Related papers: Defining equations of $X_0(2^{2n})$

We study the minimal Weierstrass equations for genus 2 curves defined over a ring of integers $\mathcal O_{\mathbb F}$. This is done via reduction theory and Julia invariant of binary sextics. We show that when the binary sextics has extra…

We show that the Mahler measure of a defining equation of the modular curve $X_1(13)$ is equal to the derivative at $s=0$ of the $L$-function of a cusp form of weight 2 and level 13 with integral Fourier coefficients. The proof combines…

We provide a programmable recursive algorithm for the $\mathbb{S}_n$-representations on the cohomology of the moduli spaces $\overline{\mathcal M}_{0,n}$ of $n$-pointed stable curves of genus 0. As an application, we find explicit inductive…

We use covariants of binary sextics to describe the structure of modules of scalar-valued or vector-valued Siegel modular forms of degree 2 with character, over the ring of scalar-valued Siegel modular forms of even weight. For a modular…

The modular discriminant $\Delta$ is known to structure the sequence of modular forms $(M_{2k}(SL_2(\mathbb{Z})))_{k\in \; \mathbb{N}^*}$ at level $1$.\\ For all positive integer $N$, we define a strong modular unit $\Delta_N$ at level $N$…

If $X$ is a variety with an additional structure $\xi$, such as a marked point, a divisor, a polarization, a group structure and so forth, then it is possible to study whether the pair $(X,\xi)$ is defined over the field of moduli. There…

We derive formulas for $(i)$ the number of toroidal $n\times n$ binary arrays, allowing rotation of rows and/or columns as well as matrix transposition, and $(ii)$ the number of toroidal $n\times n$ binary arrays, allowing rotation and/or…

We classify the three-dimensional representations of the modular group that are reducible but indecomposable, and their associated spaces of holomorphic vector-valued modular forms. We then demonstrate how such representations may be…

2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed curves where several sheets meet. They arise in the study of categorical invariants of 3-manifolds and may have applications to topological data…

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…

We define a complete invariant for doodles on a 2-sphere which takes values in series of chord diagrams of certain type. The coefficients at the diagrams with $n$ chords are finite type invariants of doodles of order at most $2n$.

This paper contains some basic results on 2-groupoids, with special emphasis on computing derived mapping 2-groupoids between 2-groupoids and proving their invariance under strictification. Some of the results proven here are presumably…

The non-linear second order Born-Infeld equation is reduced to a simpler first order complex equation, which can be trivially solved for the coordinates as functions of the field. Each solution is determined by the choice of a holomorphic…

In this paper we develop a classification of real functions based on growth rates of repeated iteration. We show how functions are naturally distinguishable when considering inverses of repeated iterations. For example, $n+2\to 2n\to 2^n\to…

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion…

We show that the defining ideal of every monomial curve in the affine or projective three-dimensional space can be set-theoretically defined by three binomial equations, two of which set-theoretically define a determinantal ideal generated…

Let $E$ be a non-CM elliptic curve defined over $\mathbb {Q}$. Fix an algebraic closure $\overline{\mathbb {Q}}$ of $\mathbb {Q}$. We get a Galois representation \[\rho_E \colon Gal(\overline{\mathbb {Q}}/\mathbb {Q}) \to GL_2(\hat{\mathbb…

We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic…

We describe a simple algorithm for computing the canonical basis of any irreducible finite-dimensional $U_{q}(so_{2n+1})$ or $U_{q}(so_{2n})$-module.

We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…