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I just started reading a paper "On the Langlands correspondence for symplectic motives" by Benedict H. Gross, which talks about Symplectic Motives. The paper starts with the following

Let $M$ be a pure motive of weight $-1$ and rank $2n$ over $\mathbb{Q}$, with a non-degenerate symplectic polarisation

$$\psi:\wedge^2 M \to \mathbb{Q}(1)$$.

I want to understand this line. I know what is a pure motive of certain weight and rank over $\mathbb{Q}$. What I don’t understand is what is this map $\psi$, which is referred to as the non-degenerate symplectic polarisation.

Thanks in advance for any kind of help.

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  • $\begingroup$ If you can tell what you know already, people can suggest what to do read next.. $\endgroup$ – Praphulla Koushik May 7 at 4:45
  • $\begingroup$ I get lots of hits here: google.com/search?q="symplectic+polarization" $\endgroup$ – David Roberts May 7 at 4:46
  • $\begingroup$ I have made my question more specific! The google hits are not answering my question. Maybe I am overlooking something, but I have no clue about the map, that I am talking about. $\endgroup$ – Kiddo May 7 at 23:56
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    $\begingroup$ Perhaps Gross means this in the sense of realizations of motives, so that this is an alternating form on the etale realization of the motive, and an alternating form with a positivity condition (i.e. a polarization in the sense of polarized Hodge structures) on the Hodge realization. $\endgroup$ – Jesse Silliman May 8 at 1:05

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