We prove that the special-value conjecture for the zeta function of a proper, regular, flat arithmetic surface formulated in [6] at $s=1$ is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of the generic fibre. There are two key results in the proof. The first is the triviality of the correction factor of [6, Conjecture 5.12], which we show for arbitrary regular proper arithmetic schemes. In the proof we need to develop some results for the eh-topology on schemes over finite fields which might be of independent interest. The second result is a different proof of a formula due to Geisser, relating the cardinalities of the Brauer and the Tate–Shafarevich group, which applies to arbitrary rather than only totally imaginary base fields.

We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a Rankin–Selberg $p$-adic $L$-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan–Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is $+1$ rather than $-1$, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a $p$-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

We prove the p-parity conjecture for elliptic curves over global fields of characteristic p>3. We also present partial results on the ℓ-parity conjecture for primes ℓ≠p.

We improve Kolyvagin’s upper bound on the order of the p-primary part of the Shafarevich–Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely that predicted by the Birch and Swinnerton-Dyer conjectural formula.

We study Rubin’s variant of the p-adic Birch and Swinnerton-Dyer conjecture for CM elliptic curves concerning certain special values of the Katz two-variable p-adic L-function that lie outside the range of p-adic interpolation.

Following Katz–Sarnak, Iwaniec–Luo–Sarnak and Rubinstein, we use the one- and two-level densities to study the distribution of low-lying zeros for one-parameter rational families of elliptic curves over $\mathbb{Q}(t)$. Modulo standard conjectures, for small support the densities agree with Katz and Sarnak's predictions. Further, the densities confirm that the curves' L-functions behave in a manner consistent with having r zeros at the critical point, as predicted by the Birch and Swinnerton-Dyer conjecture. By studying the two-level densities of some constant sign families, we find the first examples of families of elliptic curves where we can distinguish SO(even) from SO(odd) symmetry.

In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordell‘s Equation y$^2$=x$^3+$k for 0 ≠ k ∈ Z and draw some conclusions from our numerical findings. In fact we solve Mordell‘s Equation in Z for all integers k within the range 0 < | k | [les ] 10 000 and partially extend the computations to 0 < | k | [les ] 100 000. For these values of k, the constant in Hall‘s conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.