# What is the analogy between the Hilbert function and L-functions?

In his book Projective Varieties and Modular Forms, M. Eichler uses the notation $L(\lambda, M)$ for the Hilbert function of a finite graded $R=k[x_0, \dots, x_n]$-module $M$. So, $L(\lambda, M) = \dim_k M_\lambda$. By the classical theorem of Hilbert, $L(\lambda, M)$ coincides with a numerical polynomial for large $\lambda$.

On p.27 of his book Eichler writes an equation which amounts to

$$L(\lambda + \mu, M) = L(\lambda, M \otimes \mathcal O(\mu))$$

Of course, this is absolutely self-evident because tensoring with $\mathcal O(\mu)$ only shifts the degree by $\mu$.

This is very much analogous to the equation

$$L(s + \mu, V) = L(s, V\otimes \mathbf Q_{\ell}(\mu)),$$ where $V$ is a geometric representation over $\mathbf Q$.

On p.43, Eichler states in the following form a generalization of Riemann-Roch and Serre dualiy:

Theorem of Duality. Let $n>1$. For a finite, reflexive, and quasifree $R$-module $M$ the following equations hold $(i=1, \dots, n-1)$: $$L(\lambda, \text{Ext}_R^i(M, R)) = L(-\lambda -2n -2, \text{Ext}_R^{n-i}(M_*, R)).$$

This looks suspiciously like a functional equation relating the $L$-functions of $H^i_{et}(X, \mathbf Q_\ell)$ and $H^{2n-i}_{et}(X, \mathbf Q_\ell)$, for a variety of dimension $n$ over $\mathbf Q$.

What is going on here? Where does this similarity comes from? Eichler says nothing of it, but the suggestive notation is probably deliberate...

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Isn't this one of the main analogies that motivated the $\mathbb F_1$ program? – Gjergji Zaimi Jul 21 '13 at 2:55
Dear @GjergjiZaimi, would you mind elaborating? – Bruno Joyal Aug 6 '13 at 19:38