TY - CHAP
T1 - Hypergeometric Functions over Finite Fields and Modular Forms
T2 - A Survey and New Conjectures
AU - Dawsey, Madeline Locus
AU - McCarthy, Dermot
N1 - Funding Information:
Acknowledgments The first author is supported by an AMS-Simons travel grant from the American Mathematical Society and the Simons Foundation. The second author is supported by a grant from the Simons Foundation (#353329, Dermot McCarthy).
Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2021
Y1 - 2021
N2 - Hypergeometric functions over finite fields were introduced by Greene in the 1980s as a finite field analogue of classical hypergeometric series. These functions, and their generalizations, naturally lend themselves to, and have been widely used in, character sum evaluations and counting points on algebraic varieties. More interestingly, perhaps, are their links to Fourier coefficients of modular forms. In this paper, we outline the main results in this area and also conjecture 13 new relations.
AB - Hypergeometric functions over finite fields were introduced by Greene in the 1980s as a finite field analogue of classical hypergeometric series. These functions, and their generalizations, naturally lend themselves to, and have been widely used in, character sum evaluations and counting points on algebraic varieties. More interestingly, perhaps, are their links to Fourier coefficients of modular forms. In this paper, we outline the main results in this area and also conjecture 13 new relations.
KW - Hypergeometric functions
KW - Modular forms
UR - http://www.scopus.com/inward/record.url?scp=85119126070&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-75425-9_4
DO - 10.1007/978-3-030-75425-9_4
M3 - Chapter
AN - SCOPUS:85119126070
T3 - Operator Theory: Advances and Applications
SP - 41
EP - 56
BT - Operator Theory
PB - Springer Science and Business Media Deutschland GmbH
ER -