# On the Hasse-Weil L-function of $P^n$

So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p(T)=1-T$ (a polynomial of degree $1$). So the Hasse-Weil zeta function of $X_0$ is given by $$L_{X_0}(s):=\prod_{p}Z_p(p^{-s})^{-1}=\zeta(s).$$ Now if one lets $\psi(z)=\sum_{n\in\mathbf{Z}}e^{i\pi n^2z}$ then $\psi(z)$ is a modular form of weight $1/2$ (over a suitable congruence group of $SL_2(\mathbf{Z})$) in the sense that $$(-iz)^{-1/2}\psi(-1/z)=\psi(z). \;\;\;\;(*)$$ A straight forward computation which uses the definition of the Gamma function implies that $$\tilde{L}_{X_0}(s):=\int_{0}^{\infty}(\psi(it)-1)t^{s/2}\frac{dt}{t}=2\pi^{-s/2}\Gamma(s/2)\zeta(s).$$ Using the functional equation $(*)$ one may deduce the meromorphic continuation and the functional equation of $\zeta(s)$ (invariance of $\tilde{L}_{X_0}(s)$ under $s\mapsto 1-s$).

Now let us take the scheme $X_1=\mathbf{P}^1$ over $Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_1$ at a prime $p$ is given by $Z_p(T)=(1-T)(1-pT)=1-\sigma_1(p)T+pT^2$ (a polynomial of degree $2$). It thus follows that the Hasse-Weil zeta function of $X_1$ is given by $$L_{X_1}(s)=\prod_{p}Z_p(p^{-s})^{-1}=\zeta(s)\zeta(s-1).$$ Now let us look at the Eisenstein series of weight $2$ i.e. $$E_2(z)=(2\pi i)^{-2}\sum_{m,n}'\frac{1}{(mz+n)^2}:=\frac{-B_2}{2}+2\sum_{n\geq 1}\sigma_1(n)q_{z}^n,$$ where $q_{z}=e^{2\pi iz}$. (Note that I don't get any convergence issue here since I take this $q$-expansion as the definition of $E_2(z)$). Note that $E_2(z)$ is "almost" a moldular form of weight $2$ (for the full congruence group $SL_2(\mathbf{Z})$) since $$(-z)^{-2}E_2(-1/z)=E_2(z)-\frac{1}{2\pi iz} \;\;\;\; (**)$$ A straight forward computation similar to the one before implies that $$\tilde{L}_{X_1}(s):=\int_{0}^{\infty} (E_2(it)+B_2/2)t^{s}\frac{dt}{t}=2 (2\pi)^{-s}\Gamma(s)\zeta(s)\zeta(s-1).$$ As before using $(**)$ one obtains the meromorphic continuation of $L_{X_1}(s)$ and its functional equation ($\tilde{L}_{X_1}(s)=-\tilde{L}_{X_1}(2-s)$, note the appearance of the sign $-1$). Note that this could already be deduced from what we know from $L_{X_0}(s)$.

Now there is no reason to stop here. So let $X_2=\mathbf{P}^2$ over $Spec(\mathbf{Z})$. Then the Weil zeta function of $X_2$ at $p$ is $Z_p(T)=(1-T)(1-pT)(1-p^2T)$ (a polynomial of degree $3$). It thus follows that the Hasse-Weil zeta function of $X_2$ is given by $$L_{X_2}(s)=\prod_{p}Z_p(p^{-s})^{-1}=\zeta(s)\zeta(s-1)\zeta(s-2)=\sum_{n\geq 1}\frac{a_n}{n^s}$$

Q1: Is it reasonable to expect the formal $q$-expansion $f(q_z)=\sum_{n\geq 1} a_n q_z^n$ to be related in some direct way to an automorphic form w.r.t. a suitable congruence subgroup of $GL_3(\mathbf{Z})$?

Q2: What about $X_n=\mathbf{P}^n$ in general?

added: Note that in the case of $X_0$ I'm really looking at $\tilde{\zeta}(s):=\zeta(2s)$ which is an $L$-function of weight $1/2$ in the sense that $\tilde{\zeta}(s)$ is related to $\tilde{\zeta}(1/2-s)$ which is in accordance with the fact that $\psi(z)$ has weight $1/2$.

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There are no modular forms of weight 2 for the full modular group! The ring of modular forms is generated by the Eisenstein series of weight 4 and 6 (see, for example, Serre, "A course in arithmetic"). –  Victor Protsak May 21 '12 at 18:29
You are right there is a missing correcting factor that I need. –  Hugo Chapdelaine May 21 '12 at 20:18
In fact in order to get the modular invariance you need to throw in an non-holomorphic term like $\frac{C}{y}$ where $y=\Im(z)$. –  Hugo Chapdelaine May 21 '12 at 20:26


There is, of course, an autormorphic representation (Hecke character) $\chi$ for $\GL(1)$ whose $p$-adic avatar is the cyclotomic character. From this, one thus has the isobaric sum (following Langlands and Jacquet-Shalika) $$\pi = 1 \boxplus \chi \boxplus \ldots \boxplus \chi^{n-1}$$ which is an automorphic representation for $\GL(n)$ with $L(\pi,s) = L(\mathbf{P}^n,s)$.

But all of this is somewhat irrelevant to your action question, to which the answer is not really. The reason $q$-expansions arise for $\GL(2)$ has to do with the fact that $\SL_2(\R)/\mathrm{SO}_2(\R)$ is the upper half plane, and $\SL_2(\mathbf{Z})$ contains the element $z \mapsto z+1$ for which $q = e^{2 \pi i z}$ is invariant. $\SL_n(\R)/\mathrm{SO}_n(\R)$ is quite a different beast.

Edit You don't seem satisfied with my answer, but I think you seem to be missing a basic principle: if you write down something at random, there's no reason it should be interesting. The theory of automorphic forms, however labyrinthian, has an incredibly precise structure. If you take an automorphic representation $\pi$ corresponding to a classical modular form, then the representation theory of $\mathrm{GL}_2(\mathbf{R})$ will tell you that a lowest weight vector for the discrete series will be annihilated by some differential operator which one can compute to be the Cauchy-Riemann equations, and hence the corresponding function on the upper half plane will be holomorphic: this is a very specific reason why holomorphic functions might be associated to automorphic forms. If, instead, one works with an automorphic form for $\mathrm{GL}(3)$, then the representation theory will (under appropriate conditions) produce an expansion in terms of Whittaker functions which will have a completely different flavour.

It might also be worth remarking that already since Langlands time people have thought hard about the problem of transfer, that is, for example, starting with an automorphic representation for $\GL(2)$ and producing one for $\GL(3)$. The methods used to prove these results are almost exclusively via the trace formula - in particular, they proceed via harmonic analysis and representation theory, rather than explicit manipulations with functions (holomorphic or otherwise). Thus (addressing your comment) the hope that one might explicitly decompose the symmetric space of $\GL(3)$ in some way is a little too optimistic.

Finally, you can (of course) recover $L(\pi,s)$ from $f(q)$, so $f(q)$ does carry (in some sense) all the information of $L(\pi,s)$. Moreover, one can recover $f(q)$ as the inverse Mellin transform of $L(\pi,s)$ adjusted by appropriate Gamma factors. However, you will find that when $n > 2$ the relevant Gamma factors involved will not play well with respect to the functional equation, and thus there will be no reason to expect that $f(q)$ will not have any nice properties (such as a nice functional equation). Even for $n = 1$ you have to cheat a little (replacing $q^n$ by $q^{n^2}$), and moreover the theta function is (automorphically) more naturally thought of as associated to the metaplectic group rather than $\GL(2)$.

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Dear anonymous, in the case of $SL_n$ the symmetric space is $V_n=\{A\in M_{n x n}(\mathbf{R}): A=A^t, A>0, det(A)=1\}$. So this is a space of real dimension $n(n+1)/2-1$. So are you saying that there won't be a way of projecting $V_3$ to $V_2$ that would allow us to relate $f(q)$ to an automorphic form on $V_3$? –  Hugo Chapdelaine May 21 '12 at 12:01
Of course if one projects $V_3$ on $V_2$ or if one tries to embed $V_2$ in $V_3$ then these maps won't be holomorphic since $V_3$ (which is of real dimension $5$) does not even support a complex structure. –  Hugo Chapdelaine May 21 '12 at 13:24
Dear anonymous, thanks for the clarifications. Indeed, this question involved too much wishful thinking! –  Hugo Chapdelaine May 23 '12 at 21:50