Related papers: Motivic pieces of curves: $L$-functions and period…

Given an elliptic curve ${\mathcal E}$ over a field $K$ it is a challenging problem to write down explicit elements of its endomorphism ring ${\rm End}({\mathcal E});$ the problem amounts to find all possible solutions to a functional…

In this paper we introduce an automorphic variant of the Deligne conjecture for tensor product of two motives over a quadratic imaginary field. On one hand, we define some motivic periods and rewrite the Deligne conjecture in terms of these…

Let $K$ be a field of characteristic different from $2$ and let $E$ be an elliptic curve over $K$, defined either by an equation of the form $y^{2} = f(x)$ with degree $3$ or as the Jacobian of a curve defined by an equation of the form…

We construct certain elements in the integral motivic cohomology group $H^3_{{\cal M}}(E \times E',\Q(2))_{\ZZ}$, where $E$ and $E'$ are elliptic curves over $\Q$. When $E$ is not isogenous to $E'$ these elements are analogous to…

We study hyperelliptic curves C with an action of an affine group of automorphisms G. We establish a closed form expression for the quotient curve C/G and for the first etale cohomology group of C as a representation of G. The motivation…

We relate analytically defined deformations of modular curves and modular forms from the literature to motivic periods via cohomological descriptions of deformation theory. Leveraging cohomological vanishing results, we prove the existence…

We introduce a new approach of computing the automorphism group and the field of moduli of points $\p=[C]$ in the moduli space of hyperelliptic curves $\H_g$. Further, we show that for every moduli point $\p \in \H_g(L)$ such that the…

Let $A/\mathbb{Q}$ be a Jacobian variety and let $F$ be a totally real, tamely ramified, abelian number field. Given a character $\psi$ of $F/\mathbb{Q}$, Deligne's Period Conjecture asserts the algebraicity of the suitably normalised value…

Fix $d \ge 2$ and a field $k$ such that $\mathrm{char}~k \nmid d$. Assume that $k$ contains the $d$th roots of $1$. Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^d+c$…

We analyze log-algebraic power series identities for formal groups of elliptic curves over $\mathbb{Q}$ which arise from modular parametrizations. We further investigate applications to special values of elliptic curve $L$-functions.

We construct some integral elements in the motivic cohomology of the Hesse cubic curves and express their regulators in terms of generalized hypergeometric functions and Kamp\'e de F\'eriet hypergeometric functions. By using these…

In our recent work with Mat Rogers on resolving some Boyd's conjectures on two-variate Mahler measures, a new analytical machinery was introduced to write the values $L(E,2)$ of $L$-series of elliptic curves as periods in the sense of…

In recent years, Rogers and Zudilin developed a method to write $L$-values attached to elliptic curves as periods. In order to apply this method to a broader collection of $L$-values, we study Eisenstein series and determine their Fourier…

We describe a system of plane algebraic curves defined over \Z, attached naturally to the exponential function. On of these is a remarkable curve of degree 6 that has genus equal to 1. As the sectic curve has rational points, it is an…

Let $X$ be a smooth, projective, geometrically irreducible curve of genus at least two defined over a number field $K$. We prove that there is an algorithm that determines whether $X$ has a $K$-rational point if Grothendieck's section…

$L-$series attached to two classical families of elliptic curves with complex multiplications are studied over number fields, formulae for their special values at $s=1, $ bound of the values, and criterion of reaching the bound are given.…

Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the…

For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ and $K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$. We provide an explicit…

Numerical tools for computation of $\wp$-functions, also known as Kleinian, or multiply periodic, are proposed. In this connection, computation of periods of the both first and second kinds is reconsidered. An analytical approach to…

This paper is the first in a series of two dedicated to the study of period relations of the type $$ L(\frac{1}{2}+k,\Pi)\;\in\;(2\pi i)^{d\cdot k}\Omega_{(-1)^k}{\mathbb Q}(\Pi),\quad \frac{1}{2}+k\;\text{critical}, $$ for certain…