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Tetsuo Nakamura 《Proceedings of the American Mathematical Society》1999,127(6):1589-1595
Let be an elliptic curve over a number field such that
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .
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Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes.
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Yasutsugu Fujita Tetsuo Nakamura 《Transactions of the American Mathematical Society》2007,359(11):5505-5515
Let be an elliptic curve over a number field and its -isogeny class. We are interested in determining the orders and the types of torsion groups in . For a prime , we give the range of possible types of -primary parts of when runs over . One of our results immediately gives a simple proof of a theorem of Katz on the order of maximal -primary torsion in .
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Isogenies occur throughout the theory of elliptic curves. Recently,
the cryptographic protocols based on isogenies are considered as candidates
of quantum-resistant cryptographic protocols. Given two elliptic curves $E_1$,$E_2$ defined over a finite field $k$ with the same trace, there is a nonconstant isogeny $β$ from $E_2$ to $E_1$ defined over $k$. This study gives out the index of Hom$_k$($E_1$,$E_2$)$β$ as a nonzero left ideal in End$_k$($E_2$) and figures out the correspondence between
isogenies and kernel ideals. In addition, some results about the non-trivial
minimal degree of isogenies between two elliptic curves are also provided. 相似文献
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Mugurel A. Barcau 《Compositio Mathematica》2003,137(3):237-273
The purpose of this article is to develop the theory of differential modular forms introduced by A. Buium. The main points are the construction of many isogeny covariant differential modular forms and some auxiliary (nonisogeny covariant) forms and an extension of the classical theory of Serre differential operators on modular forms to a theory of -Serre differential operators on differential modular forms. As an application, we shall give a geometric realization of the space of elliptic curves up to isogeny. 相似文献
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Alexandru Buium 《Compositio Mathematica》2003,139(2):197-237
The quotient of a Shimura curve by the isogeny equivalence relation is not an object of algebraic geometry. The paper shows how this quotient space becomes a geometric object in a more general geometry obtained from 'usual algebraic geometry', by adjoining a new operation; this operation looks like a 'Fermat quotient' and should be viewed as an arithmetic analogue of usual derivations. 相似文献
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