Elliptic Curves with Complex Multiplication and a Character Sum

Padma, R.; Venkataraman, S.

doi:10.1006/jnth.1996.0148

Cited by 14 publications

(5 citation statements)

References 8 publications

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“…We mention, for example, the works of Williams [10], Joux-Morain [4] and Padma-Venkataraman [6], each of which takes slightly different approaches to such character sums, consolidates many previous results and contains comprehensive bibliographies. Similar results have been established by many different authors.…”

Section: Weighted Character Sumsmentioning

confidence: 93%

“…Similar results have been established by many different authors. We mention, for example, the works of Williams [10], Joux-Morain [4], and Padma-Venkataraman [6], which each take slightly different approaches to such character sums, consolidate many previous results, and contain comprehensive bibliographies. We briefly contrast these character sums to those contained in the present paper.…”

Section: Weighted Character Sumsmentioning

confidence: 97%

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Class number formulas via 2-isogenies of elliptic curves

McLeman¹,

Rasmussen

2012

Bulletin of the London Mathematical Society

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A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a 2-isogeny of elliptic curves.We consider the following point of view for Dirichlet's result. The F p -rational mor-p into two sets, those that are the image of an F p -rational point of the domain (that is, the quadratic residues) and those that are not (the non-residues). The character χ φ := (·/p) is now precisely the natural identification of the cokernel of φ with {±1} which makes the following sequence exact:Note that it is the properties of φ, not the underlying algebraic group G m , which allow this construction. This paper demonstrates that an analogous procedure, arising from a different morphism of algebraic groups, yields new character sums with similar arithmetic properties.

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Section: Weighted Character Sumsmentioning

confidence: 93%

Section: Weighted Character Sumsmentioning

confidence: 97%

Class number formulas via 2-isogenies of elliptic curves

McLeman¹,

Rasmussen

2012

Bulletin of the London Mathematical Society

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“…For (2.1), see also [41], the work of Rajwade et al, and/or [24,16,18]; the latter two employ ideas of Rajwade and Harold Stark.…”

Section: Theorem 25 (Gauss and Othersmentioning

confidence: 99%

Group order formulas for reductions of CM elliptic curves

Silverberg¹

2010

Arithmetic, Geometry, Cryptography and Coding Theory 2009

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“…More recently, elliptic curves have found applications in diverse areas such as the proof of Fermat's last theorem, factoring large integers, and in cryptography. An area worth particularly mentioning is the study of various character sums on the points of an elliptic curve, which have been used to sum primes, find generators for elliptic curve groups, and determine the structure of such groups by demonstrating the uniformity of the distribution of certain points (see, for example, [1], [2], [3], [4], [5], [8], [9], or [11]). A new direction in this area has been to examine integer-weighted character sums over elliptic curves [6], [7].…”

Section: Introductionmentioning

confidence: 99%