The Number of Rational Points of Two Parameter Calabi-Yau manifolds as Toric Hypersurfaces

For the elliptic curve $X(\tau)$ defined over finite field $\mathbb{F}_{p}$ with complex moduli $\tau$, the number of rational points of the curve $X(\tau)$ can be expressed as [2] :

$$\nu(\tau)=(-1)(-1)^{(p-1)/2}\sum_{n=0}^{\infty}\binom{-1/2}{r}^{2}\tau^{n}$$

This is the same as the expression of the period $\pi(\tau)$ of $X(\tau)$. For the quintic Calabi-Yau manifold $M_{3}(\psi)$ in [4], the defining equation is:

$$P(x;\psi)=\sum_{i=1}^{5}x_{i}^{5}-5\psi\prod_{i=1}^{5}x_{i}$$

Where $x\in\mathbb{F}^{5}_{p}$ , and $\psi\in\mathbb{F}_{p}$ is the complex moduli. According to (11) in Section 2, one can write the number of rational points of $M_{3}(\psi)$ as :

$\displaystyle\nu(\psi)=p^{5}-\sum_{x\in\mathbb{F}_{p}^{5}}\left(\sum_{i=1}^{5}x_{i}^{5}-5\psi\prod_{i=1}^{5}x_{i}\right)^{p-1}+o(p)$

the number of rational points can be obtained by simplification:

$\displaystyle\nu(\psi)=\sum_{n=0}^{[\frac{p}{5}]}\frac{(5n)!}{(n!)^{5}}z^{n}+o(p)$

(12)

where $z=(5\psi)^{-5}$. This is formally corresponds to the fundamental period of quintic:

$$f_{0}(\psi)=\sum_{n=0}^{\infty}\frac{(5n)!}{(n!)^{5}}z^{n}$$

(13)

the truncation of the summationthe is the only difference. It can be proved by the formula $(ap+b)!=a!b!(p!)^{a}+o(p^{2})$ in $p$-adic analysis that the truncation is generated by transferring the series $\sum_{n=0}^{\infty}a_{n}z^{n}$ to the finite field $\mathbb{F}_{p}$. Then one can find the equation between the number of rational points and fundamental period of quntic over finite field under zeroth order approximation:

$$\nu(\psi)=\ ^{[p/5]}f_{0}(\psi)+o(p)$$

(14)

where $[p/5]$ indicates the above truncation.

There is a similar discussion about first-order corrections of the number of points, and the exact expression of the number of rational points can be guess as:

$$\nu(\psi)=\sum_{k=0}^{4}\frac{1}{k!}\left(\frac{p}{1-p}\right)^{k}\ {}^{(p-1)}f^{(k)}_{k}(z^{p^{4}})\ \mathrm{mod}\ p^{5}$$

(15)

Where $f^{(k)}_{k}$ represents the $k$ logarithmic derivative $(z\frac{d}{dz})^{k}f_{k}$ of the $k$-th period $f_{k}$, and this formula is verified by numerical calculation.

This result relates the important geometric quantity, periods, for the calculation of effective superpotential in physical superstring theory to the number of rational points in arithmetic geometry for the first time, and gives a fairly accurate equation. It is still one of the few studies of Calabi-Yau manifolds, the inner spaces of superstrings, over finite fields.

In this article, Calabi-Yau manifolds will be described by the language of toric geometry [10]. Compact Calabi-Yau can be defined as hypersurfaces in toric variety [11]. Consider a Laurent polynomials:

$$f(x)=\sum c_{m}x^{m}$$

(16)

its monomial corresponds to a convex polyhedron $\Delta$ in lattice space $M_{\mathbb{Q}}$ :

$$\Delta(f)\subset M_{\mathbb{Q}}$$

which is the convex hull of monomial whose coefficients are not equal to zero. In fact, this polyhedron $\Delta$ is the same one that defines the toric variety $P_{\Delta}=\mathrm{Proj}\ S_{\Delta}$. The zeros set of each Laurent polynomial corresponds to an affine hypersurface $Z_{f,\Delta}$ in $P_{\Delta}$ :

$$Z_{f,\Delta}=\{x\in P_{\Delta}\ |\ f(x)=0\}$$

a compact Calabi-Yau manifold can then be defined. If the polyhedron $\Delta$ is a simplex, or if its vertices give only one linear relation, then the corresponding Calabi-Yau manifold has one-parameter. Otherwise, a multi-parameter object can be obtained.

In [6], some examples of the number of rational points of Calabi-Yau manifolds are examined in detail by Gauss sum. For our purposes, the relation between rational points and periods of multi-parameter hypersurfaces is expected to be given. Taking the two-parameter case as an example, for the $n$-dimensional toric variety embedded in the $N+n$-dimensional ambient space, there is a general two-parameter hypersurface equation:

$\displaystyle P(x;\phi,\psi)=\sum_{\mathbb{m}\in\Delta}\alpha_{\mathbb{m}}x^{\mathbb{m}}=\sum_{\mathbb{m}\in\Delta^{\prime}}\alpha_{\mathbb{m}}x^{\mathbb{m}}-\phi Q-\psi U$

(17)

We use Fermat’s little theorem (10) in Section 2 , instead of the Gauss sum, to compute the number of rational points in zeroth order:

	$\displaystyle\nu_{0}(\phi,\psi)$	$\displaystyle=$	$\displaystyle\sum_{x\in\mathbb{F}_{p}^{N+n}}(1-(P(x;\phi,\psi))^{p-1})$
		$\displaystyle=$	$\displaystyle p^{N+n}-\sum_{x\in\mathbb{F}_{p}^{N+n}}(\sum_{\mathbb{m}\in\Delta^{\prime}}\alpha_{\mathbb{m}}x^{\mathbb{m}}-\phi Q-\psi U)^{p-1}$

bt polynomial theorem , above equation can be expand as:

$\displaystyle\nu_{0}(\phi,\psi)$

$\displaystyle=$

$\displaystyle\sum_{x\in\mathbb{F}_{p}^{*N+n}}\sum_{n_{1},n_{2}}\frac{(p-1)!}{(p-1-n_{1}-n_{2})!n_{1}!n_{2}!}(\sum_{\mathbb{m}\in\Delta^{\prime}}\alpha_{\mathbb{m}}x^{\mathbb{m}})^{n_{1}+n_{2}}(-\phi Q)^{-n_{1}}(-\psi U)^{-n_{2}}$

considering that $p$ is infinitesimal in $p$-adic analysis, we has the following formula:

	$\displaystyle\frac{(p-1)!}{(p-1-n_{1}-n_{2})!}$	$\displaystyle=$	$\displaystyle(p-1)(p-2)...(p-1-n_{1}-n_{2})$		(18)
		$\displaystyle=$	$\displaystyle(-1)^{n_{1}+n_{1}}(n_{1}+n_{2})!+o(p)$

in terms of (18), the number of rational points can be reduced to:

	$\displaystyle\nu_{0}(\phi,\psi)$	$\displaystyle=$	$\displaystyle\sum_{x\in\mathbb{F}_{p}^{*N+n}}\sum_{n_{1},n_{2}}\frac{(-1)^{n_{1}+n_{1}}(n_{1}+n_{2})!}{n_{1}!n_{2}!}(\sum_{\mathbb{m}\in\Delta^{\prime}}\alpha_{\mathbb{m}}x^{\mathbb{m}})^{n_{1}+n_{2}}(-\phi Q)^{-n_{1}}(-\psi U)^{-n_{2}}$
		$\displaystyle=$	$\displaystyle\sum_{x\in\mathbb{F}_{p}^{*N+n}}\sum_{n_{1},n_{2}}\frac{(n_{1}+n_{2})!}{n_{1}!n_{2}!}\sum_{\sum n_{\mathbb{m}}=n_{1}+n_{2}}\frac{(n_{1}+n_{2})!}{\prod_{\mathbb{m}}n_{\mathbb{m}}!}\prod_{\mathbb{m}\in\Delta^{\prime}}(\alpha_{\mathbb{m}}x^{\mathbb{m}})^{n_{\mathbb{m}}}\frac{1}{\phi^{n_{1}}\psi^{n_{2}}Q^{n_{1}}U^{n_{2}}}$
		$\displaystyle=$	$\displaystyle\sum_{n_{1},n_{2}}\frac{[(n_{1}+n_{2})!]^{2}}{n_{1}!n_{2}!}\frac{1}{\phi^{n_{1}}\psi^{n_{2}}}\sum_{\sum n_{\mathbb{m}}=n_{1}+n_{2}}\prod_{\mathbb{m}\in\Delta^{\prime}}\frac{\alpha^{n_{\mathbb{m}}}_{\mathbb{m}}}{n_{\mathbb{m}}!}\sum_{x\in\mathbb{F}_{p}^{*N+n}}x^{\sum_{\mathbb{m}\in\Delta^{\prime}}{n_{\mathbb{m}}}\mathbb{m}-n_{1}\mathbb{q}-n_{2}\mathbb{u}}$

further expansion by polynomial theorem, finally we have:

	$\displaystyle\nu_{0}(\phi,\psi)$	$\displaystyle=$	$\displaystyle\sum_{x\in\mathbb{F}_{p}^{*N+n}}\sum_{n_{1},n_{2}}\frac{(n_{1}+n_{2})!}{n_{1}!n_{2}!}\sum_{\sum n_{\mathbb{m}}=n_{1}+n_{2}}\frac{(n_{1}+n_{2})!}{\prod_{\mathbb{m}}n_{\mathbb{m}}!}\prod_{\mathbb{m}\in\Delta^{\prime}}(\alpha_{\mathbb{m}}x^{\mathbb{m}})^{n_{\mathbb{m}}}\frac{1}{\phi^{n_{1}}\psi^{n_{2}}Q^{n_{1}}U^{n_{2}}}$
		$\displaystyle=$	$\displaystyle\sum_{n_{1},n_{2}}\frac{[(n_{1}+n_{2})!]^{2}}{n_{1}!n_{2}!}\frac{1}{\phi^{n_{1}}\psi^{n_{2}}}\sum_{\sum n_{\mathbb{m}}=n_{1}+n_{2}}\prod_{\mathbb{m}\in\Delta^{\prime}}\frac{\alpha^{n_{\mathbb{m}}}_{\mathbb{m}}}{n_{\mathbb{m}}!}\sum_{x\in\mathbb{F}_{p}^{*N+n}}x^{\sum_{\mathbb{m}\in\Delta^{\prime}}{n_{\mathbb{m}}}\mathbb{m}-n_{1}\mathbb{q}-n_{2}\mathbb{u}}$

This is the zeroth order expression of the number of rational points of $n-1$ dimensional two-parameter hypersurfaces.

For the hypersurface $Z_{f,\Delta}$ in toric variety, such as the mirror manifold $W_{3}$ in usual, which defined by the dual polyhedron $\Delta^{*}$ in mirror symmetry, the period integral is defined as:

$$\Pi_{i}=\int_{\gamma_{i}}\frac{1}{f(x)}\prod_{j=1}^{n}\frac{dx_{j}}{x_{j}}$$

(20)

These periods, as solutions of the Picard-Fuchs equation, can be obtained from the GKZ-generalized hypergeometric system related to $\Delta^{*}$ [12], [13]. But in order to compare the number of rational points in zeroth order $\nu_{0}(\phi,\psi)$ , we use the definition (20) to calculate the fundamental period $\varpi_{0}(\phi,\psi)$ of two-parameter hypersurface directly :

$\displaystyle\varpi_{0}(\phi,\psi)$

$\displaystyle=$

$\displaystyle\frac{C}{(2\pi i)^{N+n}}\int\frac{d^{N+n}x}{-P(\phi,\psi)}$

(21)

Plug the polynomial equation (17) into (21) :

	$\displaystyle\varpi_{0}(\phi,\psi)$	$\displaystyle=$	$\displaystyle\frac{C}{(2\pi i)^{N+n}}\int\frac{d^{N+n}x}{\phi Q+\psi U-\sum_{\mathbb{m}\in\Delta^{\prime}}\alpha_{\mathbb{m}}x^{\mathbb{m}}}$
		$\displaystyle=$	$\displaystyle\frac{C}{(2\pi i)^{N+n}}\int\frac{d^{N+n}x}{\phi Q+\psi U}\left(1-\frac{\sum_{\mathbb{m}\in\Delta^{\prime}}\alpha_{\mathbb{m}}x^{\mathbb{m}}}{\phi Q+\psi U}\right)^{-1}$

consider the power series expansion of the integrand:

	$\displaystyle\left(1-\frac{\sum_{\mathbb{m}\in\Delta^{\prime}}\alpha_{\mathbb{m}}x^{\mathbb{m}}}{\phi Q+\psi U}\right)^{-1}$
	$\displaystyle=\sum_{n_{1}+n_{2}}\frac{1}{(n_{1}+n_{2})!}\sum_{n_{1},n_{2}}\frac{(n_{1}+n_{2})!}{n_{1}!n_{2}!}\frac{1}{{(Q\phi)}^{n_{1}}{(U\psi)}^{n_{2}}}\partial^{n_{1}}_{1/{Q\phi}}\partial^{n_{2}}_{1/{U\psi}}\left(1-\frac{\sum_{\mathbb{m}\in\Delta^{\prime}}\alpha_{\mathbb{m}}x^{\mathbb{m}}}{\phi Q+\psi U}\right)^{-1}$
	$\displaystyle=\sum_{n_{1},n_{2}}\frac{1}{n_{1}!n_{2}!}\frac{1}{\phi^{n_{1}}\psi^{n_{2}}}(n_{1}+n_{2})!(\sum_{\mathbb{m}\in\Delta^{\prime}}\alpha_{\mathbb{m}}x^{\mathbb{m}})^{n_{1}+n_{2}}Q^{-n_{1}}P^{-n_{2}}$

expand by polynomial theorem:

	$\displaystyle\sum_{n_{1},n_{2}}\frac{1}{n_{1}!n_{2}!}\frac{1}{\phi^{n_{1}}\psi^{n_{2}}}(n_{1}+n_{2})!\sum_{\sum n_{\mathbb{m}}=n_{1}+n_{2}}\frac{(n_{1}+n_{2})!}{\prod_{\mathbb{m}}n_{\mathbb{m}}}\prod_{\mathbb{m}\in\Delta^{\prime}}\frac{(\alpha_{\mathbb{m}}x^{\mathbb{m}})^{n_{\mathbb{m}}}}{n_{\mathbb{m}}}Q^{-n_{1}}P^{-n_{2}}$
	$\displaystyle=\sum_{n_{1},n_{2}}\frac{[(n_{1}+n_{2})!]^{2}}{n_{1}!n_{2}!}\frac{1}{\phi^{n_{1}}\psi^{n_{2}}}\sum_{\sum n_{\mathbb{m}}=n_{1}+n_{2}}\prod_{\mathbb{m}\in\Delta^{\prime}}\frac{(\alpha_{\mathbb{m}}x^{\mathbb{m}})^{n_{\mathbb{m}}}}{n_{\mathbb{m}}}Q^{-n_{1}}P^{-n_{2}}$

Finally, we find that the explicit expression of the fundamental period and the number of rational points in zeroth order approximation, namely:

	$\displaystyle\nu_{0}(\phi,\psi)$	$\displaystyle=$	$\displaystyle\sum_{n_{1},n_{2}}\frac{[(n_{1}+n_{2})!]^{2}}{n_{1}!n_{2}!}\frac{1}{\phi^{n_{1}}\psi^{n_{2}}}\sum_{\sum n_{\mathbb{m}}=n_{1}+n_{2}}\prod_{\mathbb{m}\in\Delta^{\prime}}\frac{\alpha^{n_{\mathbb{m}}}_{\mathbb{m}}}{n_{\mathbb{m}}!}$
			$\displaystyle\times\sum_{x\in\mathbb{F}_{p}^{*N+n}}x^{\sum_{\mathbb{m}\in\Delta^{\prime}}{n_{\mathbb{m}}}\mathbb{m}-n_{1}\mathbb{q}-n_{2}\mathbb{u}}$
	$\displaystyle\varpi_{0}(\phi,\psi)$	$\displaystyle=$	$\displaystyle\sum_{n_{1},n_{2}}\frac{[(n_{1}+n_{2})!]^{2}}{n_{1}!n_{2}!}\frac{1}{\phi^{n_{1}}\psi^{n_{2}}}\sum_{\sum n_{\mathbb{m}}=n_{1}+n_{2}}\prod_{\mathbb{m}\in\Delta^{\prime}}\frac{\alpha_{\mathbb{m}}^{n_{\mathbb{m}}}}{n_{\mathbb{m}}!}$
			$\displaystyle\times\frac{C}{(2\pi i)^{N+n}}\int\frac{d^{N+n}x}{\phi Q+\psi U}x^{\sum_{\mathbb{m}\in\Delta^{\prime}}{n_{\mathbb{m}}}\mathbb{m}-n_{1}\mathbb{q}-n_{2}\mathbb{u}}$

Since the same restricts are given by $x$-sum and integral in the formula above [4] :

$\displaystyle\sum_{\mathbb{m}\in\Delta^{\prime}}{n_{\mathbb{m}}}\mathbb{m}=n_{1}\mathbb{q}+n_{2}\mathbb{u}$

(24)

It means that we prove that the equation of rational points and fundamental period still holds for $n-1$ dimensional two-parameter hypersurface under zeroth order approximation:

$\displaystyle\nu(\phi,\psi)=\ ^{(p-1)}\varpi_{0}(\phi,\psi)+o(p)$

(25)

In toric geometry, one way to construct a multi-parameter Calabi-Yau manifold is to embed the polyhedron $\Delta$ in a 4D lattic space corresponding to the one-parameter Calabi-Yau 3-fold $W_{3}$ into the 5D lattic space. Then add extra vertices $\{\rho_{i}\}$ to make it an enhanced polyhedron $\tilde{\Delta}$ [14]. This means that additional linear relations among vertices are introduced and therefore additional complex moduli parameters are provided.

The above configuration is called the type II/F-theory duality [9] , $\tilde{\Delta}$ relates two completely different geometry. One is the Calabi-Yau 4-fold $W_{4}$ which is related to F-theory compactification. $W_{4}$ is the multi-parameter manifold obtained by elliptic fibration of $W_{3}$ [15]. The other is the manifold pair $(W_{3},E)$, open strings is compact on $W_{3}$ , and end on the D-brane wrapping the submanifold $E$. The periods satisfies the same GKZ-system given by the same enhanced polyhedron $\tilde{\Delta}$, therefore, although the relative periods $\hat{\Pi}(W_{3},D)$ and the absolute periods $\Pi(W_{4})$ has different geometry meaning, the same low-energy effective theory can be given. Here $D$ is a divisor on $W_{3}$ and parameterizes the deformation of D-brane $E$ .

For the charge vector $l_{i}(\Delta)$ given by the linear relation $\sum_{\Delta}l_{i}(\Delta)v_{i}=0$ of the vertex $v_{i}\in\Delta$, since the linear relation is maintained in the embedding $v_{i}\mapsto u_{i}=(v_{i},0)$ from 4D to 5D , the same charge vector $\hat{l}_{i}(\tilde{\Delta})=(l_{i}(\Delta),0^{k})$ holds for extended polyhedra $\tilde{\Delta}$ , where $k=\#\{\rho_{i}\}$ . In GKZ-system, it means the same Picard-Fuchs operators:

$\displaystyle\{\prod_{l_{j}^{a}>0}(\prod_{i=0}^{l_{j}^{a}-1}(\theta_{j}-i))-\prod_{i=1}^{|l_{0}^{a}|}(i-|l_{0}^{a}|-\theta_{0})\prod_{l_{j}^{a}<0,j\neq 0}(\prod_{i=0}^{|l_{j}^{a}\mid-1}(\theta_{j}+|l_{j}^{a}|-i))z_{a}\}\Pi(z)=0$

(26)

where $\theta_{j}=z_{j}\frac{d}{dz_{j}}$ . Note that all periods of $W_{3}$ consists of a subset of the periods of $W_{4}$. In particular, the two manifolds have the same fundamental period, $\varpi_{0}(W_{3})=\varpi_{0}(W_{4})$. By the conclusion of Section 3 , the relation between complex structures of two manifolds will be expressed on the number of rational points over finite fields.

By the formula (25) at the end of Section 3, we can immediately obtain some corollaries about the relation between rational points of $W_{3}(\psi)$ and $W_{4}(\phi,\psi)$ over finite fields $\mathbb{F}_{p}$. The one-parameter polynomials and two-parameter polynomials

$$P_{3}(x;\psi)=\sum_{\mathbb{m}\in\Delta^{*}}\alpha_{\mathbb{m}}x^{\mathbb{m}}$$

$$P_{4}(x;\phi,\psi)=\sum_{\mathbb{m}\in\tilde{\Delta}^{*}}\alpha_{\mathbb{m}}x^{\mathbb{m}}$$

given by the polyhedron $\Delta^{*}$ and its enhanced $\tilde{\Delta}^{*}$ define 3D and 4D Calabi-Yau hypersurfaces are dual space in the sense of type II/F-theory duality and have the same fundamental period $\varpi_{0}=\ ^{[p/n]}f_{0}$ . Therefore, their the number of rational points are:

	$\displaystyle\nu(P_{3})$	$\displaystyle=$	$\displaystyle\#\{x\in\mathbb{F}_{p}^{n}|P_{3}(x;\psi)=0\ \mathrm{mod}\ p\}$
	$\displaystyle\nu(P_{4})$	$\displaystyle=$	$\displaystyle\#\{x\in\mathbb{F}_{p}^{\tilde{n}}|P_{4}(x;\phi,\psi)=0\ \mathrm{mod}\ p\}$

Where $n$ is the number of vertices of $\Delta$ , $\tilde{n}$ is the number of vertices of $\tilde{\Delta}$ dual of $\tilde{\Delta}^{*}$. Since they are equal to the same fundamental period in the zeroth order approximation, we can write $\nu_{0}(P_{3})=\nu_{0}(P_{4})$, or equivalent:

$\displaystyle\nu(P_{3})=\nu(P_{4})\ \mathrm{mod}\ p$

(27)

This result relate the arithmetic properties of two toric varieties with different dimensions, and is not obvious in mathematics. But a simple proof is given by type II/F-theory duality in physics. In fact, the whole reasoning does not depends on the dimension of toric hypersurfaces, so (27) can be generalized to $n$-dimension.

One can also observe the effect of elliptic fibration process from 3- to 4-dimensional manifold on the number of rational points over finite field. In toric geometry, elliptic fibration is determined by the extra vertices $\{\rho_{i}\}$, and thus is related to the divisor $D$ on the Calabi-Yau 3-fold $W_{3}$. However, over finite fields, the different geometry of $D$ , or the different elliptic fibration, has no influence on the zeroth order $p$-adic expansion of the number of rational points.

For specific models under the type II/F-theory duality, equation (27) can be tested numerically. For example, one can consider the vertices of the polyhedron $\Delta$ and its dual $\Delta^{*}$ are:

Calabi-Yau hypersurface corresponds to $\Delta^{*}$ is given by the formula:

$\displaystyle P(\Delta^{*})=\sum_{v_{j}\in\Delta^{*}}a_{i}\prod_{v_{i}\in\Delta}x_{j}^{\langle v,v_{i}^{*}\rangle+1}$

(28)

In this case it is known as the quintic :

$$P_{3}(\Delta^{*})=x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}+x_{5}^{5}+a_{0}x_{1}x_{2}x_{3}x_{4}x_{5}$$

Embedding $\Delta^{*}$ into the 5D lattic space and specify the extra vertices corresponding to the divisor $D$ as:

$$\rho_{1}=(0,0,0,0,1)\ ,\ \rho_{2}=(-1,-1,-1,-1,1)$$

Then there are extended polyhedra $\tilde{\Delta}^{*}$ and their duals $\tilde{\Delta}$ :

The defining equation of to the Calabi-Yau 4-folds of F-theory are also given by (28):

$$P_{4}(\tilde{\Delta}^{*})=x_{1}^{4}+x_{2}^{4}x_{3}^{5}+x_{4}^{4}x_{5}^{5}+x_{6}^{4}x_{7}^{5}+x_{8}^{4}x_{9}^{5}+x_{1}^{5}x_{2}x_{4}x_{6}x_{8}+b_{0}x_{3}x_{5}x_{7}x_{9}+a_{0}x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}x_{8}x_{9}$$

By selecting different complex moduli parameters $a_{0},b_{0}$, we calculate the number of rational points for the first five prime numbers, see Appendix A. For all $a_{0},b_{0}$ and prime $p$, the calculation results show that equation (27) is valid.