Abstract
Ramanujan discovered some remarkable congruences satisfied by the τ-function. In retrospect, these congruences are now better understood via the theory of ℓ-adic representations. We present a short survey of these ideas and indicate the relationship between Ramanujan’s conjecture on the growth of τ(n) with the celebrated Weil conjectures, formulated by André Weil in 1949. These conjectures inspired the rapid development of modern algebraic geometry.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
P. Deligne, Formes modulaires et représentations ℓ-adiques, in Séminaire Bourbaki, Exposé 355. Lecture Notes in Mathematics, vol. 179 (Springer, Berlin, 1971)
N. Katz, An overview of Deligne’s proof of the Riemann hypothesis for varieties over finite fields. Proc. Symp. Pure Math. 28, 275–305 (1976)
J.-P. Serre, Zeta and L-functions, in Arithmetic Algebraic Geometry. Proc. Conf. Purdue University (Harper & Row, New York, 1963), pp. 82–92
T. Shioda, On elliptic modular surfaces. J. Math. Soc. Jpn. 24, 20–59 (1972)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer India
About this chapter
Cite this chapter
Murty, M.R., Murty, V.K. (2013). Ramanujan’s Conjecture and ℓ-Adic Representations. In: The Mathematical Legacy of Srinivasa Ramanujan. Springer, India. https://doi.org/10.1007/978-81-322-0770-2_3
Download citation
DOI: https://doi.org/10.1007/978-81-322-0770-2_3
Published:
Publisher Name: Springer, India
Print ISBN: 978-81-322-0769-6
Online ISBN: 978-81-322-0770-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)