TY - JOUR
T1 - Ramanujan graphs and exponential sums over function fields
AU - Sardari, Naser T.
AU - Zargar, Masoud
N1 - Funding Information:
We are grateful to the anonymous referee for many useful comments regarding a previous version of this paper. N.T. Sardari's work was supported partially by the National Science Foundation under Grant No. DMS-2015305 and is grateful to Max Planck Institute for Mathematics in Bonn and the Institute for Advanced Study for their hospitality and financial support. M. Zargar was supported by SFB1085: Higher invariants at the University of Regensburg.
Funding Information:
We are grateful to the anonymous referee for many useful comments regarding a previous version of this paper. N.T. Sardari's work was supported partially by the National Science Foundation under Grant No. DMS-2015305 and is grateful to Max Planck Institute for Mathematics in Bonn and the Institute for Advanced Study for their hospitality and financial support. M. Zargar was supported by SFB1085 : Higher invariants at the University of Regensburg .
Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2020/12
Y1 - 2020/12
N2 - We prove that q+1-regular Morgenstern Ramanujan graphs Xq,g (depending on g∈Fq[t]) have diameter at most [Formula presented] (at least for odd q and irreducible g) provided that a twisted Linnik–Selberg conjecture over Fq(t) is true. This would break the 30 year-old upper bound of 2logq|Xq,g|+O(1), a consequence of a well-known upper bound on the diameter of regular Ramanujan graphs proved by Lubotzky, Phillips, and Sarnak using the Ramanujan bound on Fourier coefficients of modular forms. We also unconditionally construct infinite families of Ramanujan graphs that prove that [Formula presented] cannot be improved.
AB - We prove that q+1-regular Morgenstern Ramanujan graphs Xq,g (depending on g∈Fq[t]) have diameter at most [Formula presented] (at least for odd q and irreducible g) provided that a twisted Linnik–Selberg conjecture over Fq(t) is true. This would break the 30 year-old upper bound of 2logq|Xq,g|+O(1), a consequence of a well-known upper bound on the diameter of regular Ramanujan graphs proved by Lubotzky, Phillips, and Sarnak using the Ramanujan bound on Fourier coefficients of modular forms. We also unconditionally construct infinite families of Ramanujan graphs that prove that [Formula presented] cannot be improved.
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U2 - 10.1016/j.jnt.2020.05.010
DO - 10.1016/j.jnt.2020.05.010
M3 - Article
AN - SCOPUS:85087011571
SN - 0022-314X
VL - 217
SP - 44
EP - 77
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -