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Existence of solution for a system of quadratic diophantine equations / symmetric quadratic froms

This is just an extended comment, giving reformulation of the problem and reducing it to just $p-1$ unknowns and $p-1$ quadratic equations over the Gaussian integers. Consider the generating ...
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Generalization of the geometric series representation of the Kronecker delta for arbitrary lattices

The approach indicated by @AlexanderKalmynin is a good heuristic, for sure. Still, there is a somewhat larger context that may be more explanatory, depending on one's tastes. Namely, we can see that ...
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Generalization of the geometric series representation of the Kronecker delta for arbitrary lattices

If $h'=h$, then for each $h''\in L^*$ one has $\exp(2\pi iQ(h-h',h''))=1$, so your sum is equal to $$ |\det Q|^{-1} |L^*/L|=1. $$ On the other hand, if $h\neq h'$, then there is an element $h_0\in L^*/...
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Calculating the explicit constant – Siegel zeros and class numbers

One place to find this worked out in detail is the paper "On the Siegel-Tatuzawa theorem" by Jeffrey Hoffstein (published in 1980 in Acta Arithmetica). Lemma 1 of that paper states that if $\...
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Duke and Schulze-Pillot condition for equidistribution

I recommend that you study the paper by Duke and Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids (Inventiones, 1990)....
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