TY - JOUR
T1 - Hypergeometric sheaves and finite symplectic and unitary groups
AU - Katz, Nicholas M.
AU - Tiep, Pham Huu
N1 - Publisher Copyright:
© 2021, International Press, Inc.. All rights reserved.
PY - 2021
Y1 - 2021
N2 - We construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups Sp2n(q) for any odd n ≥ 3, for q any power of an odd prime p. We construct other hypergeometric sheaves whose geometric monodromy groups are the finite unitary groups GUn(q), for any even n ≥ 2, for q any power of any prime p. Suitable Kummer pullbacks of these sheaves yield local systems on A1, whose geometric monodromy groups are Sp2n(q), respectively SUn(q), in their total Weil representation of degree qn, and whose trace functions are simple-to-remember one-parameter families of two-variable exponential sums. The main novelty of this paper is three-fold. First, it treats unitary groups GUn(q) withn even via hypergeometric sheaves for the first time. Second, in both the symplectic and the unitary cases, it uses a maximal torus which is a product of two sub-tori to furnish a generator of local monodromy at 0. Third, this is the first natural occurrence of families of two-variable exponential sums in the context of finite classical groups.
AB - We construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups Sp2n(q) for any odd n ≥ 3, for q any power of an odd prime p. We construct other hypergeometric sheaves whose geometric monodromy groups are the finite unitary groups GUn(q), for any even n ≥ 2, for q any power of any prime p. Suitable Kummer pullbacks of these sheaves yield local systems on A1, whose geometric monodromy groups are Sp2n(q), respectively SUn(q), in their total Weil representation of degree qn, and whose trace functions are simple-to-remember one-parameter families of two-variable exponential sums. The main novelty of this paper is three-fold. First, it treats unitary groups GUn(q) withn even via hypergeometric sheaves for the first time. Second, in both the symplectic and the unitary cases, it uses a maximal torus which is a product of two sub-tori to furnish a generator of local monodromy at 0. Third, this is the first natural occurrence of families of two-variable exponential sums in the context of finite classical groups.
KW - Finite simple groups
KW - Hypergeometric sheaves
KW - Local systems
KW - Monodromy groups
KW - Weil representations
UR - http://www.scopus.com/inward/record.url?scp=85171966397&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85171966397&partnerID=8YFLogxK
U2 - 10.4310/CJM.2021.v9.n3.a2
DO - 10.4310/CJM.2021.v9.n3.a2
M3 - Article
AN - SCOPUS:85171966397
SN - 2168-0930
VL - 9
SP - 577
EP - 691
JO - Cambridge Journal of Mathematics
JF - Cambridge Journal of Mathematics
IS - 3
ER -