# Why should I believe in the Siegel's and Hasse's rationale ?

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.

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I might be wrong, but I think we already met around here. – quid Jun 26 '12 at 10:01
Siegel's result is known as "mass formula" and is a corollary of the Siegel-Weil formula relating theta integral to Siegel-Eisenstein series. Roughly specking, the theta integral is connected with rational solutions while the Eisenstein series is connected with local solutions both $p$-adic and $\infty$-adic. Weil 's book "Adeles and algebraic groups" and two Acta papers 1964/65 is classic on this subject. – user4245 Jun 26 '12 at 10:14
That is the original paper of Siegel and hard to read. The referene I provided is more readable. Siegel's work could be translated using adele to the "Tamagawa measure one" staement. – user4245 Jun 26 '12 at 10:54
Dear Shanmukha Srinivasan, by asking us to summarize the work of several authors, you are demanding too much of people who are working for free. Please revise your question to be more specific, e.g., following the guidelines in the MathOverflow "how to ask" page, linked at the top of this page. – S. Carnahan Jun 27 '12 at 2:10
After not receiving any reply from quid, I thought of opening a meta-discussion ( after hearing that there is a MO META, where we can post such issues ) on his weird behavior. Anyone are free to join there and comment. Thank you. – Shanmukha_Srinivasan Jun 27 '12 at 16:57

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.

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Its good to see that you work for Wikipedia. But your site don't hold an email id of you in case if I want to contact you. – Shanmukha_Srinivasan Jun 26 '12 at 16:31
I disagree with at least one of the above comments. – Charles Matthews Jun 26 '12 at 18:39
I took my comment back, you can feel free to see my new lipo-suctioned edit. ;) – Shanmukha_Srinivasan Jun 27 '12 at 17:03
But Charles, your answer is indeed nice. Only thing I wanted to hear is that I am interested in listening more about the Local-global principle, and why should one believe in it ?. I am particularly interested in knowing how can a local solution lifts to a global solution ?. @CharlesMatthews – Shanmukha_Srinivasan Jun 28 '12 at 7:32
So is everyone, however My personal view is based on my own research (actually I started with the Grothendieck–Katz p-curvature conjecture en.wikipedia.org/wiki/Grothendieck-Katz_p-curvature_conjecture) which is that the local-global view encourages new research. You are reading mathematics written about 50 years ago, and asking a historians' question. Why not find your own problem? This worked for me, and Zariski-dense subgroups. Try works by Emmanuel Kowalski for a fresher approach. – Charles Matthews Jun 28 '12 at 14:16