Related papers: Mathieu equation and Elliptic curve

We determine, for an elliptic curve $E/\mathbb{Q}$, all the possible torsion groups $E(K)_{tors}$, where $K$ is the compositum of all $\mathbb{Z}_{p}$-extensions of $\mathbb{Q}$. Furthermore, we prove that for an elliptic curve…

We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is…

Here we carefully construct an equivalence between the derived category of coherent sheaves on an elliptic curve and a version of the Fukaya category on its mirror. This is the most accessible case of homological mirror symmetry. We also…

We consider a linear differential system of Mathieu equations with periodic coefficients over periodic closed orbits and we prove that, arbitrarily close to this system, there is a linear differential system of Hamiltonian damped Mathieu…

This paper describes the Floquet theory for quaternion-valued differential equations (QDEs). The Floquet normal form of fundamental matrix for linear QDEs with periodic coefficients is presented and the stability of quaternionic periodic…

The $\varepsilon$-form of a system of differential equations for Feynman integrals has led to tremendeous progress in our abilities to compute Feynman integrals, as long as they fall into the class of multiple polylogarithms. It is…

This talk reviews Feynman integrals, which are associated to elliptic curves. The talk will give an introduction into the mathematics behind them, covering the topics of elliptic curves, elliptic integrals, modular forms and the moduli…

The modular degree m_E of an elliptic curve E/Q is the minimal degree of any surjective morphism X_0(N) -> E, where N is the conductor of E. We give a necessarily set of criteria for m_E to be odd. Specializing to N prime our results imply…

We develop an algorithm to test whether a non-CM elliptic curve $E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing…

We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the…

One of the many interesting algebraic objects associated to a given rational elliptic curve, $E$, is its full-torsion representation $\rho_E:\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\to\mathrm{GL}_2(\hat{\mathbf{Z}})$. Generalizing this…

We consider generalizations of Dunkl's differential-difference operators associated with groups generated by reflections. The commutativity condition is equivalent to certain functional equations. These equations are solved in many cases.…

We investigate a notion of "higher modularity" for elliptic curves over function fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, we say that $E$ is $r$-modular when there is an algebraic correspondence between a stack of…

We work out an example, for a CM elliptic curve E defined over a real quadratic field F, of Zagier's conjecture. This relates L(E,2) to values of the elliptic dilogarithm function at a divisor in the Jacobian of E which arises from…

For an elliptic curve E over a number field K, we prove that the algebraic rank of E goes up in infinitely many extensions of K obtained by adjoining a cube root of an element of K. As an example, we briefly discuss E=X_1(11) over Q, and…

We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contribution of the susceptibility of the Ising model for $\, n \le 6$, are operators "associated with elliptic curves". Beyond the…

In this paper, we find a power series expansion of the invariant differential $\omega_E$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we introduce…

A complex elliptic curve $E$ can be defined as the quotient of the analytic space $\mathbb{C}^*$ by a discrete action of the cyclic group $q^{\mathbb{Z}}$ for $\vert q\vert \neq 1$. We study the boundary case when $\vert q\vert =1$, which…

Let $\{E_{(p,q)}\}$ be a family of elliptic curves over a rational field such that we have $E_{(p,q)} : y^2 = x^3 - p^2x + q^2$, where $p$ and $q$ are prime numbers greater than five. Earlier work showed that the elliptic curve $E_{(p,q)}$…

Let $[K:\mathbb{Q}]=p$ be a prime number and let $E/K$ be an elliptic curve with $j(E) \in \mathbb{Q}$. We determine the all possibilities for $E(K)_{tors}$. We obtain these results by studying Galois representations of $E$ and of it's…