Higher-rank zeta functions for elliptic curves

Significance

The Riemann zeta function and the Riemann hypothesis concerning its zeros are perhaps the most famous function and most famous conjecture in all of number theory. Almost 100 years ago, Artin defined an analog, the “Artin zeta function,” for a curve (one-dimensional variety) over a finite field. In 2005, L.W. found a whole series of zeta functions for curves over finite fields, including the original Artin zeta function. The analog of the (still unsolved) classical Riemann hypothesis was proved for the Artin l function by Andre Weil, but the corresponding statement for Weng’s higher zeta functions is still open. In this paper it is proved for the simplest nontrivial case, when the curve has genus one (elliptic curve).

Abstract

In earlier work by L.W., a nonabelian zeta function was defined for any smooth curve $X$ over a finite field ${\mathbb{F}}_q$ and any integer $n\ge 1$ by

where the sum is over isomorphism classes of ${\mathbb{F}}_q$-rational semistable vector bundles $V$ of rank $n$ on $X$ with degree divisible by $n$. This function, which agrees with the usual Artin zeta function of $X/{\mathbb{F}}_q$ if $n=1$, is a rational function of $q^{-s}$ with denominator $\left(1-q^{-ns}\right)\left(1-q^{n-ns}\right)$ and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet series

where the sum is now over isomorphism classes of ${\mathbb{F}}_q$-rational semistable vector bundles $V$ of degree 0 on $X$, is equal to ${\prod }_{k=1}^{\infty }{\zeta }_{X/{\mathbb{F}}_q}\left(s+k\right),$ and use this fact to prove the Riemann hypothesis for ${\zeta }_{X,n}\left(s\right)$ for all $n$.

Let $X$ be a smooth projective curve of genus $g$ over a finite field ${\mathbb{F}}_q$. For all positive integers $n$, a “nonabelian rank $n$ zeta function” of $X/{\mathbb{F}}_q$ was defined in refs. 1 and 2 by

$${\zeta }_{X,n}\left(s\right)={\zeta }_{X/{\mathbb{F}}_q,n}\left(s\right)=\sum _{d\equiv 0\hspace{0.17em}\left(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.17em}n\right)}\hspace{0.17em}\sum _{\left[V\right]\in {\mathcal{M}}_{X,n}\left(d\right)}\frac{q^{h^0\left(X,V\right)}-1}{\left|\mathrm{A}\mathrm{u}\mathrm{t}\left(V\right)\right|}\hspace{0.17em}{q}^{-ds},$$ [1]

where ${\mathcal{M}}_{X,n}\left(d\right)$ denotes the moduli stack of ${\mathbb{F}}_q$-rational semistable vector bundles of rank $n$ and degree $d$ on $X$, and $\mathrm{A}\mathrm{u}\mathrm{t}\left(V\right)$ and $h^0\left(X,V\right)$ denote the automorphism group of $V$ and the dimension of its space of global sections. In other words, if we define, for any $d$, the $\alpha$ and $\beta$ invariants of $X/{\mathbb{F}}_q$ by

$${\alpha }_{X,n}\left(d\right)=\sum _{\left[V\right]\in {\mathcal{M}}_{X,n}\left(d\right)}\frac{q^{h^0\left(X,V\right)}-1}{\left|\mathrm{A}\mathrm{u}\mathrm{t}\left(V\right)\right|}\hspace{0.17em},\hspace{0.17em}\hspace{1em}{\beta }_{X,n}\left(d\right)=\sum _{\left[V\right]\in {\mathcal{M}}_{X,n}\left(d\right)}\frac{1}{\left|\mathrm{A}\mathrm{u}\mathrm{t}\left(V\right)\right|},$$ [2]

then ${\zeta }_{X,n}\left(s\right)$ is the generating function of the numbers ${\alpha }_{X,n}\left(mn\right)$:

$${\zeta }_{X,n}\left(s\right)\hspace{0.17em}=\sum _{d\equiv 0\hspace{0.17em}\left(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.17em}n\right)}{\alpha }_{X,n}\left(d\right)\hspace{0.17em}{t}^d=\sum _{m=0}^{\infty }{\alpha }_{X,n}\left(mn\right)\hspace{0.17em}{T}^m,$$ [3]

where $t=q^{-s},\hspace{0.17em}T=t^n=Q^{-s}$ with $Q=q^n$. Note that , so ${\alpha }_{X,n}\left(d\right)$ counts the number of isomorphism classes of pairs consisting of a semistable vector bundle $V$ of rank $n$ and degree $d$ together with an embedding , while the beta invariants were studied in the case $\left(d,n\right)=1$ (when semistability coincides with stability) by Harder and Narasimhan (3) in their famous work on moduli spaces of vector bundles. For the zeta function, on the other hand, it is crucial (for reasons indicated briefly in ref. 4, remark 1) to restrict to the opposite case $n\mid d$.

The following properties of ${\zeta }_{X,n}\left(s\right)$ were shown in ref. 2, using Riemann–Roch, duality, and vanishing for semistable bundles:

Theorem

Define ${\zeta }_{X,n}\left(s\right)$ for all $n\ge 1$ by Eq. 1. Then

• 1)The function ${\zeta }_{X,1}\left(s\right)$ equals ${\zeta }_X\left(s\right)$, the Artin zeta function of $X/{\mathbb{F}}_q$.
• 2)There exists a degree $2g$ polynomial such that

$${\zeta }_{X,n}\left(s\right)=\frac{P_{X,n}\left(T\right)}{\left(1-T\right)\left(1-QT\right)}.$$ [4]

• 3)The function ${\zeta }_{X,n}$ satisfies the functional equation

$${\zeta }_{X,n}\left(1-s\right)=Q^{\left(g-1\right)\left(2s-1\right)}\cdot {\zeta }_{X,n}\left(s\right).$$ [5]

The following conjecture, verified in many examples, was also proposed in ref. 2.

Conjecture (Riemann Hypothesis)

If ${\zeta }_{X,n}\left(s\right)=0$, then .

Of course in the classical case $n=1$ this is a famous theorem of Weil.

A further indication that the higher zeta functions defined by Eq. 1 are natural objects is that they turn out to coincide with the special case $G=S{L}_n$, $P=P_{n-\mathrm{1,1}}$ of the zeta functions ${\zeta }_X^{G,P}\left(s\right)$ defined in refs. 2 and 5 for suitable reductive algebraic groups $G$ and parabolic subgroups $P$. This equality, which was conjectured in ref. 2, is proved in the companion paper (4). In this paper we concentrate on the case of elliptic curves $X=E$, i.e., $g=1$, where we can give much more complete results. In this case we have ${\alpha }_{E,n}\left(d\right)=\left(q^d-1\right){\beta }_{E,n}\left(d\right)$ for $d>0$ (because $h^0\left(V\right)-h^1\left(V\right)=d$ by the Riemann–Roch theorem and $h^1\left(V\right)$ vanishes) and ${\beta }_{E,n}\left(mn\right)={\beta }_{E,n}\left(0\right)$ for all $m$ (because tensoring with a line bundle of degree 1 gives an isomorphism between the sets of rank $n$ semistable vector bundles of degree $d$ and degree $d+n$ for any $d\in \mathbb{Z}$), so the zeta function Eq. 3 reduces simply to

$${\zeta }_{E,n}\left(s\right)={\alpha }_{E,n}\left(0\right)+{\beta }_{E,n}\left(0\right)\hspace{0.17em}\frac{\left(Q-1\right)T}{\left(1-T\right)\left(1-QT\right)}.$$ [6]

We will give explicit formulas and generating functions for ${\alpha }_{E,n}\left(0\right)$ and ${\beta }_{E,n}\left(0\right)$ and prove the Riemann hypothesis for ${\zeta }_{E,n}\left(s\right)$. The key fact is Theorem 5, which gives an elegant expression for the generating function $\sum {\beta }_{E,n}\left(0\right)q^{-ns}$ as a product of the shifts of the zeta function of $E$.

1. Statement of Main Results

From now on $E$ denotes an elliptic curve over ${\mathbb{F}}_q$. By an “Atiyah bundle” over $E$ we mean any direct sum of the vector bundles $I_1,\hspace{0.17em}{I}_2,\hspace{0.17em}\dots \hspace{0.17em}$ over $E$ defined by Atiyah in ref. 6: is the trivial line bundle and $I_k$ for $k\ge 2$ is the unique (up to isomorphism) nontrivial extension of $I_{k-1}$ by $I_1$. For $n\ge 1$ let ${\alpha }_{E,n}^{\mathrm{\text{At}}}\left(0\right)$ and ${\beta }_{E,n}^{\mathrm{\text{At}}}\left(0\right)$ be the numbers defined as in ref. 1, but with the summations now ranging only over Atiyah bundles. We show the following:

Theorem 1.

For $n\ge 1$ we have

$${\alpha }_{E,n}^{\mathrm{\text{At}}}\left(0\right)=\sum _{\mathbf{m}}\hspace{0.17em}\left(q^{m_1+m_2+\cdots }\hspace{0.17em}-\hspace{0.17em}1\right)\hspace{0.17em}\frac{\epsilon \left(m_1\right)\hspace{0.17em}\epsilon \left(m_2\right)\cdots }{q^{N\left(m_1,m_2,\dots \right)}},$$ [7]

$${\beta }_{E,n}^{\mathrm{\text{At}}}\left(0\right)=\sum _{\mathbf{m}}\hspace{0.17em}\frac{\epsilon \left(m_1\right)\hspace{0.17em}\epsilon \left(m_2\right)\cdots }{q^{N\left(m_1,m_2,\dots \right)}},$$ [8]

where the sum is over all partitions $n=m_1+2m_2+3m_3+\cdots \hspace{0.17em}$ of $n$ and

$$\epsilon \left(m\right)=\frac{q^{m^2}}{\left|{\mathrm{G}\mathrm{L}}_m\left({\mathbb{F}}_q\right)\right|}=\frac{q^{m\left(m+1\right)/2}}{\left(q^m-1\right)\left(q^{m-1}-1\right)\cdots \left(q-1\right)},$$ [9]

$$N\left(m_1,m_2,\dots \right)=\sum _{k,\hspace{0.17em}\ell \hspace{0.17em}\ge \hspace{0.17em}1}{m}_k\hspace{0.17em}{m}_{\ell }\hspace{0.17em}min\left(k,\ell \right).$$ [10]

From this we deduce the following simple formulas for ${\alpha }_{E,n}^{\mathrm{\text{At}}}\left(0\right)$ and ${\beta }_{E,n}^{\mathrm{\text{At}}}\left(0\right)$ (“special counting miracle”) by a direct combinatorial argument:

Theorem 2.

For $n\ge 0$ we have

$${\alpha }_{E,n+1}^{\mathrm{\text{At}}}\left(0\right)={\beta }_{E,n}^{\mathrm{\text{At}}}\left(0\right)=q^{-n}\epsilon \left(n\right).$$ [11]

This in turn is used together with considerations of algebraic structures of semistable bundles of degree 0 to obtain the following intrinsic relation between $\alpha$ and $\beta$ invariants (“general counting miracle”):

Theorem 3.

For all $n\ge 0$ we have

$${\alpha }_{E,n+1}\left(0\right)={\beta }_{E,n}\left(0\right).$$ [12]

We mention that Theorem 3 has been generalized to curves of arbitrary genus by Sugahara.

The above results, whose proofs are given in Section 2, show that the higher-rank zeta functions for elliptic curves are completely determined by their beta invariants. To understand the latter, we first use results of Harder and Narasimhan (3), Desale and Ramanan (7), and Zagier (8) to get an explicit formula for ${\beta }_{E,n}\left(0\right)$ in terms of special values of the Artin zeta function of $E/{\mathbb{F}}_q$:

Theorem 4.

For $n\ge 1$ we have

$${\beta }_{E,n}\left(0\right)=\sum _{k=1}^n\hspace{0.17em}{\left(-1\right)}^{k-1}\hspace{0.17em}\hspace{0.17em}\sum _{\begin{array}{c}{n}_1+\cdots +n_k=n\\ n_1,\hspace{0.17em}\dots ,\hspace{0.17em}{n}_k>0\end{array}}\frac{v_{E{n}_1}\cdots \hspace{0.17em}{v}_{E{n}_k}}{\left(q^{n_1+n_2}-1\right)\cdots \left(q^{n_{k-1}+n_k}-1\right)},$$ [13]

where the numbers $v_{En}\hspace{0.17em}\hspace{0.17em}\left(n>0\right)$ are defined by

$$v_{E,}n={\zeta }_E^{*}\left(1\right){\zeta }_E\left(2\right)\cdots {\zeta }_E\left(n\right),\hspace{0.17em}\hspace{1em}{\zeta }_E^{*}\left(1\right)=\underset{s\to 1}{lim}\hspace{0.17em}\left(1-q^{1-s}\right)\hspace{0.17em}{\zeta }_E\left(s\right).$$ [14]

In Section 3, we will use Eq. 13 and a fairly complicated combinatorial calculation to establish the following simple formula for the generating Dirichlet series of the invariants ${\beta }_{E,n}\left(0\right)$:

Theorem 5.

Define a Dirichlet series for by

[15]

where the second sum is over isomorphism classes of ${\mathbb{F}}_q$-rational semistable vector bundles $V$ of degree 0 on $E$. Then

[16]

This formula will then be used in Section 4 to prove the following estimate:

Proposition 6.

For $n\ge 2$ we have the inequalities

$$1<\frac{{\beta }_{E,n}\left(0\right)}{{\beta }_{E,n-1}\left(0\right)}<\frac{q^{n/2}+1}{q^{n/2}\hspace{0.17em}-\hspace{0.17em}1}.$$ [17]

(In fact, we prove a stronger estimate; see Eq. 52.) Combining these bounds with Eqs. 6 and 12, we deduce the following:

Theorem 7.

The Riemann hypothesis is true for elliptic curves.

2. Calculation of the $\alpha$ and $\beta$ Invariants of Elliptic Curves

In this section we give explicit formulas for ${\alpha }_{E,n}\left(mn\right)$ and ${\beta }_{E,n}\left(mn\right)$ for an elliptic curve $E/{\mathbb{F}}_q$. By what was already explained in the Introduction, it suffices to do this for $m=0$. We will prove Theorems 1–4 as stated in Section 1.

A. Automorphisms of Atiyah Bundles.

Let $V$ be an Atiyah bundle of rank $n$ over $E$ as defined in Section 1. Then $V$ can be uniquely written in the form

$$V\hspace{0.17em}\cong \hspace{0.17em}\underset{k\ge 1}{\oplus }{I}_k^{\oplus m_k}$$ [18]

for integers $m_k\ge 0$ with ${\sum }_{k\ge 1}k{m}_k=n$, so Theorem 1 follows from the following proposition:

Proposition 8.

For $V/E$ as in Eq. 18, we have $h^0\left(V\right)={\sum }_{k\ge 1}{m}_k$ and

$$\left|\mathrm{A}\mathrm{u}\mathrm{t}\left(V\right)\right|=q^{N\left(m_1,m_2,\dots \right)}\hspace{0.17em}\prod _{k\ge 1}\hspace{0.17em}{q}^{-m_k^2}\left|{\mathrm{G}\mathrm{L}}_{m_k}\left({\mathbb{F}}_q\right)\right|,$$ [19]

where $\hspace{0.17em}N\left(m_1,m_2,\dots \right)$ is defined as in Eq. 10.

Proof:

By theorem 8 of ref. 6, for any integers $k,\hspace{0.17em}\ell \ge 1$ we have

$$I_k\otimes I_{\ell }\hspace{0.17em}\hspace{0.17em}\cong \hspace{0.17em}\underset{\genfrac{}{}{0ex}{}{|k-\ell |\hspace{0.17em}<\hspace{0.17em}m\hspace{0.17em}<\hspace{0.17em}k+\ell }{m\equiv k+\ell -1\left(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{2}\right)}}{\oplus }{I}_m$$

and consequently, since $I_k$ is self-dual,

$$\dim \mathrm{H}\mathrm{o}\mathrm{m}\left(I_k,I_{\ell }\right)=h^0\left(I_k^{\vee }\otimes I_{\ell }\right)=h^0\left(I_k\otimes I_{\ell }\right)=min\left(k,\ell \right).$$ [20]

This can be seen explicitly as follows. The bundle $I_k$ has a realization given locally away from $0\in E$ by $k$-tuples of regular functions and near 0 by $k$-tuples $\left(f_1,\dots ,f_k\right)$, where $f_1$ and all $z{f}_i-f_{i-1}$ ($2\le i\le k$) are regular, where $z$ is a local parameter at 0. (In other words, $f_i$ is allowed to have a pole of order $i-1$ at 0 and, for instance, the residue of $f_2$ equals the value of $f_1$ at 0.) The space $\mathrm{H}\mathrm{o}\mathrm{m}\left(I_k,I_{\ell }\right)$ is then spanned by the maps

$$\left(f_1,\dots ,f_k\right)\hspace{0.17em}\hspace{0.17em}\mapsto \hspace{0.17em}\hspace{0.17em}\left(\underset{\ell -s}{\underbrace{0,\dots ,0}},f_1,\dots ,f_s\right)\hspace{1em}\hspace{1em}\left(\hspace{0.17em}1\le s\le min\left(k,\ell \right)\right).$$

As a special case of Eq. 20 we have $h^0\left(I_k\right)=1$ for all $k$, making the first statement of Proposition 8 obvious. We prove the second one in several steps.

• 1)In the special case $V=I_k$, we have

$$\left|\mathrm{A}\mathrm{u}\mathrm{t}\left(I_k\right)\right|=q^{k-1}\left(q-1\right).$$

Indeed, from the above description, any endomorphism of $I_k$ has the form

$$\phi =\left(\begin{array}{ccccc}\hfill a_1\hfill & \hfill a_2\hfill & \hfill \cdots \hfill & \hfill a_{k-1}\hfill & \hfill a_k\hfill \\ \hfill 0\hfill & \hfill a_1\hfill & \hfill \cdots \hfill & \hfill a_{k-2}\hfill & \hfill a_{k-1}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill a_1\hfill & \hfill a_2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill a_1\hfill \end{array}\right)$$ [21]

for some $a_i\in {\mathbb{F}}_q$, and $\phi$ is an isomorphism if and only if $a_1\hspace{0.17em}\ne \hspace{0.17em}0$.

• 2)More generally, for any positive integers $k$ and $m$ we have

$$\left|\mathrm{A}\mathrm{u}\mathrm{t}\left(I_k^{\oplus m}\right)\right|=q^{\left(k-1\right)m^2}\hspace{0.17em}\left|{\mathrm{G}\mathrm{L}}_m\left({\mathbb{F}}_q\right)\right|,$$ [22]

because the automorphisms of $I_k^{\oplus m}$ have the same form as Eq. 21, but with each $a_i$ now being an $m\times m$ matrix over ${\mathbb{F}}_q$ and with $a_1$ invertible.

• 3)Finally, if $V$ is a general Atiyah bundle as in Eq. 18, then

$$\left|\mathrm{A}\mathrm{u}\mathrm{t}\left(V\right)\right|=\prod _{k\hspace{0.17em}\ne \hspace{0.17em}\ell }|\mathrm{H}\mathrm{o}\mathrm{m}\left(I_k,I_{\ell }\right){|}^{m_k{m}_{\ell }}\hspace{0.17em}\cdot \hspace{0.17em}\prod _k\hspace{0.17em}|\mathrm{A}\mathrm{u}\mathrm{t}\left(I_k^{\oplus m_k}\right)|,$$

and Eq. 19 then follows from Eqs. 20 and 22.$\square$

B. Proof of Theorem 2.

Introduce the three generating functions

where the notations are as in Theorem 2. We must show that they coincide.

There is a bijection between terminating sequences $\left(m_1,m_2,\dots \right)$ of nonnegative integers and monotone decreasing terminating sequences $\left(p_1,p_2,\dots \right)$ of nonnegative integers, given by setting $p_k={\sum }_{\ell \ge k}{m}_{\ell }$, $m_k=p_k-p_{k+1}$. Under this correspondence, we have

$$\begin{array}{cc}\hfill N\left(m_1,m_2,\hspace{0.17em}\dots \right)& =\sum _{k=1}^{\infty }k\hspace{0.17em}{m}_k\hspace{0.17em}\left(m_k+2m_{k+1}+2m_{k+2}+\cdots \right)\hfill \\ \hfill & =\sum _{k=1}^{\infty }k\hspace{0.17em}\left(p_k-p_{k+1}\right)\left(p_k+p_{k+1}\right)=\sum _{k=1}^{\infty }{p}_k^2.\hfill \end{array}$$

Hence Theorem 1 shows that and can be given as

[23]

with defined by

or, equivalently (set $h=p_2$ on the right-hand side), by the recursive formulas

[24]

We claim that the solution of this recursion is given by

[25]

To prove this, we denote the right-hand side of Eq. 25 by $B_p\left(x\right)$ and show that $B_p\left(x\right)$ satisfies the same recursion Eq. 24 as , i.e., that we have

[26]

where $B\left(x,t\right)$ and $\hat{B}\left(x,t\right)$ are the two generating series defined by

$$B\left(x,t\right)=\sum _{p=0}^{\infty }\hspace{0.17em}{B}_p\left(x\right)\hspace{0.17em}{t}^p,\hspace{1em}\hspace{1em}\hat{B}\left(x,t\right)=\sum _{p=0}^{\infty }\hspace{0.17em}{q}^{p^2}{x}^{-p}\hspace{0.17em}{B}_p\left(x\right)\hspace{0.17em}{t}^p.$$

But this is now fairly easy. The definition of $B_p\left(x\right)$ gives the formulas

$$\left(1-x\right)\hspace{0.17em}{B}_p\left(qx\right)=\left(q^p-x\right)\hspace{0.17em}{B}_p\left(x\right),\hspace{1em}\left(1-x\right)\left(q^p-1\right)\hspace{0.17em}{B}_p\left(qx\right)=qx\hspace{0.17em}{B}_{p-1}\left(x\right),$$

which translate into the four generating series identities

$$\begin{array}{cc}\hfill \left(1-x\right)\hspace{0.17em}B\left(qx,t\right)& =B\left(x,qt\right)\hspace{0.17em}-\hspace{0.17em}xB\left(x,t\right),\hfill \\ \hfill \left(1-x\right)\left(B\left(qx,qt\right)\hspace{0.17em}-\hspace{0.17em}B\left(qx,t\right)\right)& \hspace{0.17em}=\hspace{0.17em}qxt\hspace{0.17em}B\left(x,t\right),\hfill \end{array}$$ [27]

and

$$\begin{array}{cc}\hfill \left(1-x\right)\hspace{0.17em}\hat{B}\left(qx,qt\right)& =\hat{B}\left(x,qt\right)\hspace{0.17em}-\hspace{0.17em}x\hspace{0.17em}\hat{B}\left(x,t\right),\hfill \\ \hfill \left(1-x\right)\left(\hat{B}\left(qx,qt\right)\hspace{0.17em}-\hspace{0.17em}\hat{B}\left(qx,t\right)\right)& \hspace{0.17em}=\hspace{0.17em}qt\hspace{0.17em}\hat{B}\left(x,qt\right).\hfill \end{array}$$ [28]

Now using the identity , which follows from

we find from Eq. 27 that satisfies the same two recursions Eq. 28 as $\hat{B}\left(x,t\right)$ and hence that these two power series are equal. This proves Eq. 26 and hence also Eq. 25 and lets us rewrite Eq. 23 as

Substituting $t=q^{-1}$ into the sum of the two equations Eq. 27 now gives and hence , and then substituting $t=1$ into the first equation of Eq. 27 gives . This completes the proof.

C. Proof of Theorem 3.

Let $V$ be a semistable vector bundle of rank $n$ and degree 0 over $E/{\mathbb{F}}_q$. By the classification of indecomposable bundles on elliptic curves defined over algebraic closed fields given by Atiyah (6), we know that there are no stable bundles of rank $\ge 2$ and degree 0 over $\overline{E}\hspace{0.17em}≔\hspace{0.17em}E{\otimes }_{{\mathbb{F}}_q}\overline{{\mathbb{F}}_q}$. Consequently, over $\overline{E}$, the graded bundle $G\left(V\right)$ associated to a Jordan–Hölder filtration of $V$ decomposes as

$$G\left(V\right)=L_1\oplus L_2\oplus \cdots \oplus L_n$$

for some line bundles $L_i$ of degree 0 on $\overline{E}$. (For basics of Jordan–Hölder filtrations and their associated graded bundles for semistable bundles, see, e.g., ref. 1.) Since $L_i$ need not be defined over ${\mathbb{F}}_q$, usually it is a bit complicated to classify $V$ over $E$. (This classification problem depends on the arithmetic of the curve $E$ and, specifically, on the number of ${\mathbb{F}}_q$-rational torsion points of order $\le n$ on $E$.) Instead of doing this, we first note that to get a nontrivial contribution to $\alpha$ invariants, we must have $h^0\left(V\right)\hspace{0.17em}\ne \hspace{0.17em}0$. Guided by this, we regroup the bundles appearing in the summation defining the $\alpha$ invariant in the following way. Assume (after renumbering) that for $1\le j\le i$ and for $i<j\le n$. Since there are no nontrivial extensions of by $L_j$ for , we can uniquely decompose $V$ as $U\oplus W$, where $U$ and $W$ are ${\mathbb{F}}_q$-rational semistable bundles of degree 0 over $E$ with

Then $h^0\left(E,V\right)=h^0\left(E,U\right)$ and $\mathrm{A}\mathrm{u}\mathrm{t}\left(V\right)\cong \mathrm{A}\mathrm{u}\mathrm{t}\left(U\right)\times \mathrm{A}\mathrm{u}\mathrm{t}\left(W\right)$ (because there are no nontrivial homomorphisms among and $L_j$), and $U$ and $W$ range independently over bundles with the properties listed above. Hence

$${\alpha }_{E,n}\left(0\right)=\sum _{i=1}^n\hspace{0.17em}{\alpha }_{E,i}^{*}\hspace{0.17em}{\beta }_{E,n-i}^{*},$$

where ${\alpha }_{E,i}^{*}$ and ${\beta }_{E,k}^{*}$ are the modified $\alpha$ and $\beta$ invariants defined by

But, by using Atiyah’s classification of indecomposable bundles on elliptic curves defined over algebraic closed fields again, we know that the bundles $U$ in the sum defining ${\alpha }_{E,i}^{*}$ are precisely the Atiyah bundles $U={\oplus }_k{I}_k^{\oplus m_k}$ with $\sum k{m}_k=i$. Hence ${\alpha }_{E,i}^{*}={\alpha }_{E,i}^{\mathrm{\text{At}}}\left(0\right)$. By the same argument, of course, we have ${\beta }_{E,n}\left(0\right)={\sum }_{i=0}^n{\beta }_{E,i}^{\mathrm{\text{At}}}\left(0\right){\beta }_{E,n-i}^{*}$. (Note that this time the summation starts at $i=0$, whereas for ${\alpha }_{E,n}\left(0\right)$ we started at $i=1$ because ${\alpha }_{E,0}^{*}=0$.) Theorem 3 now follows immediately from Theorem 2.

D. Proof of Theorem 4.

In this subsection, in which $X$ is again a curve of arbitrary genus $g\ge 1$, we combine results of refs. 3, 7, and 8 to give a closed formula for ${\beta }_{X,n}\left(0\right)$ for all $n\ge 1$.

The invariant ${\beta }_{X,n}\left(d\right)$ is periodic in $d$ of period $n$ by the same argument as given for $g=1$ in the Introduction. We renormalize slightly by setting

$${\hat{\beta }}_{X,n}\left(d\right)=q^{-\left(g-1\right)\hspace{0.17em}n\left(n-1\right)/2}{\beta }_{X,n}\left(d\right),\hspace{0.17em}\hspace{1em}{\hat{\zeta }}_X\left(s\right)=q^{\left(g-1\right)s}\hspace{0.17em}{\zeta }_X\left(s\right)$$ [29]

(note that these agree with ${\beta }_{X,n}\left(d\right)$ and ${\zeta }_X\left(s\right)$ in the case when $g=1$), because this gives a simpler functional equation ${\hat{\zeta }}_X\left(1-s\right)={\hat{\zeta }}_X\left(s\right)$ and will also lead to a formula in Theorem 9 that has no explicit dependence on $g$. We also define

$${\hat{v}}_{Xn}={\hat{\zeta }}_X^{*}\left(1\right){\hat{\zeta }}_X\left(2\right)\cdots {\hat{\zeta }}_X\left(n\right),\hspace{0.17em}\hspace{1em}{\hat{\zeta }}_X^{*}\left(1\right)=\underset{s\to 1}{lim}\hspace{0.17em}\left(1-q^{1-s}\right)\hspace{0.17em}{\hat{\zeta }}_X\left(s\right)$$ [30]

instead of Eq. 14. Then the work of Harder and Narasimhan (3) and Desale and Ramanan (7) implies the following relation, involving an infinite summation:

Theorem.

For $n\ge 1$ and any $d\in \mathbb{Z}$ we have

$$\sum _{k\ge 1}\hspace{0.17em}\sum _{\begin{array}{c}{n}_1+\cdots +n_k=n\\ n_1,\dots ,n_k>0\end{array}}\hspace{0.17em}\sum _{\begin{array}{c}{d}_1+\cdots +d_k=d\\ \frac{d_1}{n_1}\hspace{0.17em}>\hspace{0.17em}\cdots \hspace{0.17em}>\hspace{0.17em}\frac{d_k}{n_k}\end{array}}\hspace{0.17em}{q}^{-{\sum }_{i<j}\hspace{0.17em}\left(d_i{n}_j-d_j{n}_i\right)}\hspace{0.17em}\prod _{j=1}^k{\hat{\beta }}_{X,n_j}\left(d_j\right)={\hat{v}}_{Xn}.$$ [31]

This Theorem is stated in ref. 7, p. 236 (beginning 9 lines from the bottom), except that there the authors use $\beta$ and $\zeta$ instead of $\hat{\beta }$ and $\hat{\zeta }$ and write the equation in the slightly different form ${\hat{\beta }}_{X,n}\left(d\right)={\hat{v}}_{Xn}\hspace{0.17em}-\hspace{0.17em}$ (sum over terms with $k\ge 2$ in Eq. 31) to make it clear that this equation gives a recursive determination of all ${\hat{\beta }}_{X,n}\left(d\right)$. This recursion relation was inverted in ref. 8. We state the result here in detail since in that paper only a corollary (namely, the application to the calculation of the Betti numbers of the moduli space ${\mathcal{M}}_{X,n}\left(d\right)$) was written out explicitly. The following theorem, however, is an immediate consequence of Eq. 31 and theorem 2 of ref. 8. Note that here the sum is finite!

Theorem 9.

For $n\ge 1$ and any $d\in \mathbb{Z}$ we have

$${\hat{\beta }}_{X,n}\left(d\right)=\sum _{k\ge 1}{\left(-1\right)}^{k-1}\hspace{0.17em}\sum _{\begin{array}{c}{n}_1+\cdots +n_k=n\\ n_1,\hspace{0.17em}\dots ,\hspace{0.17em}{n}_k>0\end{array}}\hspace{0.17em}\prod _{j=1}^k{\hat{v}}_{X{n}_j}\hspace{0.17em}\cdot \hspace{0.17em}\prod _{j=1}^{k-1}\frac{q^{\left(n_j+n_{j+1}\right)\hspace{0.17em}\left\{d\left(n_1+\cdots +n_j\right)/n\right\}}}{q^{\left(n_j+n_{j+1}\right)}\hspace{0.17em}-\hspace{0.17em}1},$$

where $\left\{t\right\}$ for $t\in \mathbb{R}$ denotes the fractional part of $t$.

Theorem 4 is just the special case $d=0$ and $X=E$, since ${\hat{\beta }}_{E,n}={\beta }_{E,n}$.

3. The Generating Series of the Beta Invariants

A. Explicit Formulas

We keep all notations as in the Introduction and Section 1. Recall that the Artin zeta function of $E$ and its renormalized special value at $s=1$ as defined by Eq. 14 are given by

$${\zeta }_{E/{\mathbb{F}}_q}\left(s\right)=\frac{1\hspace{0.17em}-\hspace{0.17em}a{q}^{-s}\hspace{0.17em}+\hspace{0.17em}{q}^{1-2s}}{\left(1-q^{-s}\right)\left(1-q^{1-s}\right)}\hspace{0.17em},\hspace{1em}\hspace{1em}{\zeta }_{E/{\mathbb{F}}_q}^{*}\left(1\right)=\frac{\left|E\left({\mathbb{F}}_q\right)\right|}{q-1},$$

where $a\in \mathbb{Z}$ is defined by $\left|E\left({\mathbb{F}}_q\right)\right|=q-a+1$ and satisfies $\left|a\right|\le 2\sqrt{q}$. For convenience, from now on we write simply ${\beta }_n$ instead of ${\beta }_{E,n}\left(0\right)$. Note that ${\beta }_n$ depends only on $q$ and $a$ and belongs to .

The closed formula for ${\beta }_n$ given in Theorem 4 has $\hspace{0.17em}\text{O}\left(2^n\right)$ terms. In this subsection we give several alternative expressions, including closed formulas with $p\left(n\right)=\text{O}\left(e^{c\sqrt{n}}\right)$ terms and with $\hspace{0.17em}\text{O}\left(n^3\right)$ terms, the generating series formula Eq. 16, and a recursion permitting the calculation of ${\beta }_1,\dots ,{\beta }_n$ in $\hspace{0.17em}\text{O}\left(n\right)\hspace{0.17em}$ steps. The proofs of these relations are given in the rest of the section.

We begin by calculating the first few values of ${\beta }_n$ from Eq. 13. Here we note that there is considerable cancellation and we always have

$${\beta }_n\hspace{0.17em}\in \hspace{0.17em}\frac{1}{\left(q^n-1\right)\cdots \left(q-1\right)}\hspace{0.17em}\mathbb{Z}\left[a,q\right],$$ [32]

even though the least common denominator of the denominators of the terms in Eq. 13 is much greater; e.g., the first few values of ${\beta }_n$ are given by

$$\begin{array}{cc}\hfill {\beta }_0& =1\hspace{0.17em},\hspace{1em}\hspace{1em}{\beta }_1=v_1=\frac{q-a+1}{q-1},\hfill \\ \hfill {\beta }_2& =v_2\hspace{0.17em}-\hspace{0.17em}\frac{v_1^2}{q^2-1}=\frac{\left(q^3-aq+1\right)\left(q-a+1\right)}{\left(q^2-1\right){\left(q-1\right)}^2}\hspace{0.17em}-\hspace{0.17em}\frac{{\left(q-a+1\right)}^2}{\left(q^2-1\right){\left(q-1\right)}^2}=\frac{\left(q-a+1\right)\left(q^2+q-a\right)}{\left(q^2-1\right)\left(q-1\right)},\hfill \\ \hfill {\beta }_3& =v_3\hspace{0.17em}-\hspace{0.17em}2\hspace{0.17em}\frac{v_1{v}_2}{q^3-1}\hspace{0.17em}+\hspace{0.17em}\frac{v_1^3}{{\left(q^2-1\right)}^2}=\frac{\left(q-a+1\right)\left(q^5+q^4-\left(a-2\right)q^3-\left(2a-1\right)q^2-\left(a+1\right)q+a^2\right)}{\left(q^3-1\right)\left(q^2-1\right)\left(q-1\right)}.\hfill \end{array}$$

Some more experimentation shows that in fact much more is true, namely

$${\beta }_1=w_1,\hspace{1em}{\beta }_2=\frac{w_1^2\hspace{0.17em}+\hspace{0.17em}{w}_2}{2},\hspace{1em}{\beta }_3=\frac{w_1^3\hspace{0.17em}+\hspace{0.17em}3w_1{w}_2+2w_3}{6}\hspace{0.17em},\hspace{1em}{\beta }_4=\frac{w_1^4\hspace{0.17em}+\hspace{0.17em}6w_1^2{w}_2\hspace{0.17em}+\hspace{0.17em}8w_1{w}_3\hspace{0.17em}+\hspace{0.17em}3w_2^2\hspace{0.17em}+\hspace{0.17em}6w_4}{24},\hspace{1em}\dots ,$$

where the numbers $w_m=w_{m,E}=w_m\left(a,q\right)\hspace{0.17em}\hspace{0.17em}\left(m\ge 1\right)$ are defined by

$$w_m={\zeta }_{E/{\mathbb{F}}_{q^m}}^{*}\left(1\right)=\frac{\left({\alpha }^m-1\right)\left({\overline{\alpha }}^m-1\right)}{q^m-1}\hspace{1em}\hspace{1em}\left(\alpha +\overline{\alpha }=a,\hspace{0.17em}\alpha \overline{\alpha }=q\right).$$ [33]

These special cases suggest that the following theorem should hold:

Theorem 10.

The numbers ${\beta }_n={\beta }_{E,n}\left(0\right)$ are given by

$${\beta }_n=\sum _{\begin{array}{c}{n}_1,n_2,\cdots \ge 0,\\ n_1+2n_2+\cdots =n\end{array}}\frac{w_1^{n_1}{w}_2^{n_2}\cdots }{1^{n_1}{2}^{n_2}\cdots \hspace{0.17em}{n}_1!\hspace{0.17em}{n}_2!\hspace{0.17em}\cdots }\hspace{1em}\hspace{1em}\left(n\ge 0\right),$$ [34]

where the numbers $w_m=w_{E,m}\hspace{0.17em}\left(m\ge 1\right)$ are defined by Eq. 33.

Eq. 34 is the promised formula expressing ${\beta }_n$ as a sum of $p\left(n\right)=\text{O}\left(e^{\pi \sqrt{2n/3}}\right)$ rather than $\hspace{0.17em}\text{O}\left(2^n\right)$ terms.

To proceed further, we introduce the generating function

$$\mathbb{B}\left(x\right)={\mathbb{B}}_{E/{\mathbb{F}}_q}\left(x\right)=\mathbb{B}\left(x;a,q\right)=\sum _{n=0}^{\infty }{\beta }_n\hspace{0.17em}{x}^n.$$ [35]

Then Eq. 34 is equivalent to the formula

$$\mathbb{B}\left(x\right)=\exp \left(\sum _{m=1}^{\infty }{w}_m\hspace{0.17em}\frac{x^m}{m}\right),$$

and substituting for $w_m$ from Eq. 33 we find

$$\begin{array}{ll}\hfill \frac{\mathbb{B}\left(qx\right)}{\mathbb{B}\left(x\right)}& =\exp \left(\hspace{0.17em}\sum _{m=1}^{\infty }\left(q^m-1\right)\hspace{0.17em}{w}_m\hspace{0.17em}\frac{x^m}{m}\right)=\exp \left(\hspace{0.17em}\sum _{m=1}^{\infty }\frac{q^m-{\alpha }^m-{\overline{\alpha }}^m+1}{m}\hspace{0.17em}{x}^m\right)\hfill \\ \hfill & =\frac{\left(1-\alpha x\right)\left(1-\overline{\alpha }x\right)}{\left(1-qx\right)\left(1-x\right)}=\frac{1-ax+q{x}^2}{\left(1-x\right)\left(1-qx\right)}.\hfill \end{array}$$ [36]

This in turn can be rewritten in three different ways, each of which is equivalent to Theorem 10. The first one is obtained by replacing $x$ by $x/q^k$ in Eq. 36 and taking the product over all $k\ge 1$ to get the following multiplicative formula for the generating function $\mathbb{B}\left(x;a,q\right)$:

Theorem 11.

The generating function $\mathbb{B}\left(x;a,q\right)$ defined in Eq. 35 has the product expansion

$$\mathbb{B}\left(x;a,q\right)=\prod _{k=1}^{\infty }\frac{1\hspace{0.17em}-\hspace{0.17em}a{q}^{-k}x\hspace{0.17em}+\hspace{0.17em}{q}^{1-2k}{x}^2}{\left(1\hspace{0.17em}-\hspace{0.17em}{q}^{-k}x\right)\left(1\hspace{0.17em}-\hspace{0.17em}{q}^{1-k}x\right)}.$$ [37]

Theorem 11 is clearly equivalent to Theorem 5 of Section 1, by setting $x=q^{-s}$.

For the second way, we recall the “$q$-Pochhammer symbol” ${\left(x;q\right)}_n$, defined for $x,\hspace{0.17em}q\in \mathbb{C}$ as ${\prod }_{m=0}^{n-1}\left(1-q^{m}x\right)$. This also makes sense for $n=\infty$ if $\left|q\right|<1$. Since our $q$ has absolute value greater rather than less than 1, we replace it by its inverse. Then the calculation in Eq. 36 is just a version of the “quantum dilogarithm identity”

$$\sum _{m=1}^{\infty }\frac{x^m}{m\hspace{0.17em}\left(q^m-1\right)}=\sum _{m,\hspace{0.17em}r\ge 1}\frac{q^{-rm}{x}^m}{m}=\log \frac{1}{{\left(q^{-1}x;\hspace{0.17em}{q}^{-1}\right)}_{\infty }}\hspace{1em}\hspace{1em}\left(\hspace{0.17em}\left|q\right|>1\right)$$

(we refer to ref. 9, pp. 28–31, for a review of the quantum dilogarithm), and Eq. 37 says simply

$$\mathbb{B}\left(x;a,q\right)=\frac{{\left(q^{-1}\alpha x;\hspace{0.17em}{q}^{-1}\right)}_{\infty }\hspace{0.17em}{\left(q^{-1}\overline{\alpha }x;\hspace{0.17em}{q}^{-1}\right)}_{\infty }}{{\left(q^{-1}x;\hspace{0.17em}{q}^{-1}\right)}_{\infty }\hspace{0.17em}{\left(x;\hspace{0.17em}{q}^{-1}\right)}_{\infty }}.$$ [38]

Together with the standard power series expansions of ${\left(x;q\right)}_{\infty }$ and $1/{\left(x;q\right)}_{\infty }$ as given in the survey paper (9) just quoted, this implies the following result, which is the abovementioned closed formula for ${\beta }_n$ with $\hspace{0.17em}\text{O}\left(n^3\right)$ terms.

Theorem 12.

The numbers ${\beta }_n={\beta }_{E,n}\left(0\right)$ are given by the sum

$${\beta }_n\left(E/{\mathbb{F}}_q\right)\hspace{0.17em}=\sum _{\begin{array}{c}{n}_1,\hspace{0.17em}{n}_2,\hspace{0.17em}{n}_3,\hspace{0.17em}{n}_4\ge 0\\ n_1+n_2+n_3+n_4=n\end{array}}\hspace{0.17em}\frac{{\left(-1\right)}^{n_1+n_2}\hspace{0.17em}{q}^{\left(\genfrac{}{}{0.0pt}{}{n_1+1}{2}\right)+\left(\genfrac{}{}{0.0pt}{}{n_2}{2}\right)}\hspace{0.17em}{\alpha }^{n_3}\hspace{0.17em}{\overline{\alpha }}^{n_4}}{{\left(q;q\right)}_{n_1}\hspace{0.17em}{\left(q;q\right)}_{n_2}\hspace{0.17em}{\left(q;q\right)}_{n_3}\hspace{0.17em}{\left(q;q\right)}_{n_4}}\hspace{1em}\hspace{1em}\left(n\ge 0\right),$$

where $\alpha$ and $\overline{\alpha }$ are defined as in Eq. 33.

Finally, multiplying both sides of Eq. 36 by their common denominator and comparing coefficients of $x^n$, we obtain the following:

Theorem 13.

The numbers ${\beta }_n$ satisfy, and are uniquely determined by, the recursion relation

$$\left(q^n-1\right)\hspace{0.17em}{\beta }_n=\left(q^n+q^{n-1}-a\right)\hspace{0.17em}{\beta }_{n-1}\hspace{0.17em}-\hspace{0.17em}\left(q^{n-1}-q\right)\hspace{0.17em}{\beta }_{n-2}$$ [39]

together with the initial conditions ${\beta }_0=1$ and ${\beta }_{-1}=0$.

Theorem 13 gives the simplest algorithm for computing ${\beta }_n$ of all of the formulas we have given, since, as already mentioned, it calculates each ${\beta }_n$ in time O(1) from its predecessors and hence requires time only $\hspace{0.17em}\text{O}\left(n\right)$ to calculate all of the numbers ${\beta }_1,\dots ,{\beta }_n$. We also remark that Eq. 39 immediately implies the assertion Eq. 32 by induction on $n$.

B. Proof of Theorem 5: First Part.

In the previous subsection we formulated four theorems, found experimentally, each of which was equivalent to the others and to Theorem 5. Of these, the simplest by far is the recursion relation Eq. 39. Unfortunately, we were not able to find a direct proof that the numbers defined by Eq. 13 satisfy this recursion, and the proof of Theorem 5 that we give will be indirect and fairly complicated.

There are two main ideas. The first one is to replace the “closed formula” Eq. 13 for ${\beta }_n$ by a recursive formula (thus in some sense undoing the calculation in ref. 8 that led to that formula, which began with the recursion Eq. 31 and then inverted it). To do this, we break up the sum Eq. 13 into $n$ pieces according to the value of the last $n_i$; i.e., we decompose ${\beta }_n$ as

$${\beta }_0=b_0^{\left(0\right)},\hspace{1em}\hspace{1em}{\beta }_n=\sum _{m=1}^n\hspace{0.17em}{\beta }_n^{\left(m\right)}\hspace{1em}\hspace{1em}\left(n\ge 1\right),$$ [40]

where ${\beta }_n^{\left(m\right)}={\beta }_n^{\left(m\right)}\left(E/{\mathbb{F}}_q\right)={\beta }_n^{\left(m\right)}\left(a,q\right)$ is the partial sum defined by

$${\beta }_n^{\left(m\right)}=\sum _{k=1}^{\infty }\sum _{\begin{array}{c}{n}_1,\dots ,n_{k-1}\ge 1,\hspace{0.17em}{n}_k=m\\ n_1+\cdots +n_k=n\end{array}}\frac{{\left(-1\right)}^{k-1}\hspace{0.17em}{v}_{n_1}\cdots v_{n_k}}{\left(q^{n_1+n_2}-1\right)\cdots \left(q^{n_{k-1}+n_k}-1\right)}\hspace{1em}\left(n\ge m\ge 1\right).$$

Denoting the last-but-one variable $n_{k-1}$ in this sum by $p$ whenever $k$ is at least 2, we find

$${\beta }_n^{\left(m\right)}=v_m\cdot \hspace{0.17em}\left\{\begin{array}{cc}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}1\hspace{1em}\hfill & \hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.17em}m=n,\hfill \\ -\hspace{0.17em}\sum _{p=1}^{n-m}\frac{{\beta }_{n-m}^{\left(p\right)}}{q^{m+p}-1}\hspace{1em}\hfill & \hspace{1em}\mathrm{i}\mathrm{f}\hspace{0.17em}1\le m\le n-1,\hfill \end{array}\right.$$

which defines all of the numbers ${\beta }_n^{\left(m\right)}$ (and hence also all of the numbers ${\beta }_n$) by recursion. Multiplying this formula by $x^n$ and summing over all $n\ge 0$, we find that the generating functions

$${\mathbb{B}}^{\left(m\right)}\left(x\right)={\mathbb{B}}_{E/{\mathbb{F}}_q}^{\left(m\right)}\left(x\right)={\mathbb{B}}^{\left(m\right)}\left(x;a,q\right)=\sum _{n=0}^{\infty }{\beta }_n^{\left(m\right)}\hspace{0.17em}{x}^n$$ [41]

of the ${\beta }_n^{\left(m\right)}$ (observe that the sum here actually starts at $n=m$, so that ${\mathbb{B}}^{\left(m\right)}\left(x\right)=\text{O}\left(x^m\right)$, and also that ${\mathbb{B}}^{\left(0\right)}\left(x\right)\equiv 1$) satisfy the identity

$${\mathbb{B}}^{\left(m\right)}\left(x\right)=v_m\hspace{0.17em}{x}^m\hspace{0.17em}\left(1\hspace{0.17em}-\hspace{0.17em}\sum _{p=1}^{\infty }\frac{{\mathbb{B}}^{\left(p\right)}\left(x\right)}{q^{m+p}-1}\right)\hspace{1em}\hspace{1em}\left(m\ge 1\right).$$ [42]

A natural strategy of proof would therefore be to guess a closed formula for the individual series ${\mathbb{B}}^{\left(m\right)}\left(x\right)$ that satisfies the same recursion and that gives Eq. 37 when summed over $m\ge 0$. Unfortunately, we were not able to do this here, so that we have to argue indirectly. The second idea is therefore to prove the identity for special values of the parameter $a$. Since the desired formula Eq. 37 is equivalent to the recursion Eq. 39, which is an identity among polynomials in $a$ and therefore is true if it can be verified for infinitely many values of the argument $a$ for each $n$, it is enough to prove the identity Eq. 37 only for the special values

$$a=a_k\hspace{0.17em}≔\hspace{0.17em}{q}^{k+1}\hspace{0.17em}+\hspace{0.17em}{q}^{-k}\hspace{1em}\hspace{1em}\left(k\in \mathbb{Z},\hspace{0.17em}\hspace{0.17em}k\ge 0\right).$$ [43]

We denote by ${\beta }_{n,k}$ and ${\beta }_{n,k}^{\left(m\right)}$ the specializations of ${\beta }_n$ and ${\beta }_n^{\left(m\right)}$ to this value of $a$ and by ${\mathbb{B}}_k\left(x\right)$ and ${\mathbb{B}}_k^{\left(m\right)}\left(x\right)$ the corresponding generating series. Then Eq. 42 specializes to the identity

$${\mathbb{B}}_k^{\left(m\right)}\left(x\right)=v_{m,k}\hspace{0.17em}{x}^m\hspace{0.17em}\left(1\hspace{0.17em}-\hspace{0.17em}\sum _{p=1}^{\infty }\frac{{\mathbb{B}}_k^{\left(p\right)}\left(x\right)}{q^{m+p}-1}\right)\hspace{1em}\hspace{1em}\left(m\ge 1,\hspace{0.17em}k\ge 0\right),$$ [44]

where $v_{m,k}$ denotes the specialization of $v_m=v_m\left(a,q\right)$ to $a=a_k$, so that if we can guess some other collection of numbers ${\stackrel{\sim }{\beta }}_{n,k}^{\left(m\right)}$ whose generating functions satisfy the same identity, then we automatically have ${\beta }_{n,k}^{\left(m\right)}={\stackrel{\sim }{\beta }}_{n,k}^{\left(m\right)}$. The reason for looking at the special value Eq. 43 is that Eq. 37 for this value of $a$ says that the generating function ${\mathbb{B}}_k\left(x\right)$ ($k\ge 0$) is given by

$${\mathbb{B}}_k\left(x\right)=\prod _{r=1}^{\infty }\frac{\left(1-q^{-k-r}x\right)\left(1-q^{k+1-r}x\right)}{\left(1-q^{-r}x\right)\left(1-q^{1-r}x\right)}=\prod _{j=1}^k\frac{1\hspace{0.17em}-\hspace{0.17em}{q}^{j}x}{1\hspace{0.17em}-\hspace{0.17em}{q}^{-j}x}$$ [45]

(in particular, it is a rational function of $x$) and also that the numbers $v_{m,k}$ are given by

$$v_{m,k}=\left\{\begin{array}{cc}{\left(-1\right)}^{m-1}\hspace{0.17em}{q}^{\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)-km}\hspace{0.17em}\frac{{\left(q\right)}_{k+m}}{{\left(q\right)}_m\hspace{0.17em}{\left(q\right)}_{m-1}\hspace{0.17em}{\left(q\right)}_{k-m}}\hspace{1em}\hfill & \hspace{1em}\text{if}1\le m\le k,\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0\hspace{1em}\hfill & \hspace{1em}\text{if}m>k,\hfill \end{array}\right.$$ [46]

as one checks easily. (Here and for the rest of the section we use the notation ${\left(x\right)}_n$ for the $q$-Pochhammer symbol $\left(1-x\right)\left(1-qx\right)\cdots \left(1-q^{n-1}x\right)$, with ${\left(x\right)}_0=1$.) After a considerable amount of computer experimentation, we found that the generating function ${\mathbb{B}}_k^{\left(m\right)}\left(x\right)$ is given by the following closed formula:

Proposition 14.

For $k\ge 0$ and $m\ge 1$ the generating function ${\mathbb{B}}_k^{\left(m\right)}\left(x\right)={\mathbb{B}}^{\left(m\right)}\left(x;a_k,q\right)$ is a rational function of $x$, equal to 0 if $m>k$ and otherwise given by

$${\mathbb{B}}_k^{\left(m\right)}\left(x\right)={\left(-1\right)}^{m-1}\hspace{0.17em}\frac{{\left(q\right)}_{m+k}}{{\left(q\right)}_k{\left(q\right)}_{m-1}}\hspace{0.17em}\frac{x^m\hspace{0.17em}{Y}_k^{\left(m\right)}\left(x\right)}{D_k\left(x\right)},$$ [47]

where $D_k\left(x\right)\in \mathbb{Z}\left[q,x\right]$ is defined by the product expansion

$$D_k\left(x\right)=\prod _{j=1}^k\left(q^j-x\right)$$

and where $Y_k^{\left(m\right)}\left(x\right)\in \mathbb{Z}\left[q,x\right]$ is the polynomial of degree $k-m$ defined by

$$Y_k^{\left(m\right)}\left(x\right)=\sum _{r=0}^{k-m}{q}^{\left(\genfrac{}{}{0.0pt}{}{r+1}{2}\right)+\left(\genfrac{}{}{0.0pt}{}{k-m-r+1}{2}\right)}\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k}{r}\right]\hspace{0.17em}\left[\begin{array}{c}\hfill k\hfill \\ \hfill k-m-r\hfill \end{array}\right]\hspace{0.17em}{x}^r\hspace{0.17em}=\hspace{0.17em}\mathrm{Coefficient}\mathrm{o}\mathrm{f}\hspace{0.17em}{T}^{k-m}\hspace{0.17em}\text{in}\hspace{0.17em}\prod _{j=1}^k\left(1+q^{j}T\right)\left(1+q^{j}Tx\right).$$ [48]

The symbol $\left[\genfrac{}{}{0ex}{}{k}{r}\right]$ used in Eq. 48 is the $q$-binomial coefficient $\frac{{\left(q\right)}_k}{{\left(q\right)}_r{\left(q\right)}_{k-r}}$, which occurs in the following two well-known $q$ versions of the binomial theorem

$$\sum _{r=0}^k\hspace{0.17em}{\left(-1\right)}^r{q}^{\left(\genfrac{}{}{0.0pt}{}{r}{2}\right)}\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k}{r}\right]\hspace{0.17em}{x}^r={\left(x\right)}_k,\hspace{1em}\hspace{1em}\sum _{r=0}^{\infty }\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k+r-1}{r}\right]\hspace{0.17em}{x}^r=\frac{1}{{\left(x\right)}_k},$$ [49]

where $k$ denotes an integer $\ge 0$. The equality of the two expressions in Eq. 48 follows from the first of these formulas.

The proof of Proposition 14 is given in the next subsection. Here we show that it implies our main identity Eq. 37. For this, as we have already explained, it suffices to show that the sum over $m\ge 1$ of the rational functions Eq. 47 coincides with the right-hand side of Eq. 45. Combining Eq. 47 with the second part of Eq. 48 and the second part of Eq. 49, we find

$$\frac{1}{x}\hspace{0.17em}\frac{D_k\left(x\right)}{1-q^{k+1}}\hspace{0.17em}\sum _{m=1}^{\infty }{\mathbb{B}}_k^{\left(m\right)}\left(x\right)=\sum _{m=1}^k\hspace{0.17em}{\left(-x\right)}^{m-1}\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k+m}{k+1}\right]\hspace{0.17em}{Y}_k^{\left(m\right)}\left(x\right)\hspace{0.17em}=\hspace{0.17em}\mathrm{Coefficient}\hspace{0.17em}\mathrm{o}\mathrm{f}\hspace{0.17em}{T}^{k-1}\hspace{0.17em}\text{in}\hspace{0.17em}\frac{{\prod }_{j=1}^k\left(1+q^{j}T\right)}{\left(1+xT\right)\left(1+q^{k+1}xT\right)}.$$

But by comparing poles and residues (partial fractions decomposition), we see that

$$\frac{{\prod }_{j=1}^k\left(1+q^{j}T\right)}{\left(1+xT\right)\left(1+q^{k+1}xT\right)}=\frac{{\prod }_{j=1}^k\left(1-q^j{x}^{-1}\right)}{\left(1-q^{k+1}\right)\left(1+xT\right)}\hspace{0.17em}+\hspace{0.17em}\frac{{\prod }_{j=1}^k\left(1-q^{j-k-1}{x}^{-1}\right)}{\left(1-q^{-k-1}\right)\left(1+q^{k+1}xT\right)}\hspace{0.17em}+\hspace{0.17em}{P}_{k-2}\left(T\right),$$

where $P_{k-2}\left(T\right)$ is a polynomial of degree $\le k-2$ in $T$. It follows that

$$\sum _{m=1}^{\infty }{\mathbb{B}}_k^{\left(m\right)}\left(x\right)=\frac{{\left(-x\right)}^k}{D_k\left(x\right)}\hspace{0.17em}\left(-\prod _{j=1}^k\left(1-q^j{x}^{-1}\right)+\hspace{0.17em}{q}^{k\left(k+1\right)}\hspace{0.17em}\prod _{j=1}^k\left(1-q^{j-k-1}{x}^{-1}\right)\right)=-1+\hspace{0.17em}\hspace{0.17em}\prod _{j=1}^k\frac{1\hspace{0.17em}-q^{j}x}{1\hspace{0.17em}-q^{-j}x}\hspace{0.17em}=-1\hspace{0.17em}+{\mathbb{B}}_k\left(x\right),$$

where the final equality is Eq. 45. Since ${\mathbb{B}}_k^{\left(0\right)}\left(x\right)=1$, this completes the proof that the sum of the functions ${\mathbb{B}}^{\left(m\right)}\left(x\right)$ defined recursively by Eq. 42 coincides with the right-hand side of Eq. 37 and hence, by what has already been said, completes the proof of Theorem 11.

C. Completion of the Proof.

It remains to prove Proposition 14. For this purpose we reverse the order of the logic, taking Eq. 47 (with $Y_k^{\left(m\right)}\left(x\right)$ defined as 0 if $m>k$) as the definition of the power series ${\mathbb{B}}_k^{\left(m\right)}\left(x\right)$ for all $m\ge 1$ and $k\ge 0$ and then proving that these power series satisfy the identity Eq. 44. Inserting Eqs. 45, 47, and 48 into Eq. 44, we see after multiplying both sides by a common factor that the identity to be proved is

$$q^{km-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)}\hspace{0.17em}{Y}_k^{\left(m\right)}\left(x\right)=\left[\genfrac{}{}{0ex}{}{k}{m}\right]\hspace{0.17em}{D}_k\left(x\right)\hspace{0.17em}+\hspace{0.17em}\frac{{\left(q\right)}_{k+1}}{{\left(q\right)}_m{\left(q\right)}_{k-m}}\hspace{0.17em}\sum _{p=1}^k\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k+p}{k+1}\right]\hspace{0.17em}{Y}_k^{\left(p\right)}\left(x\right)\hspace{0.17em}\frac{{\left(-x\right)}^p}{q^{m+p}-1}.$$ [50]

But by a simpler version of the same partial fraction argument as the one that was used above we see that $D_k\left(x\right)$ is the coefficient of $T^k$ in ${\left(1+xT\right)}^{-1}{\prod }_{j=1}^k\left(1+q^{j}T\right)$, and one also sees without difficulty that $Y_k^{\left(m\right)}\left(x\right)$ equals ${\left(q^{k+1}x\right)}^{-m}$ times the coefficient of $T^{k+m}$ in the same product ${\prod }_{j=1}^k\left(1-q^{j}T\right)\left(1-q^{j}Tx\right)$ as the one used in the original definition Eq. 48, so that the left-hand side of Eq. 50 can be written, using the first equation in Eq. 49, as

$$q^{km-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)}\hspace{0.17em}{Y}_k^{\left(m\right)}\left(x\right)=\text{Coefficient of}{\hspace{0.17em}T}^k\text{in}\hspace{0.17em}\prod _{j=1}^k\left(1+q^{j}T\right)\hspace{0.17em}\cdot \hspace{0.17em}\sum _{s=0}^{k-m}{q}^{\left(\genfrac{}{}{0.0pt}{}{s+1}{2}\right)+ms}\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k}{m+s}\right]\hspace{0.17em}{\left(xT\right)}^s.$$

The identity Eq. 50 then follows immediately from the following lemma by replacing $x$ by $xT$, multiplying both sides by ${\prod }_{j=1}^k\left(1+q^{j}T\right)$, and comparing the coefficients of $T^k$ on both sides.

Lemma 15.

For fixed $k\ge 0$ and $m\ge 1$, define two power series ${\mathcal{F}}_1\left(x\right)$ and ${\mathcal{F}}_2\left(x\right)$ by

$$\begin{array}{cc}\hfill {\mathcal{F}}_1\left(x\right)& ={\left(-qx\right)}_k\hspace{0.17em}\sum _{p=1}^{\infty }\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k+p}{k+1}\right]\hspace{0.17em}\frac{{\left(-x\right)}^p}{q^{m+p}-1},\hfill \\ \hfill {\mathcal{F}}_2\left(x\right)& =\left[\genfrac{}{}{0ex}{}{k}{m}\right]\hspace{0.17em}{\left(1+x\right)}^{-1}\hspace{0.17em}-\hspace{0.17em}\sum _{s=0}^{k-m}{q}^{\left(\genfrac{}{}{0.0pt}{}{s+1}{2}\right)+ms}\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k}{m+s}\right]\hspace{0.17em}{x}^s.\hfill \end{array}$$

Then

$${\mathcal{F}}_2\left(x\right)\hspace{0.17em}=\hspace{0.17em}-\hspace{0.17em}\frac{{\left(q\right)}_{k+1}}{{\left(q\right)}_m{\left(q\right)}_{k-m}}{\mathcal{F}}_1\left(x\right).$$ [51]

Proof:

The power series ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ satisfy the functional equations

$$\left(1+qx\right)\hspace{0.17em}{q}^m{\mathcal{F}}_1\left(qx\right)\hspace{0.17em}-\hspace{0.17em}\left(1+q^{k+1}x\right)\hspace{0.17em}{\mathcal{F}}_1\left(x\right)\hspace{0.17em}=\hspace{0.17em}{\left(-qx\right)}_{k+1}\hspace{0.17em}\sum _{p=1}^{\infty }\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k+p}{p-1}\right]\hspace{0.17em}{\left(-x\right)}^p\hspace{0.17em}=\hspace{0.17em}\frac{-x}{1+x}$$

(here we have used the second equation of Eq. 49) and

$$\begin{array}{cc}\hfill & \left(1+qx\right)\hspace{0.17em}{q}^m{\mathcal{F}}_2\left(qx\right)\hspace{0.17em}-\hspace{0.17em}\left(1+q^{k+1}x\right)\hspace{0.17em}{\mathcal{F}}_2\left(x\right)\hfill \\ \hfill & =\left[\genfrac{}{}{0ex}{}{k}{m}\right]\hspace{0.17em}\left(q^m-\frac{1+q^{k+1}x}{1+x}\right)\hspace{0.17em}-\hspace{0.17em}\sum _{s=0}^{k-m}\hspace{0.17em}{q}^{\left(\genfrac{}{}{0.0pt}{}{s+1}{2}\right)+ms}\left[\genfrac{}{}{0ex}{}{k}{m+s}\right]\hspace{0.17em}\left\{\left(q^{m+s}-1\right)-\left(q^{k-m-s}-1\right)q^{s+1+m}x\right\}\hspace{0.17em}{x}^s\hfill \\ \hfill & \hspace{1em}\hspace{1em}=\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k}{m}\right]\hspace{0.17em}\left(\left(1-q^{k+1}\right)\hspace{0.17em}\frac{x}{1+x}\hspace{0.17em}-\hspace{0.17em}\left(1-q^m\right)\right)\hspace{0.17em}+\hspace{0.17em}\left(1-q^k\right)\hspace{0.17em}\sum _{s=0}^{k+m}\left[\genfrac{}{}{0ex}{}{k-1}{m+s-1}\right]\hspace{0.17em}{q}^{\left(\genfrac{}{}{0.0pt}{}{s+1}{2}\right)+ms}{x}^s\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}-\hspace{0.17em}\left(1-q^k\right)\sum _{s=0}^{k+m-1}\hspace{0.17em}\left[\genfrac{}{}{0ex}{}{k-1}{m+s}\right]\hspace{0.17em}{q}^{\left(\genfrac{}{}{0.0pt}{}{s+2}{2}\right)+m\left(s+1\right)}{x}^{s+1}\hfill \\ \hfill & =\frac{{\left(q\right)}_{k+1}}{{\left(q\right)}_m{\left(q\right)}_{k-m}}\hspace{0.17em}\frac{x}{1+x}\hspace{1em}\hspace{1em}\hspace{1em}\left(\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{g}\hspace{0.17em}\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{s}\right).\hfill \end{array}$$

Together these imply Eq. 51, since it is easily seen that a power series $\mathcal{F}\left(x\right)$ satisfying $\left(1+qx\right)q^m\mathcal{F}\left(qx\right)=\left(1+q^{k+1}x\right)\mathcal{F}\left(x\right)$ for some integers $k\ge 0$ and $m\ge 1$ must vanish identically.

This completes the proof of Lemma 15, Proposition 14, and hence also of Theorem 5.

4. The Riemann Hypothesis

A. Proof of Proposition 6.

We can use the recursion relation Eq. 39 to give an easy inductive proof of the inequality Eq. 17, which, as we will see in a moment, implies the Riemann hypothesis for our zeta functions. Indeed, Eq. 17 holds for $n=2$ since

$$\frac{{\beta }_2}{{\beta }_1}=\frac{q^2+q-a}{q^2-1}=1\hspace{0.17em}+\hspace{0.17em}\frac{N}{q^2-1},$$

where $N=q-a+1=\left|E\left({\mathbb{F}}_q\right)\right|$ satisfies $0<N<2q+2$, and if $n\ge 3$ and we assume by induction on $n$ that Eq. 17 holds for $n-1$, then Eq. 39 gives

$$\frac{{\beta }_n}{{\beta }_{n-1}}\hspace{0.17em}>\hspace{0.17em}\frac{q^n+q^{n-1}-a\hspace{0.17em}-\hspace{0.17em}\left(q^{n-1}-q\right)}{q^n-1}=1\hspace{0.17em}+\hspace{0.17em}\frac{N}{q^n-1}\hspace{0.17em}>\hspace{0.17em}1$$

and

$$\begin{array}{cc}\hfill \left(q^n-1\right)\hspace{0.17em}\left(\frac{q^{n/2}+1}{q^{n/2}-1}\hspace{0.17em}-\hspace{0.17em}\frac{{\beta }_n}{{\beta }_{n-1}}\right)& ={\left(q^{n/2}+1\right)}^2\hspace{0.17em}-\hspace{0.17em}\left(q^n+q^{n-1}-a\right)\hspace{0.17em}+\hspace{0.17em}\left(q^{n-1}-q\right)\hspace{0.17em}\frac{{\beta }_{n-2}}{{\beta }_{n-1}}\hfill \\ \hfill & \hspace{0.17em}>\hspace{0.17em}2q^{n/2}+1-q^{n-1}-\left(q+1\right)+\left(q^{n-1}-q\right)\hspace{0.17em}\frac{q^{\left(n-1\right)/2}-1}{q^{\left(n-1\right)/2}+1}\hfill \\ \hfill & =\frac{2\left(q^{n-1}-q^{n/2}\right)\left(q^{1/2}-1\right)}{q^{\left(n-1\right)/2}+1}\hspace{0.17em}>\hspace{0.17em}0\hfill \end{array}$$

(where we have again used only $\left|a\right|<q+1$, and not the stronger estimate $\left|a\right|\le 2\sqrt{q}$ given by the usual Riemann hypothesis of $E/{\mathbb{F}}_q$), completing the proof of Eq. 17 by induction.

In fact, the estimates Eq. 17 are quite wasteful, and by a more careful analysis one finds that

$$\frac{{\beta }_n}{{\beta }_{n-1}}=1\hspace{0.17em}+\hspace{0.17em}\frac{\left(n-1\right)\left(q-a+1\right)\hspace{0.17em}-\hspace{0.17em}c\left(q\right)}{q^n}\hspace{0.17em}+\hspace{0.17em}\text{O}\left(\frac{n^2}{q^{2n-2}}\right)$$ [52]

uniformly as $q^n\to \infty$, where $c\left(q\right)=2+3\left(a-2\right)/q\hspace{0.17em}+\hspace{0.17em}\cdots \hspace{0.17em}$ is independent of $n$. (Recall that $a=\text{O}\left(\sqrt{q}\right)$.) We also remark that the bounds Eq. 17 together with the initial value ${\beta }_0=1$ give upper and lower estimates for each ${\beta }_n$. In particular, we have the uniform estimate

$${\beta }_n=1\hspace{0.17em}+\hspace{0.17em}\text{O}\left(1/\hspace{0.17em}\sqrt{q}\right),$$ [53]

where the implicit constant is universal and can be taken, e.g., to be 3.

B. Proof of the Riemann Hypothesis.

By Eqs. 6 and 12, the polynomial $P_{E,n}\left(T\right)$ appearing in Eq. 4 is given by

$$\frac{1}{{\alpha }_{E,n}\left(0\right)}\hspace{0.17em}{P}_{E,n}\left(T\right)=1-\left(\left(Q+1\right)-\left(Q-1\right)\frac{{\beta }_{E,n}\left(0\right)}{{\beta }_{E,n-1}\left(0\right)}\right)T+Q{T}^2,$$ [54]

and by the inequalities Eq. 17 the coefficient of $T$ in the second factor lies between $-2$ and $2\sqrt{Q}$. Theorem 7 follows immediately.

Note that this argument gives much more than just the Riemann hypothesis, for which we would only need that the coefficient of $T$ is between $-2\sqrt{Q}$ and $2\sqrt{Q}$. In fact, inserting Eq. 52 into Eq. 54, we see that the reciprocal roots of $P_{E,n}\left(T\right)$, divided by $q^{n/2}$, are not uniformly distributed on the unit circle, but are actually very near to $i$ and $-i$ for $n$ large. In a related direction, we mention that, since each ${\beta }_{E,n}\left(a\right)$ is completely determined by $n$, $q$, and $a$, the usual Sato–Tate distribution property for the roots of the local zeta functions of the reductions $Ē/{\mathbb{F}}_p$ at varying primes $p$ of an elliptic curve $E$ defined over implies a corresponding explicit Sato–Tate distribution for the roots of the higher zeta functions ${\zeta }_{Ē/{\mathbb{F}}_p,\hspace{0.17em}n}$ as $p$ varies with $n$ fixed and also, after a suitable renormalization, as $n\to \infty$.

5. Complements

The most important consequence of Theorem 5, of course, is the Riemann hypothesis for the higher-rank zeta functions ${\zeta }_{E,n}\left(s\right)$, but Theorem 5 has several other corollaries that seem to be of independent interest. We end the paper by listing some of these.

The first statement concerns the analytic continuation and functional equation of the Dirichlet series defined by Eq. 15.

Corollary 16.

The function continues meromorphically to the entire complex plane and satisfies the functional equation

[55]

Proof:

The meromorphic continuation is obvious from Eq. 16, since ${\zeta }_E\left(s\right)$ is meromorphic and tends rapidly to 1 as . The functional equation Eq. 55 then follows tautologically from Eq. 16.$\square$

Corollary 17.

The meromorphic function defined by

[56]

is invariant under $s\mapsto s+1$.

Proof:

This follows from Eq. 55 and the functional equation of ${\zeta }_{E/{\mathbb{F}}_q}\left(s\right)$:

Alternatively, we could apply the functional equation of ${\zeta }_{E/{\mathbb{F}}_q}\left(s\right)$ to each factor of the infinite product defining to write as the absolutely convergent doubly infinite product ${\prod }_{n\in \mathbb{Z}}{\zeta }_{E/{\mathbb{F}}_q}\left(s+n\right)$, from which the periodicity is obvious.$\square$

There is a curious relation between Corollary 17 and the theory of elliptic curves over $\mathbb{C}$. Denote by $\theta \left(x;q^{-1}\right)$ the Jacobi theta function

$$\theta \left(x;\hspace{0.17em}{q}^{-1}\right)=\sum _{n\in \mathbb{Z}+\frac{1}{2}}{\left(-1\right)}^{\left[n\right]}\hspace{0.17em}{q}^{-n^2/2}\hspace{0.17em}{x}^n\hspace{1em}\hspace{1em}\left(q,\hspace{0.17em}x\in {\mathbb{C}}^{*},\hspace{0.17em}\hspace{0.17em}\left|q\right|>1\right).$$

It has the well-known elliptic transformation property

$$\theta \left(qx;\hspace{0.17em}{q}^{-1}\right)=-\hspace{0.17em}{q}^{1/2}\hspace{0.17em}x\hspace{0.17em}\theta \left(x;\hspace{0.17em}{q}^{-1}\right)$$ [57]

saying that the function $\theta \left(e^{2\pi iz};e^{2\pi i\tau }\right)$ is doubly periodic, up to simple nonvanishing factors, with respect to translation of $z\in \mathbb{C}$ by the lattice $\mathbb{Z}\tau +\mathbb{Z}$. The Jacobi triple-product formula is the formula

$$\theta \left(x;q^{-1}\right)=q^{-1/8}{x}^{1/2}\hspace{0.17em}{\left(q^{-1};\hspace{0.17em}{q}^{-1}\right)}_{\infty }\hspace{0.17em}{\left(q^{-1}x;\hspace{0.17em}{q}^{-1}\right)}_{\infty }\hspace{0.17em}{\left(x^{-1};\hspace{0.17em}{q}^{-1}\right)}_{\infty }$$

expressing $\theta \left(x;q^{-1}\right)$ as a product of three infinite $q$-Pochhammer symbols. Combining this with Eq. 38, we find that the symmetrized zeta function Eq. 56 is related to the Jacobi theta function by

[58]

so that the periodicity statement of Corollary 17 can also be seen as a consequence of the elliptic transformation property Eq. 57 of $\theta \left(x;q^{-1}\right)$. This gives some kind of connection between the zeta function of an elliptic curve $E$ over ${\mathbb{F}}_q$ and the theory of elliptic functions for the elliptic curve ${\mathbb{C}}^{*}/q^{\mathbb{Z}}$ over $\mathbb{C}$.

In the next statement, our result for elliptic curves over finite fields is used to motivate the definition of a zeta function for elliptic curves defined over and to prove a factorization result for this function.

Corollary 18.

Let $E$ be an elliptic curve over , and define as the multiplicative function with . Then the Dirichlet series defined for $s\in \mathbb{C}$ with by

continues meromorphically to all $s$ and has the product expansion

We do not know whether these higher global zeta functions have other interesting properties.

The final corollary of Theorem 5 that we give concerns the limiting values of the invariants we have been studying. To explain it properly, we must first recall the geometric meaning of the numbers $v_{E,n}$ occurring in Theorem 4. In Eq. 14 these numbers were simply defined as the products of the values of ${\zeta }_{E/{\mathbb{F}}_q}\left(s\right)$ at $s=1,\dots ,n$ (with the value at the pole $s=1$ being replaced by a suitable limit), because this was all that was necessary for our purposes. But that formula is actually a theorem, due to Desale and Ramanan in the paper (7) already quoted, rather than a definition. In fact, $v_{E,n}$ is $v_{E,n}\left(0\right)$, where $v_{X,n}\left(d\right)$ is defined, for all curves $X/{\mathbb{F}}_q$ and for all integers $n>0$ and $d$, by

$$v_{X,n}\left(d\right)=\sum _{\mathrm{a}\mathrm{l}\mathrm{l}\left[V\right]}\frac{1}{\left|\mathrm{A}\mathrm{u}\mathrm{t}\left(V\right)\right|},$$

i.e., by the same summation as ${\beta }_{X,n}\left(d\right)$ in Eq. 2, but with the summation now ranging over all isomorphism classes of ${\mathbb{F}}_q$-rational vector bundles of rank $n$ and degree $d$ rather than just the semistable ones. Using the fact that the Tamagawa number of $\hspace{0.17em}\mathrm{S}\mathrm{L}\left(n\right)$ equals 1, one shows (proposition 1 of ref. 7, summed over all $\hspace{0.17em}\left|{\mathrm{P}\mathrm{i}\mathrm{c}}^0\left(X\right)\left({\mathbb{F}}_q\right)\right|=\left(q-1\right){\hat{\zeta }}_X^{*}\left(1\right)$ possible values of the determinant) that $v_{X,n}\left(d\right)=q^{\left(g-1\right)\hspace{0.17em}n\left(n-1\right)/2}\hspace{0.17em}{\hat{v}}_{Xn}$ (independent of $d\hspace{0.17em}$!) with ${\hat{v}}_{Xn}$ defined as in Eq. 30; i.e., $v_{X,n}\left(d\right)$ is related to ${\hat{v}}_{Xn}$ in the same way as ${\beta }_{X,n}\left(d\right)$ and ${\hat{\beta }}_{X,n}\left(d\right)$ are related in Eq. 29. Note that this formula includes as a special case the formula $v_{E/{\mathbb{F}}_q}\left(n,0\right)=v_{E,n}$ mentioned above. Also, since semistable bundles form a subset of all bundles, it is clear from the geometric definition that ${\beta }_{X,n}\left(d\right)\le v_{X,n}\left(d\right)$ and ${\hat{\beta }}_{X,n}\left(d\right)\le {\hat{v}}_{Xn}$ for all $n$, with equality if $n=1$.^† (This is also visible in the Harder–Narasimhan–Desale–Ramanan recursion Eq. 31, in which the $k=1$ term on the left equals ${\hat{\beta }}_{X,n}\left(d\right)$.) Therefore the following result can be interpreted as saying that, at least in the case of elliptic curves, “almost all bundles of large rank are semistable.”

Corollary 19.

The limiting values ${\beta }_{E,\infty }≔\underset{n}{lim}{\beta }_{E,n}$ and $v_{E,\infty }≔\underset{n}{lim}{v}_{E,n}$ of the sequences $\left\{{\beta }_{E,n}\left(0\right)\right\}$ and $\left\{v_{E,n}\right\}$ exist and coincide, with the value

$${\beta }_{E,\infty }=v_{E,\infty }={\zeta }_E^{*}\left(1\right)\hspace{0.17em}{\zeta }_E\left(2\right)\hspace{0.17em}{\zeta }_E\left(3\right)\hspace{0.17em}\cdots \hspace{0.17em}.$$ [59]

Of course the uniform bound for the numbers ${\beta }_n$ that we gave in Eq. 53 also holds for the limiting value ${\beta }_{\infty }$.

Just for fun, we mention that the analog of the product appearing on the right-hand side of Eq. 59 when the function field ${\mathbb{F}}_q\left(E\right)$ is replaced by the number field is the number ${\prod }_{n=2}^{\infty }\zeta \left(n\right)=2.2948565916733\cdots \hspace{0.17em}$, which has a well-known interpretation as the average number of abelian groups of given order. It would be interesting to know whether the product occurring in Eq. 59, or its global analog

has any similar geometrical or arithmetical interpretation. In particular, one can ask whether there is any connection with the famous Cohen–Lenstra class number heuristics. (Compare Eq. 16 with theorem 3.2 (ii) of ref. 10, or Corollary 17 with theorem 7.1 of ref. 11.)

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