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Let $X$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ and consider $\lambda:X\rightarrow \hat X$ a polarization of $X$ of degree $d$. Assume that $d$ is prime to the characteristic of $k$. Then it is known that the kernel $\mathrm{Ker}(\lambda)$ is an étale, constant group scheme over $k$. Moreover because $\lambda$ is symmetric, its kernel also has the structure of a symplectic module. We deduce the existence of a unique sequence of integers $d_1|\ldots |d_n$ such that $d_1\geq 2$ and $$\mathrm{Ker}(\lambda)\simeq \left( \mathbb Z/d_1\mathbb Z \times \ldots \times \mathbb Z/d_n\mathbb Z\right)^2$$ as group schemes over $k$. In particular, $d=(d_1\ldots d_n)^2$.

On many occasions in the litterature, I see that the integer $n$ is taken to be equal to the dimension $g$ of $X$, up to adding some $1$ at the beginning of the sequence $(d_1,\ldots ,d_n)$. I am all fine with that when $n\leq g$, but wouldn't it be possible for $n$ to actually be bigger than $g$ in the first place ? I can't find any reference discussing this, which I find puzzling. Would somebody have an explanation ? Am I missing something obvious ?

Edit: I kept thinking about it and believe that $n \leq g$ may hold. I denote by $\hat X$ the dual of $X$. Composing the natural pairings $X[N]\times \hat{X}[N]\rightarrow \mu_N$ with the polarization $\lambda:X[N]\rightarrow \hat X[N]$, we obtain the Weil pairings on $X[N]\times X[N]$. Taking $N=l^m$ where $l$ is a prime different from $\mathrm{char}(k)$, these pairings go to the projective limit and give an alternate form $E_l:\mathrm T_l(X)\times \mathrm T_l(X)\rightarrow \mathbb Z_l(1)$, where on the target we have the Tate twist and $\mathrm T_l(X)$ denotes the Tate module of $X$ : it is a free $\mathbb Z_l$-module of rank $2g$.

Up to choosing a basis of the Tate twist, I wonder if the type of this perfect pairing relates to the type of $\lambda$. Namely, I denote by $D_l$ the $l$-part of $D$, ie. $D_l:=(l^{v_1},\ldots ,l^{v_n})$ where $v_i$ is the $l$-adic valuation of $d_i$. I suspect that up to a unit scalar in $\mathbb Z_l$, there may be a subspace of $\mathrm T_l(X)$ with a basis so that the restriction of $E_l$ to this subspace would be described by the matrix

$$\left( \begin{matrix} 0 & \mathrm{Diag}(D_l) \\ -\mathrm{Diag}(D_l) & 0 \end{matrix} \right)$$

In particular, the above subspace would have dimension $2n$, that is then less than $2g$. I would obtain $n\leq g$. I have not unveiled the construction of the pairings yet to try and check if this holds. It seems believable though, as it may correspond to the conditions which are imposed in PEL moduli problem.


Some references where $n$ is taken to the dimension $g$ of $X$ without any specific explanation:

  • Genestier and Ngo's lecture on Shimura varieties, available here. See the definition of the moduli problem in 2.3 page 13. The condition (3) implicitly implies that $n=g$. This moduli problem corresponds to that studied in Mumford's GIT, where no such condition was imposed to my understanding.
  • Olsson's workshop notes on abelian varieties, available here. See remark 6.13.
  • Hulek and Sankaran's paper on the geometry of Siegel modular varieties, available here. See p.93 (ie. p.5 of the pdf). In the case of abelian varieties over $\mathbb C$ described as projective tori, the definition of a polarization seems to be slightly different, and there it is clear that the number of integers in the sequence $(d_1,\ldots,d_n)$ is precisely the dimension of $X$.
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