# Definition of Symplectic Motive

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.

• If you can tell what you know already, people can suggest what to do read next.. – Praphulla Koushik May 7 at 4:45
• I get lots of hits here: google.com/search?q="symplectic+polarization" – David Roberts May 7 at 4:46
• 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. – Kiddo May 7 at 23:56
• 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. – Jesse Silliman May 8 at 1:05