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 PARAMETRIZING ELLIPTIC CURVES BY MODULAR UNITS

2015 ◽  
Vol 100 (1) ◽  
pp. 33-41 ◽  
Author(s):  
FRANÇOIS BRUNAULT
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Recent Work ◽  
Short Note ◽  
Torsion Subgroup ◽  
Modular Functions ◽  
Open Questions ◽  
Modular Units

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

scholarly journals The distribution of the maximum of partial sums of Kloosterman sums and other trace functions

2021 ◽  
Vol 157 (7) ◽  
pp. 1610-1651
Author(s):  
Pascal Autissier ◽  
Dante Bonolis ◽  
Youness Lamzouri
Keyword(s):  
Distribution Function ◽  
Recent Work ◽  
Character Sums ◽  
Kloosterman Sums ◽  
Partial Sums ◽  
Optimal Range ◽  
Uniform Estimates ◽  
The Third ◽  
Distribution Of The Maximum ◽  
Complex Valued

In this paper, we investigate the distribution of the maximum of partial sums of families of $m$ -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of $m$ -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.

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]

scholarly journals Characteristic varieties and logarithmic differential 1-forms

2010 ◽  
Vol 146 (1) ◽  
pp. 129-144 ◽  
Author(s):  
Alexandru Dimca
Keyword(s):  
Elliptic Curve ◽  
Recent Work ◽  
Configuration Space ◽  
Zero Set ◽  
Characteristic Variety ◽  
Local Systems ◽  
Complex Variety ◽  
Rank One ◽  
Characteristic Varieties ◽  
Resonance Varieties

AbstractWe introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety M. A logarithmic resonance variety is also considered and, as an application, we determine the first characteristic variety of the configuration space of n distinct labeled points on an elliptic curve. Finally, for a logarithmic 1-form α on M we investigate the relation between the resonance degree of α and the codimension of the zero set of α on a good compactification of M. This question was inspired by the recent work by Cohen, Denham, Falk and Varchenko.

scholarly journals Seventh power moments of Kloosterman sums

2010 ◽  
Vol 175 (1) ◽  
pp. 349-362 ◽  
Author(s):  
Ronald Evans
Keyword(s):  
Kloosterman Sums ◽  
Power Moments

scholarly journals Galois representations attached to moments of Kloosterman sums and conjectures of Evans

2014 ◽  
Vol 151 (1) ◽  
pp. 68-120 ◽  
Author(s):  
Zhiwei Yun ◽  
Christelle Vincent
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Kloosterman Sums ◽  
Symmetric Power ◽  
Reductive Groups ◽  
Local Systems ◽  
Symmetric Powers ◽  
Exterior Powers ◽  
Power Moments ◽  
Tensor Powers

AbstractKloosterman sums for a finite field $\mathbb{F}_{p}$ arise as Frobenius trace functions of certain local systems defined over $\mathbb{G}_{m,\mathbb{F}_{p}}$. The moments of Kloosterman sums calculate the Frobenius traces on the cohomology of tensor powers (or symmetric powers, exterior powers, etc.) of these local systems. We show that when $p$ ranges over all primes, the moments of the corresponding Kloosterman sums for $\mathbb{F}_{p}$ arise as Frobenius traces on a continuous $\ell$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ that comes from geometry. We also give bounds on the ramification of these Galois representations. All of this is done in the generality of Kloosterman sheaves attached to reductive groups introduced by Heinloth, Ngô and Yun [Ann. of Math. (2) 177 (2013), 241–310]. As an application, we give proofs of conjectures of Evans [Proc. Amer. Math. Soc. 138 (2010), 517–531; Israel J. Math. 175 (2010), 349–362] expressing the seventh and eighth symmetric power moments of the classical Kloosterman sum in terms of Fourier coefficients of explicit modular forms. The proof for the eighth symmetric power moment conjecture relies on the computation done in Appendix B by C. Vincent.

scholarly journals Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals

2011 ◽  
Vol 150 (3) ◽  
pp. 439-458 ◽  
Author(s):  
KEVIN JAMES ◽  
ETHAN SMITH
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Fixed Number ◽  
Prime Ideals ◽  
Galois Extensions ◽  
Trace Of Frobenius

AbstractLet K be a fixed number field, assumed to be Galois over ℚ. Let r and f be fixed integers with f positive. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r. Except in the case f = 2, we show that ‘on average,’ the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

Codes Associated with and Power Moments of Kloosterman Sums

Integers ◽  
2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Dae San Kim
Keyword(s):  
Linear Codes ◽  
Kloosterman Sums ◽  
Binary Linear Codes ◽  
Power Moments

Abstract.We shall construct three binary linear codes

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