Non‐vanishing theorems for central L‐values of some elliptic curves with complex multiplication

@article{Coates2018NonvanishingTF,
  title={Non‐vanishing theorems for central L‐values of some elliptic curves with complex multiplication},
  author={John Coates and Yongxiong Li},
  journal={arXiv: Number Theory},
  year={2018}
}
Let $q$ be any prime $\equiv 7 \mod 16$, $K = \mathbb{Q}(\sqrt{-q})$, and let $H$ be the Hilbert class field of $K$. Let $A/H$ be the Gross elliptic curve defined over $H$ with complex multiplication by the ring of integers of $K$. We prove the existence of a large explicit infinite family of quadratic twists of $A$ whose complex $L$-series does not vanish at $s=1$. This non-vanishing theorem is completely new when $q > 7$. Its proof depends crucially on the results established in our earlier… Expand
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References

SHOWING 1-10 OF 43 REFERENCES
Critical $L$-values for some quadratic twists of gross curves
Let $K=\Bbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-\beta)}$ denote its quadratic twist, with $\beta=\sqrt{-q}$. The curveExpand
Analogues of Iwasawa's $\mu=0$ conjecture and the weak Leopoldt conjecture for a non-cyclotomic $\mathbb{Z}_2$-extension
Let $K = \mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, and let $\mathcal{O}$ be the ring of integers of $K$. The prime $2$ splits in $K$, say $2\mathcal{O} =Expand
Iwasawa theory of quadratic twists of X0(49)
The field $$K = \mathbb{Q}\left( {\sqrt { - 7} } \right)$$K=ℚ(−7) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lyingExpand
On the p-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of $\mathbbQ(\sqrt-3)$
  • Yukako Kezuka
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2016
Abstract We study infinite families of quadratic and cubic twists of the elliptic curve E = X 0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of theExpand
A criterion for elliptic curves with second lowest 2-power in L(1)
  • C. Zhao
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2001
Let D = p1 … pm, where p1, …, pm are distinct rational primes ≡ 1(mod 8), and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the Hecke L-functionExpand
Infinite Descent on Elliptic Curves with Complex Multiplication
It is a pleasure to dedicate this paper to I. R. Safarevic, in recognition of his important work on the arithmetic of elliptic curves. As anyone who has worked on the arithmetic of elliptic curves isExpand
Lectures on the Birch-Swinnerton-Dyer Conjecture
In 1839, Dirichlet gave his remarkable proof that there are infinitely many primes of the form an+b (n = 1,2, . . .), where a,b are any pair of positive integers with (a,b) = 1. His proof usedExpand
Arithmetic on elliptic curves with complex multiplication. II
0. In this paper we will continue to study the arithmetic of elliptic curves with complex multiplication by Q ( 1 / ~ ) , which we began in [5]. Chapter I reviews the basic facts on Q-curves, andExpand
Quadratic twists of elliptic curves
In this paper, we give the method of constructing non-torsion points on elliptic curves, which generalizes the classical Birch lemma. As an application, we get more quadratic twist families of theExpand
Cyclotomic Fields and Zeta Values
Cyclotomic fields have always occupied a central place in number theory, and the so called "main conjecture" on cyclotomic fields is arguably the deepest and most beautiful theorem known about them.Expand
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