147 research outputs found
On the Mahler measure of hyperelliptic families
We prove Boyd’s “unexpected coincidence” of the Mahler measures for two families of two-variate polynomials defining curves of genus 2. We further equate the same measures to the Mahler measures of polynomials y³ − y + x³ − x + kxy whose zero loci define elliptic curves for k ≠ 0, ± 3
Positivity of rational functions and their diagonals
The problem to decide whether a given rational function in several variables
is positive, in the sense that all its Taylor coefficients are positive, goes
back to Szeg\H{o} as well as Askey and Gasper, who inspired more recent work.
It is well known that the diagonal coefficients of rational functions are
-finite. This note is motivated by the observation that, for several of the
rational functions whose positivity has received special attention, the
diagonal terms in fact have arithmetic significance and arise from differential
equations that have modular parametrization. In each of these cases, this
allows us to conclude that the diagonal is positive.
Further inspired by a result of Gillis, Reznick and Zeilberger, we
investigate the relation between positivity of a rational function and the
positivity of its diagonal.Comment: 16 page
A -microscope for supercongruences
By examining asymptotic behavior of certain infinite basic (-)
hypergeometric sums at roots of unity (that is, at a "-microscopic" level)
we prove polynomial congruences for their truncations. The latter reduce to
non-trivial (super)congruences for truncated ordinary hypergeometric sums,
which have been observed numerically and proven rarely. A typical example
includes derivation, from a -analogue of Ramanujan's formula of the two supercongruences valid
for all primes , where denotes the truncation of the infinite sum
at the -th place and stands for the quadratic character
modulo .Comment: 26 page
Ramanujan-type formulae for : -analogues
The hypergeometric formulae designed by Ramanujan more than a century ago for
efficient approximation of , Archimedes' constant, remain an attractive
object of arithmetic study. In this note we discuss some -analogues of
Ramanujan-type evaluations and of related supercongruences.Comment: 8 page
A modular supercongruence for : an Ap\'ery-like story
We prove a supercongruence modulo between the th Fourier coefficient of a weight 6 modular form and a truncated -hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence between the Apéry numbers and another Apéry-like sequence
On the transcendence degree of the differential field generated by Siegel modular forms
It is a classical fact that the elliptic modular functions satisfies an
algebraic differential equation of order 3, and none of lower order. We show
how this generalizes to Siegel modular functions of arbitrary degree. The key
idea is that the partial differential equations they satisfy are governed by
Gauss--Manin connections, whose monodromy groups are well-known. Modular theta
functions provide a concrete interpretation of our result, and we study their
differential properties in detail in the case of degree 2.Comment: 21 pages, AmSTeX, uses picture.sty for 1 LaTeX picture; submitted for
publicatio
A correspondence of modular forms and applications to values of L-series
An interpretation of the Rogers–Zudilin approach to the Boyd conjectures is established. This is based on a correspondence of modular forms which is of independent interest. We use the reinterpretation for two applications to values of L-series and values of their derivatives
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