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.



Why should I believe in the Siegel's and Hasse's rationale ? 
The short answer is that you shouldn't believe implicitly in it: like most local-global principles it is a heuristic until it is a theorem. Birch and S-Dyer used it that way.


Voskresenskii in his book *Algebraic Groups and Their Birational Invariants*, p. 138 of English translation, commented "One badly needs a monograph with a detailed exposition of all known results on Tamagawa numbers." If first written 20 years ago, that is probably true today still; especially if the scope is all algebraic groups, not just affine ones. (The whole point, really.)


See the Wikipedia articles on


\*Glossary of arithmetic and Diophantine geometry


\*Weil conjecture on Tamagawa numbers


\*Smith–Minkowski–Siegel mass formula ‎ (recommended)


\*Siegel–Weil formula


\*Special values of L-functions


for decent coverage (by WP standards) bringing one up to the L-function theory in contemporary terms.


