TY - JOUR
T1 - On the singularity of the demjanenko matrix of quotients of fermat curves
AU - Fité, Francesc
AU - Shparlinski, Igor E.
PY - 2016/1/1
Y1 - 2016/1/1
N2 - Given a prime ℓ ≥3 and a positive integer k ≤ ℓ −2, one can define a matrix Dk,ℓ, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian Jac(Ck,ℓ) of a certain quotient of the Fermat curve of exponent ℓ. For a fixed ℓ, the existence of k for which Dk,ℓ is singular (equivalently, for which the rank of the Hodge group of Jac(Ck,ℓ) is not maximal) has been extensively studied in the literature. We provide an asymptotic formula for the number of such k when ℓ tends to infinity.
AB - Given a prime ℓ ≥3 and a positive integer k ≤ ℓ −2, one can define a matrix Dk,ℓ, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian Jac(Ck,ℓ) of a certain quotient of the Fermat curve of exponent ℓ. For a fixed ℓ, the existence of k for which Dk,ℓ is singular (equivalently, for which the rank of the Hodge group of Jac(Ck,ℓ) is not maximal) has been extensively studied in the literature. We provide an asymptotic formula for the number of such k when ℓ tends to infinity.
KW - Demjanenko matrix
KW - Fermat curve
KW - Sato-tate conjecture
UR - http://www.scopus.com/inward/record.url?scp=84945275089&partnerID=8YFLogxK
UR - http://purl.org/au-research/grants/arc/DP130100237
U2 - 10.1090/proc12717
DO - 10.1090/proc12717
M3 - Article
AN - SCOPUS:84945275089
VL - 144
SP - 55
EP - 63
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
SN - 0002-9939
IS - 1
ER -