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Published in:

2021 | OriginalPaper | Chapter

# Linear Dependence Among Hecke Eigenvalues

Author : Dohyeong Kim

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## Abstract

We prove an explicit upper bound on the absolute value of the coefficients of a non-trivial integral linear relation among Hecke eigenvalues of a given cuspidal eigenform. Our motivation lies in its algorithmic application. For any fixed positive integer n, the bound established here yields an algorithm that computes cuspidal Hecke eigenforms with a given weight k whose Hecke eigenvalues generate a number field of degree n. The resulting algorithm reduces to Cremona’s when n = 1 and k = 2.
Footnotes
1
Indeed, it is isomorphic to the Hecke field of any eigenform in $$W \otimes _{\mathbb {Q}}\mathbb {C}$$.

2
As explained in the preceding paragraph, the kernel of T λ may be strictly bigger than W, while being harmless to the purpose.

Literature
[1]
H. F. Blichfeldt. The minimum value of quadratic forms, and the closest packing of spheres. Math. Ann., 101(1):605–608, 1929.
[2]
Henri Cohen and Fredrik Strömberg. Modular forms, volume 179 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2017. A classical approach.
[3]
Henry Cohn and Noam Elkies. New upper bounds on sphere packings. I. Ann. of Math. (2), 157(2):689–714, 2003.
[4]
J. H. Conway and N. J. A. Sloane. Sphere packings, lattices and groups, volume 290 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, third edition, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov.
[5]
J. E. Cremona. Algorithms for modular elliptic curves. Cambridge University Press, Cambridge, second edition, 1997. MATH
[6]
John Cremona. The elliptic curve database for conductors to 130000. In Algorithmic number theory, volume 4076 of Lecture Notes in Comput. Sci., pages 11–29. Springer, Berlin, 2006.
[7]
J. C. Lagarias, H. W. Lenstra, Jr., and C.-P. Schnorr. Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica, 10(4):333–348, 1990.
[8]
Phong Q. Nguyen and Brigitte Valle. The LLL Algorithm: Survey and Applications. Springer Publishing Company, Incorporated, 1st edition, 2009.
[9]
A. M. Odlyzko. Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Sém. Théor. Nombres Bordeaux (2), 2(1):119–141, 1990.
[10]
B. L. van der Waerden. Die Reduktionstheorie der positiven quadratischen Formen. Acta Math., 96:265–309, 1956.