50 research outputs found
Double Shuffle and Kashiwara-Vergne Lie algebras
We prove that the double shuffle Lie algebra ds, dual to the space of new
formal multiple zeta values, injects into the Kashiwara-Vergne Lie algebra krv
defined and studied by Alekseev-Torossian. The proof is based on a
reformulation of the definition of krv, and uses a theorem of Ecalle on a
property of elements of ds.Comment: 18 page
Period polynomial relations between double zeta values
The even weight period polynomial relations in the double shuffle Lie algebra
were discovered by Ihara, and completely classified by the
second author by relating them to restricted even period polynomials associated
to cusp forms on . In an article published in the
same year, Gangl, Kaneko and Zagier displayed certain linear combinations of
odd-component double zeta values which are equal to scalar multiples of simple
zeta values in even weight, and also related them to restricted even period
polynomials. In this paper, we relate the two sets of relations, showing how
they can be deduced from each other by duality.Comment: 13 page
Explicit realisations of subgroups of GL2(F3) as Galois groups
AbstractLet F be a number field and K an extension of F with Galois group D4 (resp. A4 or S4). In this article we explicitly construct all of the quadratic extensions L of K having Galois group D̃4, the Sylow subgroup of GL2(F3) (resp. SL2(F3) or GL2(F3)) over F, whenever such extensions exist
Elliptic multizetas and the elliptic double shuffle relations
We define an elliptic generating series whose coefficients, the elliptic
multizetas, are related to the elliptic analogues of multiple zeta values
introduced by Enriquez as the coefficients of his elliptic associator; both
sets of coefficients lie in , the ring of functions
on the Poincar\'e upper half-plane . The elliptic multizetas
generate a -algebra which is an elliptic analogue of
the algebra of multiple zeta values. Working modulo , we show that the
algebra decomposes into a geometric and an arithmetic part and
study the precise relationship between the elliptic generating series and the
elliptic associator defined by Enriquez. We show that the elliptic multizetas
satisfy a double shuffle type family of algebraic relations similar to the
double shuffle relations satisfied by multiple zeta values. We prove that these
elliptic double shuffle relations give all algebraic relations among elliptic
multizetas if (a) the classical double shuffle relations give all algebraic
relations among multiple zeta values and (b) the elliptic double shuffle Lie
algebra has a certain natural semi-direct product structure analogous to that
established by Enriquez for the elliptic Grothendieck-Teichm\"uller Lie
algebra.Comment: major revision, to appear in: Int. Math. Res. No