Elliptic Curves with no Rational Points
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Publication:3813910
DOI10.2307/2047452zbMath0663.14023OpenAlexW4244964921MaRDI QIDQ3813910
Publication date: 1988
Full work available at URL: https://doi.org/10.2307/2047452
elliptic curveorder of the 3-primary component of the ideal class group of quadratic fieldstrivial Mordell-Weil group
Quadratic extensions (11R11) Elliptic curves over global fields (11G05) (L)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture (11G40) Density theorems (11R45) Global ground fields in algebraic geometry (14G25)
Related Items (37)
Families of elliptic curves with trivial Mordell-Weil group ⋮ A positive proportion of cubic curves over Q admit linear determinantal representations ⋮ Class numbers of quadratic fields ${\mathbb Q}(\sqrt {D})$ and ${\mathbb Q}(\sqrt {tD})$ ⋮ Unnamed Item ⋮ A \(p\)-adic analytic family of the \(D\)-th Shintani lifting for a Coleman family and congruences between the central \(L\)-values ⋮ A note on the Iwasawa \(\lambda\)-invariants of real quadratic fields ⋮ Existence of an infinite family of pairs of quadratic fields \(\mathbb{Q}(\sqrt{m_1D})\) and \(\mathbb{Q}(\sqrt{m_2D})\) whose class numbers are both divisible by 3 or both indivisible by 3 ⋮ Indivisibility of class numbers of imaginary quadratic fields ⋮ On the Iwasawa \(\lambda_2\)-invariants of certain families of real quadratic fields ⋮ Elliptic curves of rank zero satisfying the \(p\)-part of the Birch and Swinnerton-Dyer conjecture ⋮ Secondary terms in counting functions for cubic fields ⋮ Simultaneous indivisibility of class numbers of pairs of real quadratic fields ⋮ Triquadratic 3-rational fields ⋮ The mean number of 3-torsion elements in ray class groups of quadratic fields ⋮ Heegner points on elliptic curves with a rational torsion point ⋮ GOLDFELD’S CONJECTURE AND CONGRUENCES BETWEEN HEEGNER POINTS ⋮ Class numbers and Iwasawa invariants of certain totally real number fields ⋮ Canonical periods and congruence formulae ⋮ Infinitely many hyperelliptic curves with exactly two rational points ⋮ Construction of elliptic curves with large Iwasawa \(\lambda\)-invariants and large Tate-Shafarevich groups ⋮ Odd degree number fields with odd class number ⋮ Shintani’s zeta function is not a finite sum of Euler products ⋮ L-series with nonzero central critical value ⋮ Note on the rank of quadratic twists of Mordell equations ⋮ The torsion of $p$-ramified Iwasawa modules II ⋮ Elliptic curves of rank 1 satisfying the 3-part of the Birch and Swinnerton-Dyer conjecture ⋮ Trace formulae and imaginary quadratic fields ⋮ Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function) ⋮ Sur les propriétés de divisibilité des nombres de classes des corps quadratiques ⋮ Class number divisibility for imaginary quadratic fields ⋮ There are genus one curves of every index over every number field ⋮ The torsion of \(p\)-ramified Iwasawa modules for \(\mathbb Z^2_p\)-extensions ⋮ Rank-one quadratic twists of an infinite family of elliptic curves ⋮ Elliptic curves satisfying the Birch and Swinnerton-Dyer conjecture mod 3 ⋮ Mordell-Weil ranks of quadratic twists of pairs of elliptic curves. ⋮ A note on rank one quadratic twists of elliptic curves and the non-degeneracy of 𝑝-adic regulators at Eisenstein primes ⋮ On families of imaginary abelian fields with pseudo-null unramified Iwasawa modules
Cites Work
- Unnamed Item
- A note on basic Iwasawa \(\lambda\)-invariants of imaginary quadratic fields
- Sur les coefficients de Fourier des formes modulaires de poids demi-entier
- Proof of the existence of infinitely many imaginary quadratic fields whose class number is not divisible by 3
- On the conjecture of Birch and Swinnerton-Dyer
- Some results on the Mordell-Weil group of the Jacobian of the Fermat curve
- On the possibility of \(x^4+y^4=z^4\) in quadratic fields
- On the Density of Discriminants of Cubic Fields
- On the density of discriminants of cubic fields. II
- Sur la résolubilité de l'équation $x^2 + y^2 + z^2 = 0$ dans un corps quadratique
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