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...

Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform such that the special L-value of A (at s = 1) is non-zero. Suppose that the algebraic part of the special L-value of A is divisible by an odd prime q such that q does not divide the numerator of N−1 12 . Then the Birch and Swinnerton-Dyer conjecture predicts that the q-adic valuations of the algebraic part of the s...

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 ...

Let $$E/{\mathbb {Q}}$$ be an elliptic curve and p odd prime where E has good reduction, assume that admits a rational p-isogeny. In this paper we study the anticyclotomic Iwasawa theory of over imaginary quadratic field in which splits, relate to characters by variation method Greenberg–Vatsal. As result our obtain proofs (under relatively mild hypotheses) Perrin-Riou’s Heegner point main conj...

Let E be an elliptic curve over Q and ` be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at `. Under certain assumptions, we prove the vanishing of the Galois cohomology group H1(Gal(K(E[`i])/K), E[`i]) for all i ≥ 1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyv...

Let F be a number field with ring of integers OF , and let E/OF be an abelian scheme of arbitrary dimension. In this paper, we study the class invariant homomoprhisms on E with respect to powers of a prime p of ordinary reduction of E. Our main result implies that if the p-adic Birch and Swinnerton-Dyer conjecture holds for E, then the kernels of these homomorphisms are of bounded order. It fol...

We study zeros of the L-functions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the L-functions at the central critical point s = 1 2 is absolutely bounded. On the Birch and Swinnerton-Dyer conjecture, this provides an upper bound for the rank of the Jacobian J0(q) of the modular curve X0(q), which is of the sam...