For primes $$q \equiv 7 \ \mathrm {mod}\ 16$$ , the present manuscript shows that elementary methods enable one to prove surprisingly strong results about Iwasawa theory of Gross family elliptic curves with complex multiplication by ring integers field $$K = {\mathbb {Q}}(\sqrt{-q})$$ which are in perfect accord predictions conjecture Birch and Swinnerton-Dyer. We also some interesting phenomen...

Later on in [2], Grekos refined the previous results, in some cases, by introducing the infimum of the radii of curvature r(Γ ) of the curve. He succeeded in showing an upper bound of the shape N(Γ ) l(Γ )r(Γ )−1/3 and conversely constructed a family of curves Γ0 with N(Γ0) l(Γ0)r(Γ0)−1/3. Naturally, Grekos’ results suppose that the curve has at least C2-regularity. In fact, this is the maximal...

Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1...

This survey paper contains two parts. The first one is a written version of a lecture given at the “Random Matrix Theory and L-functions” workshop organized at the Newton Institute in July 2004. This was meant as a very concrete and down to earth introduction to elliptic curves with some description of how random matrices become a tool for the (conjectural) understanding of the rank of MordellW...

We prove a fundamental conjecture of Rubin on the structure local units in anticyclotomic $\mathbb{Z}_p$-extension unramified quadratic extension $\mathbb{Q}_p$ for $p\geq 5$ prime. Rubin's underlies Iwasawa theory deformation CM elliptic curve over field at primes $p$ good supersingular reduction, notably main terms $p$-adic $L$-function. As consequence, we an inequality Birch and Swinnerton-D...

We prove the Gross–Zagier–Zhang formula over global function fields of arbitrary characteristics. It is an explicit which relates Néron-Tate heights CM points on abelian varieties and central derivatives associated quadratic base change L-functions. Our proof based arithmetic variant a relative trace identity Jacquet. This approach proposed by Zhang. apply our results to Birch Swinnerton–Dyer c...

In the early 90’s, Perrin-Riou (Ann Inst Fourier 43(4):945–995, 1993) introduced an important refinement of Mazur–Swinnerton-Dyer p-adic L-function elliptic curve E over $$\mathbb {Q}$$ , taking values in its de Rham cohomology. She then formulated a analogue Birch and Swinnerton-Dyer conjecture for this L-function, which formal group logarithms global points on make intriguing appearance. The ...