A rigid local system with monodromy group the big Conway group $2.\mathsf {Co}_1$ and two others with monodromy group the Suzuki group $6.{{Suz}}$

Katz, Nicholas M.; Rojas‐León, Antonio; Tiep, Pham Huu

doi:10.1090/tran/7967

Cited by 8 publications

(15 citation statements)

References 15 publications

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“…A hypergeometric sheaf

scriptH

is said to be geometrically induced if the representation of the geometric monodromy group

G_{geom}

scriptH

can be induced from a proper subgroup. We note by [10, Proposition 1.2] that this occurs precisely when

scriptH

is either Kummer induced or Belyi induced , as described in (i), respectively, in (ii), of [10, Proposition 1.2]. In particular, we have the following result: Proposition If a geometrically irreducible hypergeometric sheaf

scriptH

of type

(n, m)

with

n > m > 0

, or with

m > n > 0

, is Belyi induced, then

n - m

is prime to

p

, and is divisible by

p - 1

.…”

Section: (Ab) Generalitiesmentioning

confidence: 99%

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Exponential sums and total Weil representations of finite symplectic and unitary groups

Katz

Tiep

2020

Proc. London Math. Soc.

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We construct explicit local systems on the affine line in characteristic p>2, whose geometric monodromy groups are the finite symplectic groups normalSp2nfalse(qfalse) for all n⩾2, and others whose geometric monodromy groups are the special unitary groups normalSUnfalse(qfalse) for all odd n⩾3, and q any power of p, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one‐parameter families of exponential sums of a very simple, that is, easy to remember, form. We also exhibit hypergeometric sheaves on Gm, whose geometric monodromy groups are the finite symplectic groups normalSp2nfalse(qfalse) for any n⩾2, and others whose geometric monodromy groups are the finite general unitary groups normalGUnfalse(qfalse) for any odd n⩾3.

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“…A hypergeometric sheaf

scriptH

is said to be geometrically induced if the representation of the geometric monodromy group

G_{geom}

scriptH

can be induced from a proper subgroup. We note by [10, Proposition 1.2] that this occurs precisely when

scriptH

scriptH

of type

(n, m)

with

n > m > 0

, or with

m > n > 0

, is Belyi induced, then

n - m

is prime to

p

, and is divisible by

p - 1

.…”

Section: (Ab) Generalitiesmentioning

confidence: 99%

“…Proof In the notations of [10, Proposition 1.2] (whose

A

and

B

have nothing to do with ours), when

n > m > 0

, we have

n = A + B

and either

A + B

A

B

d_{0} p^{r}

with

s r ⩾ 1

and

d_{0}

prime to

p

. In these cases,

m

is either

d_{0}

, or

d_{0} + B

, or

A + d_{0}

.…”

Section: (Ab) Generalitiesmentioning

confidence: 99%

Exponential sums and total Weil representations of finite symplectic and unitary groups

Katz

Tiep

2020

Proc. London Math. Soc.

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“…In the case of a hypergeometric sheaf H with m > 0, primitivity is less easy to determine at first glance, because there is also the possibility of Belyi induction, cf. [KRLT3,Proposition 1.2]. It is known that an H of type (D, 1) is primitive unless D is a power of p, cf.…”

Section: Introductionmentioning

confidence: 99%

“…It is known that an H of type (D, 1) is primitive unless D is a power of p, cf. [KRLT3,Cor 1.3]. It is also known [KRLT3,Proposition 1.4] that an H of type (D, m), with D > m ≥ 2 and D a power of p, is primitive.…”

Section: Introductionmentioning

confidence: 99%