In this paper, we study bounds for the number of rational points on twists C' of a fixed curve C over a number field K, under the condition that the group of K-rational points on the Jacobian J' of C' has rank smaller than the genus of C'.
The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J'(K) and c is a constant depending on C.
For the proof, we use a refinement of the method of Chabauty-Coleman; the main new ingredient is to use it for an extension field of K_v, where v is a place of bad reduction for C'.
We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao's method for constructing elements in the finite field $\mathbb{F}_{q^n}$ whose orders are larger than any polynomial in $n$ when $n$ becomes large. Additionally, we discuss the finiteness of polynomials which translate a given finite set of polynomials to become multiplicatively dependent.
Given any two rational numbers $r_1$ and $r_2$, a necessary and sufficient condition is established for the three numbers $1$, $\cos (πr_1)$, and $\cos (πr_2)$ to be rationally independent. Extending a classical fact sometimes attributed to I. Niven, the result even yields linear independence over larger number fields. The tools employed in the proof are applicable also in the case of more than two trigonometric numbers. As an application, a complete classification is given of all planar triangles with rational angles and side lengths each containing at most one square root. Such a classificat…
Given a family of quadratic twists of a fixed elliptic curve defined over $\mathbb{Q}$, we investigate the average rank in the subfamily of twists having a nontorsion rational point of almost minimal height. We show in particular that the average analytic rank is greater than one.
We introduce and study algebraic dynamical systems generated by triangular systems of rational functions. We obtain several results about the degree growth and linear independence of iterates as well as about possible lengths of trajectories generated by such dynamical systems over finite fields. Some of these results are generalisations of those known in the polynomial case, some are new even in this case.
Katz showed that the L-functions of all Dirichlet characters of F_q(t), with conductor a fixed power of a degree one prime, are equidistributed in the limit as q goes to infinity. We generalize this statement to the L-functions of twists of an arbitrary Galois representation by Dirichlet characters, including independence of the L-functions of twists of different representations by the same Dirichlet character. A similar generalization, without the independence statement, for characters with squarefree conductor, was proven by Hall, Keating, and Roditty-Gershon.
We formulate a conjecture about the distribution of the canonical height of the lowest non-torsion rational point on a quadratic twist of a given elliptic curve, as the twist varies. This conjecture seems to be very deep and we can only prove partial results in this direction.
We generalize a result of Frey [Fre88] on the Selmer group of twists of elliptic curves over Q with Q-rational torsion points to elliptic curves defined over number fields of small degree K with a K-rational point. We also provide examples of elliptic curves coming from [Zyw15] that satisfy the conditions of our Theorem B.
We provide an alternative proof that the finite rational linear combination of radicals, under certain constraint, are linearly independent over $\mathbb{Q}$.
We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.
This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion o…
We prove a weighted analogue of the Khintchine-Groshev Theorem, where the distance to the nearest integer is replaced by the absolute value. This is subsequently applied to proving the optimality of several linear independence criteria over the field of rational numbers.
A set of real $n$th roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a new technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Section 2 and 3, using the technique together with Mahler's method, we can reduce the al…
Weyl group multiple Dirichlet series are Dirichlet series in $r$ complex variables, with analytic continuation to $\mathbb{C}^r$ and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank $r$. Such series may be defined for any global field $K$, but in the case when $K$ is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when $K=\mathbb{F}_q(T)$, and describe the denominators and support of the numerators.
On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point is the image of another rational point under a certain 3-isogeny, all terms beyond the first fail to be primes.
The Linear Independence hypothesis (LI), which states roughly that the imaginary parts of the critical zeros of Dirichlet L-functions are linearly independent over the rationals, is known to have interesting consequences in the study of prime number races, as was pointed out by Rubinstein and Sarnak. In this paper, we prove that a function field analogue of LI holds generically within certain families of elliptic curve L-functions and their symmetric powers. More precisely, for certain algebro-geometric families of elliptic curves defined over the function field of a fixed curve over a finite …
We compute a lower bound of the canonical height on quadratic twists of certain elliptic curves. Also we show a simple method for constructing families of quadratic twists with an explicit rational point. % from cubic polynomials. Using the above lower bound, we show that the explicit rational point is primitive as an element of the Mordell--Weil group.
We investigate the arithmetic nature of special values of Thakur's function field Gamma function at rational points. Our main result is that all linear independence relations over the field of algebraic functions are consequences of the known relations of Anderson and Thakur arising from the functional equations of the Gamma function.
This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]), then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r_2, the number of complex places of K. We also obtain results abo…
Let $n\geq 2$ be an integer, $F$ a number field, $O_F$ the integral closure of $\mathbb{Z}$ in $F$ and $N$ a positive multiple of $n$. The paper deals with degree $N$ polynomials $P(T) \in O_F[T]$ such that the superelliptic curve $Y^n=P(T)$ has twists $Y^n=d\cdot P(T)$ without $F$-rational points. We show that this condition holds if the Galois group of $P(T)$ over $F$ has an element which fixes no root of $P(T)$. Two applications are given. Firstly, we prove that the proportion of degree $N$ polynomials $P(T) \in O_F[T]$ with height bounded by $H$ and such that the associated curve satisfies…
We present some results about the number of rational points on a certain family of curves defined over a finite field. In a small number of cases the curves have more rational points than expected. Fibonacci numbers make an appearance, as do cyclotomic polynomials.
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 $\geq 5$) for twists by Dirichlet characters of prime order $\ell \geq 3$, 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 th…
Given a pair of elliptic curves $E_1$ and $E_2$ over the rational field $\mathbb Q$ whose $j$-invariants are not simultaneously 0 or 1728, Kuwata and Wang proved the existence of infinitely many square-free rationals $d$ such that the $d$-quadratic twists of $E_1$ and $E_2$ are both of positive rank. We construct infinite families of pairs of elliptic curves $E_1$ and $E_2$ over $\mathbb Q$ such that for each pair there exist infinitely many square-free rationals $d$ for which the $d$-quadratic twists of $E_1$ and $E_2$ are both of rank at least 2.
Let $K$ denote a number field and a finite set $S$ of places of $K$ and $ϕ:\PP^n\rightarrow\PP^n$ be rational morphism defined over $K$. The main result of this paper proves that there are only finitely many twists of $ϕ$ defined over $K$ which have good reduction at all places outside $S$. This answers a question of Silverman in the affirmative.
Using a construction of Hassett--Várilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, $\mathbb{Q}_2$, and $\mathbb{R}$, where one has a rational point and the other does not. This answers negatively a question recently raised by Hassett and Tschinkel.
Conditionally on the $abc$ conjecture, we generalize previous work of Clark and the author to show that a superelliptic curve $C: y^n = f(x)$ of sufficiently high genus has infinitely many twists violating the Hasse Principle if and only if $f(x)$ has no $\mathbb Q$-rational roots. We also show unconditionally that a curve defined by $C: y^{pN}=f(x)$ has infinitely many twists violating the Hasse Principle over any number field $k$ such that $k$ contains the $p$th roots of unity and $f(x)$ has no $k$-rational roots.
In studying rational points on elliptic K3 surfaces of the form $f(t)y^2=g(x)$, where $f,g$ are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell-Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels-Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.
On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. It is shown that there are finitely many perfect powers in such a sequence whose first term is greater than 1. Moreover, if the first term is divisible by 6 and the generating point is triple another rational point then there are no perfect powers in the sequence except possibly an lth power for some l dividing the order of 2 in the first term.
Given an elliptic curve E over the rational with no rational 2-torsion points, we prove the existence of a quadratic twist of E for which the 2-Selmer rank is less than or equal to 1. By the author's earlier result, we establish a lower bound on the number of D's for which the twists E(D) have 2-Selmer rank <= 1. We include in the introduction our (brief) opinion about why it is supposed to be hard to push our technique to make the Selmer group trivial.
We evaluate in closed form series of the type $\sum u(n) R(n)$, where $(u(n))_n$ is a strongly $B$-multiplicative sequence and $R(n)$ a (well-chosen) rational function. A typical example is: $$ \sum_{n \geq 1} (-1)^{s_2(n)} \frac{4n+1}{2n(2n+1)(2n+2)} = -\frac{1}{4} $$ where $s_2(n)$ is the sum of the binary digits of the integer $n$. Furthermore closed formulas for series involving automatic sequences that are not strongly $B$-multiplicative, such as the regular paperfolding and Golay-Shapiro-Rudin sequences, are obtained; for example, for integer $d \geq 0$: $$ \sum_{n \geq 0} \frac{v(n)}{(n…
In this paper, we consider a family of twists of a superelliptic curve over a global field, and obtain results on the distribution of the Mordell-Weil rank of these twists. Our results have applications to the distribution of the number of rational points.
This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having…
We investigate the average rank in the family of quadratic twists of a given elliptic curve defined over $\mathbb{Q}$, when the curves are ordered using the canonical height of their lowest non-torsion rational point.
Using Serre's proposed complement to Shih's Theorem, we obtain PSL_2(F_p) as a Galois group over Q for at least 614 new primes p. Under the assumption that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for 3/8 of the primes not covered by previous results; it would also suffice to assume a certain (plausible, and perhaps tractable) conjecture concerning class numbers of quadratic fields. The key issue is to understand rational points on Atkin-Lehner twists of X_0(N). In an appendix, we explore the existence of local points on these curves.
For every n there exists an elliptic curve E over the rational numbers and an n-dimensional subspace V of non-zero rationals modulo squares such that for all v in V, the quadratic twist of E by v has positive rank.
We conjecture a lower bound for the minimal canonical height of non-torsion rational points on a natural density 1 subset of the sextic twist family of Mordell curves. We then establish a lower bound that yields a partial result towards this conjecture.
Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively independent algebraic integers $γ_1, \dots, γ_s \in K^*$ that fulfil some technical requirements, which can be vastly simplified to $\mathbb{Q}$-linearly independence, given Schanuel's conjecture. We then consider the twisted Thue equation \[ \left|N_{K/\mathbb{Q}}\left(X-γ_1^{t_1}\cdotsγ_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots, t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( γ_1^{t_1}\cdots γ_s^{t_s} \right) = K$, all of which are effectively computable.
Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral point.
Using the sieve for Frobenius, we show that, in a certain sense, the roots of the L-functions of "most" algebraic curves over finite fields do not satisfy any non-trivial (linear or multiplicative) rational dependency relations. This can be seen as an analogue of conjectures of linear independence among ordinates of zeros of L-functions over number fields. As a corollary, we find, for "most" pairs of distinct algebraic curves over a finite field, the limiting distribution of the (suitably normalized) difference between the number of rational points over extensions of the ground field. The meth…
Let $k$ be the rational function field over the finite field of $q$ elements and $\bar{k}$ its fixed algebraic closure. In this paper, we study algebraic relations over $\bar{k}$ among the fundamental period $\widetildeπ$ of the Carlitz module and the positive characteristic multizeta values $ζ(n)$ and $ζ(n,n)$ for an "odd" integer $n$, where we say that $n$ is "odd" if $q-1$ does not divide $n$. We prove that either they are algebraically independent over $\bar{k}$ or satisfy some simple relation over $k$. We also prove that if $2n$ is "odd" then they are algebraically independent over $\bar{…
Watkins conjectured that for an elliptic curve $E$ over $\mathbb{Q}$ of Mordell-Weil rank $r$, the modular degree of $E$ is divisible by $2^r$. If $E$ has non-trivial rational $2$-torsion, we prove the conjecture for all the quadratic twists of $E$ by squarefree integers with sufficiently many prime factors.
In this paper we deduce a lower bound for the rank of a family of $p$ vectors in $\R^k$ (considered as a vector space over the rationals) from the existence of a sequence of linear forms on $\R^p$, with integer coefficients, which are small at $k$ points. This is a generalization to vectors of Nesterenko's linear independence criterion (which corresponds to $k=1$), used by Ball-Rivoal to prove that infinitely many values of Riemann zeta function at odd integers are irrational. The proof is based on geometry of numbers, namely Minkowski's theorem on convex bodies.
Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon as certain local points on rational elliptic curves, conjecturally defined over abelian extensions of real quadratic fields. The rationality conjecture for these points is only known in the unramified case, namely, when these points are specializations of global points defined over the strict Hilbert class field $H^+_F$ of the real quadratic field $F$ and twisted by (unramified) quadratic characters of $Gal(H_c^+/F)$. We extend these results to the situation of ramified quadratic characters; more precisely, we show…
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.
Let $D > 546$ be the discriminant of an indefinite rational quaternion algebra. We show that there are infinitely many imaginary quadratic fields $l/\mathbb Q$ such that the twist of the Shimura curve $X^D$ by the main Atkin-Lehner involution $w_D$ and $l/\mathbb Q$ violates the Hasse Principle over $\mathbb Q$.
For every point on the Jacobian of the modular curve $X_0(l)$ we define and study certain twisted holomorphic Eisenstein series. These are particular cases of a more general notion of twisted modular forms which correspond to sections on the modular curve $X_1(l)$ of the degree zero twists of line bundles of usual modular forms. We conjecture that a point on the Jacobian is rational if and only if the ratios of these twisted Eisenstein series of the same weights have rational coefficients.
Let phi be a morphism of projective N-space defined over a number field K. We prove that there is a bound B depending only on phi such that every twist of phi has no more than B K-rational preperiodic points. (This result is analagous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over Q, we find the bound B explicitly.
Let K be a number field and {V_l} be a rational strictly compatible system of semisimple Galois representations of K arising from geometry. Let G_l and V_l^ab be respectively the algebraic monodromy group and the maximal abelian subrepresentation of V_l for all l. We prove that the system {V_l^ab} is also a rational strictly compatible system under some group theoretic conditions, e.g., when G_l' is connected and satisfies Hypothesis A for some prime l'. As an application, we prove that the Tate conjecture for abelian variety X/K is independent of l if the algebraic monodromy groups of the Gal…
We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve X. To restrict the set of relevant twists, we exploit the isomorphism between X and the modular curve X(7), and use modularity of elliptic curves and level lowering. This leaves 10 genus-3 curves, whose rational points are found by a combination of methods.
For a tuple $(θ_1,..,θ_M)$ of complex number, buliding on the approximation techniques in earlier papers of this series, this paper engages in deducing lower estimates on the transcendence degree of the field generated by $θ_1, ..., θ_M$ over the field of rational numbers from the approximability of the point $θ=(1,θ_1,...,θ_M)$ in projective space by hypersurfaces. The first given result is an new proof of an algebraic independence criterion, that was formerly proved by Laurent and Roy, and generalizes the Philippon criterion by introducing also evaluations of derivatives of global sections. …
Let X^d(N) be the quadratic twist of the modular curve X_0(N) through the Atkin-Lehner involution w_N and a quadratic extension Q(\sqrt{d})/Q. The points of X^d(N)(Q) are precisely the Q(\sqrt{d})-rational points of X_0(N) that are fixed by σcomposition w_N, where σis the generator of Gal(Q(\sqrt{d})/Q).Ellenberg asked the following question: For which d and N does X^d(N) have rational points over every completion of Q? Given (N,d,p) we give necessary and sufficient conditions for the existence of a Q_p-rational point on X^d(N), whenever p is not simultaneously ramified in Q(\sqrt{d}) and Q(\s…
We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2,3,5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological obstruction that measures if a given twist is birational to a moduli space of abelian varieties. This obstruction has implications for the rational points on these varieties. As a result, we see that all possible 3-level structures can be realized by abelian surfaces, whereas Kummer 3-level structures that group-theoretically may be admissible, may not be realizable…
Let L(E/Q,s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L(E,1,χ) of the twisted L-function as χranges over Dirichlet characters of given order.
Manin's conjecture predicts the distribution of rational points on Fano varieties. Using explicit parameterizations of rational points by integral points on universal torsors and lattice-point-counting techniques, it was proved for several specific varieties over $\mathbb{Q}$, in particular del Pezzo surfaces. We show how this method can be implemented over arbitrary number fields $K$, by proving Manin's conjecture for a singular quartic del Pezzo surface of type $\mathbf{A}_3+\mathbf{A}_1$. The parameterization step is treated in high generality with the help of twisted integral models of uni…
We give an explicit description of all 168 quartic curves over the field of two elements that are isomorphic to the Klein curve over an algebraic extension. Some of the curves have been known for their small class number, others for attaining the maximal number of rational points.
