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2. On the ranks of elliptic curves in families of quadratic twists over number fields
- Creator:
- Lee, Jung-Jo
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- elliptic curve, rank, and quadratic twist
- Language:
- English
- Description:
- A conjecture due to Honda predicts that given any abelian variety over a number field $K$, all of its quadratic twists (or twists of a fixed order in general) have bounded Mordell-Weil rank. About 15 years ago, Rubin and Silverberg obtained an analytic criterion for Honda's conjecture for a family of quadratic twists of an elliptic curve defined over the field of rational numbers. In this paper, we consider this problem over number fields. We will prove that the existence of a uniform upper bound for the ranks of elliptic curves in this family is equivalent to the convergence of a certain infinite series. This result extends the work of Rubin and Silverberg over $\mathbb Q$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Small discriminants of complex multiplication fields of elliptic curves over finite fields
- Creator:
- Shparlinski, Igor E
- Format:
- print, bez média, and svazek
- Type:
- model:article and TEXT
- Subject:
- matematika, mathematics, elliptic curve, complex multiplication field, Frobenius discriminant, 13, and 51
- Language:
- English
- Description:
- We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves E over a prime finite field F_{p} of p elements, such that the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over F_{p} is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I.E. Shparlinski (2007)., Igor E. Shparlinski., and Obsahuje seznam literatury
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4. The integral points on elliptic curves $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$
- Creator:
- Yang, Hai and Fu, Ruiqin
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- elliptic curve, integral point; quadratic, and diophantine equation
- Language:
- English
- Description:
- Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n>1$ and both $6n^2-1$ and $12n^2+1$ are odd primes, then the general elliptic curve $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$ has only the integral point $(x, y)=(2, 0)$. By this result we can get that the above elliptic curve has only the trivial integral point for $n=3, 13, 17$ etc. Thus it can be seen that the elliptic curve $y^2=x^3+27x-62$ really is an unusual elliptic curve which has large integral points.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
5. Why is the class number of ℚ(3√11) even?
- Creator:
- Lemmermeyer, F.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- class number, pure cubic field, and elliptic curve
- Language:
- English
- Description:
- In this article we will describe a surprising observation that occurred in the construction of quadratic unramified extensions of a family of pure cubic number fields. Attempting to find an explanation will lead us on a magical mystery tour through the land of pure cubic number fields, Hilbert class fields, and elliptic curves.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public