40 research outputs found
Decomposition of elliptic multiple zeta values and iterated Eisenstein integrals
We describe a decomposition algorithm for elliptic multiple zeta values,
which amounts to the construction of an injective map from the algebra
of elliptic multiple zeta values to a space of iterated Eisenstein integrals.
We give many examples of this decomposition, and conclude with a short
discussion about the image of . It turns out that the failure of
surjectivity of is in some sense governed by period polynomials of
modular forms.Comment: v2, minor change
On the algebraic structure of iterated integrals of quasimodular forms
We study the algebra of iterated integrals of quasimodular
forms for , which is the smallest extension of
the algebra of quasimodular forms, which is closed under
integration. We prove that is a polynomial algebra in
infinitely many variables, given by Lyndon words on certain monomials in
Eisenstein series. We also prove an analogous result for the
-subalgebra of of iterated
integrals of modular forms.Comment: v2, minor changes, to appear in Algebra and Number Theor
Elliptic Double Zeta Values
We study an elliptic analogue of multiple zeta values, the elliptic multiple
zeta values of Enriquez, which are the coefficients of the elliptic KZB
associator. Originally defined by iterated integrals on a once-punctured
complex elliptic curve, it turns out that they can also be expressed as certain
linear combinations of indefinite iterated integrals of Eisenstein series and
multiple zeta values. In this paper, we prove that the -span of
these elliptic multiple zeta values forms a -algebra, which is
naturally filtered by the length and is conjecturally graded by the weight. Our
main result is a proof of a formula for the number of -linearly
independent elliptic multiple zeta values of lengths one and two for arbitrary
weight.Comment: 22 page
On Ecalle's and Brown's polar solutions to the double shuffle equations modulo products
Two explicit sets of solutions to the double shuffle equations modulo
products were introduced by Ecalle and Brown respectively. We place the two
solutions into the same algebraic framework and compare them. We find that they
agree up to and including depth four but differ in depth five by an explicit
solution to the linearized double shuffle equations with an exotic pole
structure.Comment: 22 pages, final version, to appear in Kyushu J. Mat
Relations between elliptic multiple zeta values and a special derivation algebra
We investigate relations between elliptic multiple zeta values and describe a
method to derive the number of indecomposable elements of given weight and
length. Our method is based on representing elliptic multiple zeta values as
iterated integrals over Eisenstein series and exploiting the connection with a
special derivation algebra. Its commutator relations give rise to constraints
on the iterated integrals over Eisenstein series relevant for elliptic multiple
zeta values and thereby allow to count the indecomposable representatives.
Conversely, the above connection suggests apparently new relations in the
derivation algebra. Under https://tools.aei.mpg.de/emzv we provide relations
for elliptic multiple zeta values over a wide range of weights and lengths.Comment: 43 pages, v2:replaced with published versio
An algebraic characterization of the Kronecker function
We characterize the generating series of extended period polynomials of
normalized Hecke eigenforms for studied by
Zagier in terms of the period relations and existence of a suitable
factorization. For this we prove a characterization of the Kronecker function
as the `fundamental solution' to the Fay identity.Comment: 11 pages, final version, minor differences in abstract and
acknowledgements compared to published versio
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
Towards algebraic iterated integrals on elliptic curves via the universal vectorial extension
For an elliptic curve defined over a field , we study
iterated path integrals of logarithmic differential forms on , the
universal vectorial extension of . These are generalizations of the
classical periods and quasi-periods of , and are closely related to multiple
elliptic polylogarithms and elliptic multiple zeta values. Moreover, if is
a finite extension of , then these iterated integrals along paths
between -rational points are periods in the sense of Kontsevich--Zagier.Comment: 12 pages; for proceedings of workshop "Various aspects of multiple
zeta values", RIMS, Kyoto, Japan, 18th-22nd. November. 201