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.

UR - http://www.scopus.com/inward/record.url?scp=85087011571&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85087011571&partnerID=8YFLogxK

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 -