# 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

#### 6 Citations

Critical $L$-values for some quadratic twists of gross curves

- Mathematics
- 2019

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 curve… Expand

Classical Iwasawa theory and infinite descent on a family of abelian varieties

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- 2020

For primes $$q \equiv 7 \ \mathrm {mod}\ 16$$ q ≡ 7 mod 16 , the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross… Expand

On the $2$-adic logarithm of units of certain totally imaginary quartic fields

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- 2020

In this paper, we prove a result on the $2$-adic logarithm of the fundamental unit of the field $\mathbb{Q}(\sqrt[4]{-q}) $, where $q\equiv 3\bmod 4$ is a prime. When $q\equiv 15\bmod 16$, this… Expand

Practice of the Incomplete $p$-Ramification Over a Number Field -- History of Abelian $p$-Ramification

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- 2019

The theory of p-ramification in the maximal prop extension of a number field K, unramified outside p and $\infty$, is well known including numerical experiments with PARI/GP programs. The case of… Expand

On the $\lambda$-invariant of Selmer groups arising from certain quadratic twists of Gross curves

- Mathematics
- 2021

Let q be a prime with q ≡ 7 mod 8, and let K = Q( √ −q). Then 2 splits in K, and we write p for either of the primes K above 2. Let K∞ be the unique Z2-extension of K unramified outside p with n-th… Expand

A classical family of elliptic curves having rank one and the $2$-primary part of their Tate-Shafarevich group non-trivial

- Mathematics
- 2019

We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the… Expand

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Critical $L$-values for some quadratic twists of gross curves

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- 2019

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 curve… Expand

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