Functional equations for Mahler measures of genus-one curves

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 $y^3-y+x^3-x+kxy$ whose zero loci define elliptic curves for $k\ne0,\pm3$.

On linear independence measures of the values of Mahler functions

In this paper, we estimate the linear independence measures for the values of a class Mahler functions of degree one and two. For the purpose, we study the determinants of suitable Hermite-Padé approximation polynomials. Based on the non-vanishing of these determinants, we apply the functional equations to get an infinite sequence of approximations which is used to produce the linear independence measures.

一族非回火多项式的马勒测度与博伊德猜想

Mahler measures of a family of non-tempered polynomials and Boyd's conjectures

本文通过建立非回火多项式与回火多项式马勒测度之间的新恒等式，结合 Brunault-Mellit-Zudilin 公式及附加泛函恒等式，证明了导子为 30 的椭圆曲线的博伊德猜想。

From $L$-series of elliptic curves to Mahler measures

We prove the conjectural relations between Mahler measures and $L$-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for $L$-values of CM elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family $(1+X)(1+Y)(X+Y)-αXY$, $α\in\mathbb R$.

Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture

In the first part we construct algorithms which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we u…

CM点处Mahler测度恒等式的认证延拓机器：Samart猜想的十二个新证明

A certified continuation machine for Mahler measure identities at CM points, with twelve new proofs of conjectures of Samart

该论文将微分比较延拓方法公理化为一台认证机器，用于在CM点证明Mahler测度恒等式，为Samart猜想提供了十二个新证明，并发现其适用性边界并非拓扑边界。

The functional equation for the twisted spinor L-series of genus 2

We prove the functional equation for the twisted spinor L-series of a cuspidal, holomorphic Siegel eigenform for the full modular group of genus 2. It follows from a more general functional equation, valid for Rankin convolutions of paramodular cuspforms. A non-vanishing result for Fourier-Jacabi coefficients of the eigenforms in question is the central pillar of the deduction of the former from the latter functional equation.

Zero Order Estimates for Analytic Functions

The primary goal of this paper is to provide a general multiplicity estimate. Our main theorem allows to reduce a proof of multiplicity lemma to the study of ideals stable under some appropriate transformation of a polynomial ring. In particular, this result leads to a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. On the other hand, it allows to establish an improvement of Nesterenko's conditional result on solutions of systems of differential equations. We also deduce, under some condition on stable varieties, the optimal multiplicity e…

线性 Mahler 算子的一阶因子

First-order factors of linear Mahler operators

本文提出并比较两种计算线性 Mahler 算子一阶右因子的算法，分别基于 Petkovšek 经典递推算法和广义幂级数基结合 Hermite-Padé 逼近，并展示了其在证明微分超越性中的应用。

Functional equations for Selberg zeta functions with Tate motives

For a compact Riemann surface $M$ of genus $g\ge 2$, we study the functional equations of the Selberg zeta functions attached with the Tate motives $f$. We prove that certain functional equations hold if and only if $f$ has the absolute automorphy.

一类与有限域上曲线相关的函数恒等式

A class of functional identities associated to curves over finite fields

本文证明有限域上射影曲线的 Pellarin L-级数函数恒等式可推广至任意亏格，方法是利用除子拓扑性质并引入伴随 shtuka 函数。

Abelian Surfaces over totally real fields are Potentially Modular

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

The functional equation for L-functions of hyperelliptic curves

We compute the $L$-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local $L$-factor and the conductor exponent at the primes of bad reduction. Most of our examples are hyperelliptic curves of genus $g\geq 2$ defined over $\mathbb{Q}$ which have semistable reduction at every prime $p$. We also treat a few more general examples of superelliptic curves.

Functional equations for Rogers dilogarithm

This paper proves a "new" family of functional equations (Eqn) for Rogers dilogarithm. These equations rely on the combinatorics of dihedral coordinates on moduli spaces of curves of genus 0, M 0,n. For n = 4 we find back the duality relation while n = 5 gives back the 5 terms relation. It is then proved that the whole family reduces to the 5 terms relation. In the author's knownledge, it is the first time that an infinite family of functional equations for the dilogarithm with an increasing number of variables (n -- 3 for (Eqn)) is reduced to the 5 terms relation.

Some functional transcendence results around the Schwarzian differential equation

This paper centers around proving variants of the Ax-Lindemann-Weierstrass (ALW) theorem for analytic functions which satisfy Schwarzian differential equations. In previous work, the authors proved the ALW theorem for the uniformizers of genus zero Fuchsian groups, and in this work, we generalize that result in several ways using a variety of techniques from model theory, galois theory and geometry.