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I am trying to understand the genus of a lattice from Conway and Sloane textbook. They said two quadratic forms $Q_1$ and $Q_2$ lie in the same genus if they are equivalent over $\mathbb{R}$ and over the $p$-adic integers $\mathbb{Z}_p$ for all primes $p$. I don't understand that definition, especially when they define the Jordan decomposition of a quadratic form f at a a prime p as: $f = f_1 \oplus pf_p \oplus p^2f_{p^2} \oplus \ldots$

Can you give me some examples for this definition or any textbook or references about quadratic form, and genus of a lattice that I learn by myself?

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    $\begingroup$ in the column on the right of the page are links to related questions; see some of those $\endgroup$
    – Will Jagy
    Jul 17 at 2:52
  • $\begingroup$ next day: I would not begin with SPLAG. If you like that approach, buy two books; The first is From Error Correcting Codes Through Sphere Packings to Simple Groups, by Thomas M. Thompson. Next is Lattices and Codes, by Wolfgang Ebeling. In the language of quadratic forms, Rational Quadratic Forms by Cassels $\endgroup$
    – Will Jagy
    Jul 17 at 17:31
  • $\begingroup$ Thank you Will, I will check your suggestions. $\endgroup$
    – vent
    Jul 18 at 21:21

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