Related papers: A construction of single-valued elliptic polylogar…

In this work, singular surfaces are obtained from smooth orientable closed surfaces by applying three basic simple loop operations, collapsing operation, zipping operation and double loop identification, each of which produces different…

We prove that under certain explicit conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of elliptic curve $L$-values and Bloch-Wigner dilogarithmmic values, conditionally on Beilinson's conjecture. In…

We formulate a multi-variable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curves A over number fields K. It generalises the one-variable conjecture of Mazur-Tate-Teitelbaum, who studied the case K=Q and the…

The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$…

For an elliptic curve $E$ over $K$, the Birch and Swinnerton-Dyer conjecture predicts that the rank of Mordell-Weil group $E(K)$ is equal to the order of the zero of $L(E_{/ K},s)$ at $s=1$. In this paper, we shall give a proof for elliptic…

For each $t\in\mathbb{Q}\setminus\{-1,0,1\}$, define an elliptic curve over $\mathbb{Q}$ by \begin{align*} E_t:y^2=x(x+1)(x+t^2). \end{align*} Using a formula for the root number $W(E_t)$ as a function of $t$ and assuming some standard…

A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in block diagonal matrix form (resulting in the Wedderburn decomposition), a general form of polyadic…

We relate one-loop scattering amplitudes of massless open- and closed-string states at the level of their low-energy expansion. The modular graph functions resulting from integration over closed-string punctures are observed to follow from…

Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the…

A simple property of the integrals over the hyperelliptic surfaces of arbitrary genus is observed. Namely, the derivatives of these integrals with respect to the branching points are given by the linear combination of the same integrals. We…

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.…

Orbifold elliptic genus and elliptic genus of singular varieties are introduced and relation between them is studied. Elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of…

We study integrals appearing in intermediate steps of one-loop open-string amplitudes, with multiple unintegrated punctures on the $A$-cycle of a torus. We construct a vector of such integrals which closes after taking a total differential…

We generalize the definition of orbifold elliptic genus, and introduce orbifold genera of chromatic level h, using h-tuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove…

Multiple zeta values arise as special values of polylogarithms defined on Riemann surfaces of various genera. Building on the vast knowledge for classical and elliptic multiple zeta values, we explore a canonical extension of the formalism…

We study uniqueness of Dirichlet problems of second order divergence-form elliptic systems with transversally independent coefficients on the upper half-space in absence of regularity of solutions. To this end, we develop a substitute for…

The Deligne-Ogus-Shioda theorem guarantees the existence of isomorphisms between products of supersingular elliptic curves over finite fields. In this paper, we present methods for explicitly computing these isomorphisms in polynomial time,…

This paper explores a construction of the elliptic classes of the Springer resolution using the periodic Hecke module. The module is established by employing the Poincar\'e line bundle over the product of the abelian variety of elliptic…

Let $k$ denote an algebraically closed field. We revisit a construction of the author of families of elliptic curves over the rational function field $k(t)$. Combining a combinatorial analysis with a rank formula of Ulmer we prove that, for…

We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an…