Chapter Four. Computing the Trace of Frobenius

2019 ◽  
pp. 204-238
Keyword(s):  
Trace Of Frobenius

scholarly journals Shintani descent for algebraic groups and almost characters of unipotent groups

2016 ◽  
Vol 152 (8) ◽  
pp. 1697-1724 ◽  
Author(s):  
Tanmay Deshpande
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Algebraic Groups ◽  
Unipotent Group ◽  
Connected Component ◽  
Unipotent Groups ◽  
Character Sheaves ◽  
Treat All ◽  
Trace Of Frobenius

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

Supercharacters and mixed moments of Kloosterman sums

2018 ◽  
Vol 14 (04) ◽  
pp. 1023-1032 ◽  
Author(s):  
Ángel Chávez ◽  
George Todd
Keyword(s):  
Elliptic Curve ◽  
Recent Work ◽  
Kloosterman Sums ◽  
Fourth Degree ◽  
Mixed Power ◽  
Power Moments ◽  
Supercharacter Theory ◽  
Trace Of Frobenius

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on [Formula: see text]. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

scholarly journals Local models for Weil-restricted groups

2016 ◽  
Vol 152 (12) ◽  
pp. 2563-2601 ◽  
Author(s):  
Brandon Levin
Keyword(s):  
Hecke Algebra ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Local Models ◽  
Central Function ◽  
Nearby Cycles ◽  
Weil Restriction ◽  
Trace Of Frobenius ◽  
Invent Math

We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math.194(2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when$p\geqslant 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.

scholarly journals Constructing eliptic curves with zero trace of Frobenius endomorphism

2018 ◽  
Vol 20 (1) ◽  
Author(s):  
Руслан Вячеславович Скуратовський
Keyword(s):  
Frobenius Endomorphism ◽  
Trace Of Frobenius

scholarly journals 3F2 HYPERGEOMETRIC SERIES AND PERIODS OF ELLIPTIC CURVES

2010 ◽  
Vol 06 (03) ◽  
pp. 461-470 ◽  
Author(s):  
DERMOT McCARTHY
Keyword(s):  
Elliptic Curves ◽  
Hypergeometric Series ◽  
The Real ◽  
Gaussian Hypergeometric Series ◽  
Trace Of Frobenius

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.

scholarly journals Elliptic curve variants of the least quadratic nonresidue problem and Linnik’s theorem

2017 ◽  
Vol 14 (01) ◽  
pp. 255-288
Author(s):  
Evan Chen ◽  
Peter S. Park ◽  
Ashvin A. Swaminathan
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Complex Multiplication ◽  
Symmetric Power ◽  
Explicit Result ◽  
Generalized Riemann Hypothesis ◽  
Quadratic Nonresidue ◽  
Trace Of Frobenius

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-nonisogenous, semistable elliptic curves over [Formula: see text], having respective conductors [Formula: see text] and [Formula: see text] and both without complex multiplication. For each prime [Formula: see text], denote by [Formula: see text] the trace of Frobenius. Assuming the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power [Formula: see text]-functions [Formula: see text] where [Formula: see text], we prove an explicit result that can be stated succinctly as follows: there exists a prime [Formula: see text] such that [Formula: see text] and [Formula: see text] This improves and makes explicit a result of Bucur and Kedlaya. Now, if [Formula: see text] is a subinterval with Sato–Tate measure [Formula: see text] and if the symmetric power [Formula: see text]-functions [Formula: see text] are functorial and satisfy GRH for all [Formula: see text], we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime [Formula: see text] such that [Formula: see text] and [Formula: see text]

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