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Let $P$ and $Q$ be two even, unimodular, positive definite quadratic forms of rank $n$. Let $r_{k}(P)$ be the number of vectors of norm $k$, in symbols:

$$ r_k(P)=\textrm{cardinality of }\{v\in \mathbb{Z}^n \; | \; P(v,v)=k\} $$

It is well known that there exists a constant $c$, which is explicitly known and depends just on the rank $n$, such that if $$ r_k(P)=r_k(Q) \quad \forall \, k \leq c $$ then $$ r_k(P)=r_k(Q) \quad \forall \, k $$

The proof I know relies on the theory of modular forms. In particular, one looks at the associated theta series, and uses the fact that the dimension of the vector space of modular forms of given weight is known, actually one uses even an explicit basis of this vector space.

I would like to know if there exists an alternative proof which does not use modular forms.

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It is known at least since Hermite that, given integers $N$ and $D$ there are only finitely many equivalence classes of positive definite quadratic modules rank $n<N$ and discriminant $d<D$. (His proof (as well as the modern proof using Minkowski's convex body theorem) doesn't even allude to theta series.)

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  • $\begingroup$ thanks, this is very interesting. Can you give me a reference please? Anyway, this is proving less than what I asked, right? $\endgroup$
    – Giulio
    Commented Jun 26, 2016 at 19:56
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    $\begingroup$ In fact it shows much more : it often arises that an equivalence invariant on a certain class of objects takes only a finite number of values, whilst the number of equivalence classes is infinite. Here the theorem says that there are finitely many equivalent classes, and hence finitely many possible theta series, so that you can be certain that there exists a $c$ as in your question (but its value is undetermined ... maybe that's what you want to say by "proving less" in which case I seriously doubt you will have a positive answer). $\endgroup$
    – few_reps
    Commented Jun 28, 2016 at 10:25
  • $\begingroup$ this is a great answer actually: if you have a bound on the number of isomorphism classes of quadratic forms, you also do get a bound on the constant $c$ in my question (proably not a sharp one...). Anyway, do you have a reference? $\endgroup$
    – Giulio
    Commented Jun 28, 2016 at 10:32
  • $\begingroup$ Sorry, I guess you do not get any bound because you do not know if the relations $r_k(P)=r_k(Q)$ are independent. $\endgroup$
    – Giulio
    Commented Jun 28, 2016 at 10:33

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