Hello everyone,

I was deeply attracted by the Hasse and Siegel's theorems while studying $p$-adic analysis. While reading a paper B.J. Birch and H.P.F. Swinnerton-Dyer - Notes on elliptic curves. I, Journ. reine u. angewandte Math. 212 (1963), 7-25 the authors emphasize that their central conjecture related to some cubic curves ( more popularly called as elliptic curves ) is based upon the work of Siegel and Tamagawa.

In that paper Peter points out that C.L. Siegel has done some work showing that "densities of rational points on a quadric surface can be expressed in terms of the densities of $p$ -adic points ". I was totally surprized that how can one get an intuition in what the author is pointing to.

More-over the Hasse-principle also tells the same thing. Existence of global solutions can be decided by looking at the local solutions. I can't understand the whole point. Suppose let us take a polynomial $f(x,y)=0$.

So my question is how can you estimate and find the solution set $(x,y)$ to $f$ seeing the solution set $(x^{\prime},y^{\ \prime})$ of $f(x,y)=0 \mod p$ . So I am sure that $(x,y) \neq (x^{\prime},y^{\ \prime})$ in all cases. So are we merging all the obtained local $(x^{\prime},y^{\ \prime})$ solutions and thereby constructing $(x,y)$ ?.

Is there some other way of looking at local - global lift ? . I do read Tate-Shafarevich group that measures the failure. We all know that if the $\mathbb{Ш}$ is trivial then local solution will lift to global solution. How is that possible ?.

Please post a precise answer stating the gist of Siegel's work ( about the densities ) and the reinterpretation of the same work by Tamagawa in terms of volumes.

Thank you.

Hello everyone,

I was deeply attracted by the Hasse and Siegel's theorems while studying $p$-adic analysis. While reading a paper B.J. Birch and H.P.F. Swinnerton-Dyer - Notes on elliptic curves. I, Journ. reine u. angewandte Math. 212 (1963), 7-25 the authors emphasize that their central conjecture related to some cubic curves ( more popularly called as elliptic curves ) is based upon the work of Siegel and Tamagawa.

In that paper Peter points out that C.L. Siegel has done some work showing that "densities of rational points on a quadric surface can be expressed in terms of the densities of $p$ -adic points ". I was totally surprized that how can one get an intuition in what the author is pointing to.

More-over the Hasse-principle also tells the same thing. Existence of global solutions can be decided by looking at the local solutions. I can't understand the whole point. Suppose let us take a polynomial $f(x,y)=0$.

So my question is how can you estimate and find the solution set $(x,y)$ to $f$ seeing the solution set $(x^{\prime},y^{\ \prime})$ of $f(x,y)=0 \mod p$ . So I am sure that $(x,y) \neq (x^{\prime},y^{\ \prime})$ in all cases. So are we merging all the obtained local $(x^{\prime},y^{\ \prime})$ solutions and thereby constructing $(x,y)$ ?.

Thank you.