arXiv subjects
Publications and source records attributed to Paloma Bengoechea.
We refine a previous construction by Akhtari and Bhargava so that, for every positive integer m, we obtain a positive proportion of Thue equations F(x; y) = h that fail the integral Hasse principle simultaneously for every positive integer h less than m. The binary forms F have fixed degree >2 and are ordered by the absolute value of the maximum of the coefficients.
We prove two of Kaneko's conjectures on the "values" $\mathrm{val}(w)$ of the modular $j$ function at real quadratic irrationalities: we prove the lower bound $\mathrm{Re}(\mathrm{val}(w))\geq \mathrm{val}\left(\frac{1+\sqrt{5}}{2}\right)$ for all real quadratics $w$ and the upper bound $\mathrm{Re}(\mathrm{val}(w))\leq \mathrm{val}\left(1+\sqrt{2}\right)$ for all Markov irrationalities $w$. These results generalize to the "values" at quadratic irrationalities of any weakly holomorphic modular function $f$ such that $f(e^{it})$ is real, non-negative and increasing for $t\in [π/3,π/2]$.
To each weakly holomorphic modular function $f\not \equiv 0$ for $\mathrm{SL}(2,\mathbb{Z})$, which is non-negative on the geodesic arc $\{e^{it} : π/3\leq t\leq 2π/3\}$, we attach a $\mathrm{GL}(2,\mathbb{Z})$-invariant map $Λ_f:\mathbb{P}^1(\mathbb{R})\to \mathbb{R}$ that generalizes the Lyapunov exponent function introduced by Spalding and Veselov. We prove that it takes every value between $0$ and $Λ_f\left(\frac{1+\sqrt{5}}{2}\right)$ and it gives an increasing convex function on the Markov irrationalities when ordered using their parametrization by Farey fractions in $[0,1/2]$. In the case of quadratic irrationals $w$ with purely periodic continued fraction expansion, the value $Λ_f(w)$ equals the real part of the cycle integral of $f$ along the associated geodesic $C_w$ on the modular surface, normalized with the word length of the associated hyperbolic matrix $A_w$ as a word in the generators $T=\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ and $V=\left(\begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix}\right)$. These results are related to conjectures of Kaneko who observed several similar behavior for the cycle integrals of the modular $j$ function when normalized by the hyperbolic length of the geodesic $C_w$.
We will give new upper bounds for the number of solutions to the inequalities of the shape $|F(x , y)| \leq h$, where $F(x , y)$ is a sparse binary form, with integer coefficients, and $h$ is a sufficiently small integer in terms of the absolute value of the discriminant of the binary form $F$. Our bounds depend on the number of non-vanishing coefficients of $F(x , y)$. When $F$ is really sparse, we establish a sharp upper bound for the number of solutions that is linear in terms of the number of non-vanishing coefficients. This work will provide affirmative answers to a number of conjectures posed by Mueller and Schmidt in 1988, for special but important cases.
We give asymptotic upper and lower bounds for the real and imaginary parts of cycle integrals of the classical modular j-function along geodesics that correspond to Markov irrationalities.
Let $F(x, y)$ be a binary form with integer coefficients, degree $n\geq 3$ and irreducible over the rationals. Suppose that only $s + 1$ of the $n + 1$ coefficients of $F$ are nonzero. We show that the Thue inequality $|F(x,y)|\leq m$ has $\ll sm^{2/n}$ solutions provided that the absolute value of the discriminant $D(F)$ of $F$ is large enough. We also give a new upper bound for the number of solutions of $|F(x,y)|\leq m$, with no restriction on the discriminant of $F$ that depends mainly on $s$ and $m$, and slightly on $n$. Our bound becomes independent of $m$ when $m<|D(F)|^{2/(5(n-1))}$, and also independent of $n$ if $|D(F)|$ is large enough.
Let T_{n,k}(X) be the characteristic polynomial of the n-th Hecke operator acting on the space of cusp forms of weight k for the full modular group. We show that if there exists n>1 such that T_{n,k}(X) is irreducible and has the full symmetric group as Galois group, then the same is true of T_{p,k}(X) for all primes p.
In 2008, M. Kaneko made several interesting observations about the values of the modular j invariant at real quadratic irrationalities. The values of modular functions at real quadratics are defined in terms of their cycle integrals along the associated geodesics. In this paper we prove some of the conjectures of M. Kaneko for a general modular function.
In this paper we study the values of modular functions at the Markov quadratics which are defined in terms of their cycle integrals along the associated closed geodesics. These numbers are shown to satisfy two properties that were conjectured by Kaneko. More precisely we show that the values of a modular function f, along any branch B of the Markov tree, converge to the value of f at the Markov number which is the predecessor of the tip of B. We also prove an interlacing property for these values.
We discuss the approximation of real numbers by Fourier coefficients of newforms, following recent work of Alkan, Ford and Zaharescu. The main tools used here, besides the (now proved) Sato-Tate Conjecture, come from metric number theory.
This paper is motivated by Davenport's problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidian space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate C^1 submanifold of R^n has full dimension. In the second approach we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.
For any j_1,...,j_n>0 with j_1+...+j_n=1 and any x \in R^n, we consider the set of points y \in R^n for which max_{1\leq i\leq n}(||qx_i-y_i||^{1/j_i})>c/q for some positive constant c=c(y) and all q\in N. These sets are the `twisted' inhomogeneous analogue of Bad(j_1,...,j_n) in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e provided that j_i=1/n, and in the weighted setting when x is chosen from Bad(j_1,...,j_n). We generalise these results proving the full Hausdorff dimension in the weighted setting without any condition on x.
A classical theorem in continued fractions due to Serret shows that for any two irrational numbers x and y related by a transformation $γ$ in PGL(2,Z) there exist s and t for which the complete quotients x_s and y_t coincide. In this paper we give an upper bound in terms of $γ$ for the smallest indices s and t.
We study the meromorphic modular forms defined as sums of -k (k>1) powers of integral quadratic polynomials with negative discriminant. These functions can be viewed as meromorphic analogues of the holomorphic modular forms defined in the same way with positive discriminant, first investigated by Zagier in connection with the Doi-Naganuma map and then by Kohnen and Zagier in connection with the Shimura-Shintani lifts. We compute the Fourier coefficients of these meromorphic modular forms and we show that they split into the sum of a meromorphic modular form with computable algebraic Fourier coefficients and a holomorphic cusp form.
Using Shimura's reciprocity law, we calculate the Galois action on the special values of theta functions associated to primitive Dirichlet characters of odd conductor, normalised by the Dedekind eta function, at the point i. As a consequence, we prove some experimental results of Cohen and Zagier and we deduce a partial result on the non-vanishing of these special theta values with prime conductor.
We show how a recent result by Aka and Shapira on the evolution of continued fractions in a fixed quadratic field implies the classic result of de Mathan and Teullié on the Mixed Littlewood
In this paper we study certain real functions defined in a very simple way by Zagier as sums of infinite powers of quadratic polynomials with integer coefficients. These functions give the even parts of the period polynomials of the modular forms which are the coefficients in Fourier expansion of the kernel function for Shimura-Shintani correspondence. We prove two conjectures of Zagier showing that the sums converge exponentially. We also prove unexpected results on the representation of these functions as sums over simple or reduced quadratic forms and the positive or negative continued fraction of the variable. These arise from more general results on polynomials of even degree. Especially we give the even part of the Eichler integral on a real number x of any cusp form for PSL(2,Z) in terms of the even part of its period polynomial and the continued fraction of x.