Universität zu Köln

Lobrich, Steffen, Ma, Wenjun and Thorner, Jesse ORCID: 0000-0003-0623-3473 (2017). Special values of motivic L-functions and zeta-polynomials for symmetric powers of elliptic curves. Res. Math. Sci., 4. CHAM: SPRINGER INTERNATIONAL PUBLISHING AG. ISSN 2197-9847

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Abstract

Let M be a pure motive over Q of odd weight w >= 3, even rank d >= 2, and global conductor N whose L-function L(s, M) coincides with the L-function of a self-dual algebraic tempered cuspidal symplectic representation of GL(d)(A(Q)). We show that a certain polynomial which generates special values of L(s, M) (including all of the critical values) has all of its zeros equidistributed on the unit circle, provided that N or w are sufficiently large with respect to d. These special values have arithmetic significance in the context of the Bloch-Kato conjecture. We focus on applications to symmetric powers of semistable elliptic curves over Q. Using the Rodriguez-Villegas transform, we use these results to construct large classes of zeta-polynomials (in the sense of Manin) arising from symmetric powers of semistable elliptic curves; these polynomials have a functional equation relating s ->. 1 -s, and all of their zeros on the line Kappa(s) = 1/2.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Lobrich, Steffen UNSPECIFIED UNSPECIFIED UNSPECIFIED Ma, Wenjun UNSPECIFIED UNSPECIFIED UNSPECIFIED Thorner, Jesse UNSPECIFIED orcid.org/0000-0003-0623-3473 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-208138
DOI:	10.1186/s40687-017-0114-0
Journal or Publication Title:	Res. Math. Sci.
Volume:	4
Date:	2017
Publisher:	SPRINGER INTERNATIONAL PUBLISHING AG
Place of Publication:	CHAM
ISSN:	2197-9847
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language PERIOD POLYNOMIALS; EXTERIOR SQUARE; ZEROS Multiple languages Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/20813