. For large $X$ the asymptotics of these cardinalities have been investigated by Schanuel, Schmidt, Gao, Masser and Vaaler, and the author. In this paper we study the case where the coordinates of the points are restricted to algebraic integers, and we derive the analogues of Schanuel's, Schmidt's, Gao's and the author's results.
Let $F$ be a non-zero polynomial with integer coefficients in $N$ variables of degree $M$. We prove the existence of an integral point of small height at which $F$ does not vanish. Our basic bound depends on $N$ and $M$ only. We separately investigate the case when $F$ is decomposable into a product of linear forms, and provide a more sophisticated bound. We also relate this problem to a certain extension of Siegel's Lemma as well as to Faltings' version of it. Finally we exhibit an application of our results to a discrete version of the Tarski plank problem.
We study the asymptotic distribution of integral points of bounded height on partial bi-equivariant compactifications of semi-simple groups of adjoint type.
Given an isotropic quadratic form over a number field which assumes a value $t$, we investigate the distribution of points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of quadratic forms, we produce bounds on height of points outside of some algebraic sets in a quadratic space at which the form assumes the value $t$. Our bounds on height are explicit in terms of the heights of the form, the space, the algebraic set and the value $t$.
A new proof is given for the explicit formulae for the non-archimedean canonical height on an elliptic curve. This arises as a direct calculation of the Haar integral in the elliptic Jensen formula.
The present note is devoted to an amendment to a recent paper of Ellenberg, Lawrence and Venkatesh. Roughly speaking, the main result here states the subpolynomial growth of the number of integral points with bounded height of a variety over a number field whose fundamental group is large. Such an improvement, i.e. requiring large fundamental group as opposed to the existence of a geometric variation of pure Hodge structures, was already asked in op.cit..
We study heights of motives with integral coefficients over number fields introduced by Kato. It is a generalization of the Faltings height of an abelian variety and we establish generalizations of some properties of the Faltings height in our context as conjectured by Kato. This sheds some light on integral structures of motives.
We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell-Weil group tensored with the rationals.
Bombieri and Zannier established lower and upper bounds for the limit infimum of the Weil height in fields of totally p-adic numbers and generalizations thereof. In this paper, we use potential theoretic techniques to generalize the upper bounds from their paper and, under the assumption of integrality, to improve slightly upon their bounds.
We describe an algorithm to compute the local component at p of the Coleman-Gross p-adic height pairing on divisors on hyperelliptic curves. As the height pairing is given in terms of a Coleman integral, we also provide new techniques to evaluate Coleman integrals of meromorphic differentials and present our algorithms as implemented in Sage.
We consider a system of homogeneous quadratic forms with congruence conditions in $n\geq 3$ variables and prove the existence of two linearly independent integral solutions of bounded height. We also show the existence of small height integral zeros of this system avoiding a given set of hyperplanes.
We use Arakelov theory to define a height on divisors of degree zero on a hyperelliptic curve over a global field, and show that this height has computably bounded difference from the Néron-Tate height of the corresponding point on the Jacobian. We give an algorithm to compute the set of points of bounded height with respect to this new height. This provides an `in principle' solution to the problem of determining the sets of points of bounded Néron-Tate heights on the Jacobian. We give a worked example of how to compute the bound over a global function field for several curves, of genera up t…
In this paper, we give an explicit bound for the height of integral points on $X_0(p)$ by using a very explicit version of the Chevalley-Weil principle. We improve the bound given by Sha in \cite{sha2014bounding1}.
Let $P_N(R)$ be the space of all real polynomials in $N$ variables with the usual inner product $<, >$ on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form representing this inner product on the space of coefficient vectors of all polynomials in $P_N(R)$ of degree $\leq M$. We exhibit two applications of this formula. First, given a finite dimensional subspace $V$ of $P_N(R)$ defined over $Q$, we prove the existence of an orthogonal basis for $(V, <, >)$, consisting of polynomials of small height with integer coefficient…
We prove that the number of rational points of bounded height on certain del Pezzo surfaces of degree 1 defined over Q grows linearly, as predicted by Manin's conjecture. Along the way, we investigate the average number of integral points of small naive height on quadratic twists of a fixed elliptic curve with full rational 2-torsion.
We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.
The Arakelov-Zhang pairing $\langleψ,ϕ\rangle$ is a measure of the "dynamical distance" between two rational maps $ψ$ and $ϕ$ defined over a number field $K$. It is defined in terms of local integrals on Berkovich space at each completion of $K$. We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference be…
We prove that, when elliptic curves $E/\mathbb{Q}$ are ordered by height, the average number of integral points $\#|E(\mathbb{Z})|$ is bounded, and in fact is less than $66$ (and at most $\frac{8}{9}$ on the minimalist conjecture). By "$E(\mathbb{Z})$" we mean the integral points on the corresponding quasiminimal Weierstrass model $E_{A,B}: y^2 = x^3 + Ax + B$ with which one computes the na\"ıve height. The methods combine ideas from work of Silverman, Helfgott, and Helfgott-Venkatesh with work of Bhargava-Shankar and a careful analysis of local heights for "most" elliptic curves. The same met…
Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one integral multiple [n]P such that n > C. The proof is a modification of a proof of Ingram giving an unconditional but not uniform bound. The new ingredient is a collection of explicit formulae for the sequence of valuations of the division polynomials. For P of non-singular reduction, such sequences are already well described in most cases, but for P of singul…
Let X be a smooth projective curve of positive genus defined over a number field K. Assume given a Galois covering map x from X to the projective line over K and a place v of K. We introduce a local canonical height on the set of K_v-valued points of X associated to x as an integral with logarithmic integrand, generalizing Tate's local Neron function on an elliptic curve. The resulting global height can be viewed as a 'Mahler measure' associated to x. We prove that the local canonical height can be obtained by averaging, and taking a limit, over divisors of higher order Weierstrass points on X…
Let f: A^N \to A^N be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the v-adic Green functions G_{f,v} and G_{f^{-1},v} (i.e., the v-adic canonical height functions) for f and f^{-1}. Next we introduce for f the notion of good reduction at v, and using this notion, we show that the sum of v-adic Green functions over all v gives rise to a canonical height function for f that satisfies the Northcott-type finiteness property. Using previous results, we recover results on arithmetic properties of f-periodic points and non f-periodic points. We…
We apply the theory of Borcherds products to calculate arithmetic volumes (heights) of Shimura varieties of orthogonal type up to contributions from very bad primes. The approach is analogous to the well-known computation of their geometric volume by induction, using special cycles. A functorial theory of integral models of toroidal compactifications of those varieties and a theory of arithmetic Chern classes of integral automorphic vector bundles with singular metrics are used. We obtain some evidence in the direction of Kudla's conjectures on relations of heights of special cycles on these v…
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.
In this paper we construct parameterizations of elliptic curves over the rationals which have many consecutive integral multiples. Using these parameterizations, we perform searches in GMP and Magma to find curves with points of small height, curves with many integral multiples of a point, curves with high multiples of a point integral, and over two hundred curves with more than one hundred integral points. In addition, a novel and complete classification of "self-descriptive numbers" is constructed by bounding the number of zeros such a number must contain.
Let $P$ be a polynomial that depends on two variables $X$ and $Y$ and has algebraic coefficients. If $x$ and $y$ are algebraic numbers with $P(x,y)=0$, then by work of Néron $h(x)/q$ is asymptotically equal to $h(y)/p$ where $p$ and $q$ are the partial degrees of $P$ in $X$ and $Y$, respectively. In this paper we compute a completely explicit bound for $|h(x)/q-h(y)/p|$ in terms of $P$ which grows asymptotically as $\max\{h(x),h(y)\}^{1/2}$. We apply this bound to obtain a simple version of Runge's Theorem on the integral solutions of certain polynomial equations.
Let $\mathbb{F}_q(T)$ be the field of rational functions in one variable over a finite field. We introduce the notion of a totally $T$-adic function: one that is algebraic over $\mathbb{F}_q(T)$ and whose minimal polynomial splits completely over the completion $\mathbb{F}_q(\!(T)\!)$. We give two proofs that the height of a nonconstant totally $T$-adic function is bounded away from zero, each of which provides a sharp lower bound. We spend the majority of the paper providing explicit constructions of totally $T$-adic functions of small height (via arithmetic dynamics) and minimum height (via …
Let C be an affine, plane, algebraic curve of degree d with integer coefficients. In 1989, Bombieri and Pila showed that if one takes a box with sides of length N then C can obtain no more than O_{d,ε}(N^{1/d+ε}) integer points within the box. Importantly, the implied constant makes no reference to the coefficients of the curve. Examples of certain rational curves show that this bound is tight but it has long been thought that when restricted to non-rational curves an improvement should be possible whilst maintaining the uniformity of the bound. In this paper we consider this problem restricte…
An approach to the calculation of local canonical morphic heights is described, motivated by the analogy between the classical height in Diophantine geometry and entropy in algebraic dynamics. We consider cases where the local morphic height is expressed as an integral average of the logarithmic distance to the closure of the periodic points of the underlying morphism. The results may be thought of as a kind of morphic Jensen formula.
We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local obstructions) integer is the curvature of some circle in the gasket.
We show that the height of a variety over a finitely generated field of characteristic zero can be written as an integral of local heights over the set of places of the field. This allows us to apply our previous work on toric varieties and extend our combinatorial formulae for the height to compute some arithmetic intersection numbers of non toric arithmetic varieties over the rational numbers.
Let $K$ be a global field or $\overline{\mathbb Q}$, $F$ a nonzero quadratic form on $K^N$, $N \geq 2$, and $V$ a subspace of $K^N$. We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of $(V,F)$ such that each such family spans $V$ as a $K$-vector space. This result generalizes and extends a well known theorem of J. Vaaler and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels' theorem on small zeros of quadratic forms. All bounds on height are explicit.
