Draft:Weil Conjectures - Abelian Surfaces

Abelian surfaces
An Abelian surface is a two-dimensional Abelian variety. This is, they are projective varieties that also have the structure of a group, in a way that is compatible with the group composition and taking inverses. Elliptic curves represent one-dimensional Abelian varieties. As an example of an Abelian surface defined over a finite field, consider the Jacobian variety $$X:=\text{Jac}(C/F_{41})$$ of the genus 2 curve
 * $$C/F_{41}: y^2 + y=x^5,$$

which was introduced in the section on hyperelliptic curves. The dimension of $$X$$ equals the genus of $$C$$, so $$n=2$$. There are algebraic integers $$\alpha_1,\ldots,\alpha_4$$ such that
 * 1) the polynomial $$P(x)=\prod_{j=1}^4 (x-\alpha_j)$$ has coefficients in $$\mathbb{Z}$$;
 * 2) $$M_k:=|\text{Jac}(C/F_{41^k})|=\prod_{j=1}^4 (1-\alpha_j^k)$$ for all $$k\ge 1$$; and
 * $$|\alpha_j|=\sqrt{41}$$ for $$j=1,\ldots,4$$.

The zeta-function of $$X$$ is given by
 * $$\zeta(X,s)=\prod_{i=0}^{4} P_i(q^{-s})^{(-1)^{i+1}} = \frac{P_1(T)\cdot P_3(T)}{P_0(T)\cdot P_2(T)\cdot P_{4}(T)},

$$ where $$q=41$$, $$T=q^{-s}\,\stackrel{\rm def}{=}\,\text{exp}(-s\cdot\text{log}(41))$$, and $$s$$ represents the complex variable of the zeta-function. The Weil polynomials $$P_i(T)$$ have the following specific form :
 * $$P_i(T)=\prod_{1 \leq j_1<j_2<\ldots<j_{i-1}<j_i \leq 4} (1-\alpha_{j_1}\cdot\ldots\cdot\alpha_{j_i}T)$$

for $$i=0,1,\ldots,4$$, and
 * $$P_1(T)=\prod_{j=1}^{4} (1-\alpha_{j} T)=1 - 9\cdot T + 71\cdot T^2 - 9\cdot 41\cdot T^3 + 41^2\cdot T^4$$

is the same for the curve $$C$$ (see section above) and its Jacobian variety $$X$$. This is, the inverse roots of $$P_i(T)$$ are the products $$\alpha_{j_1}\cdot\ldots\cdot\alpha_{j_i} $$ that consist of $$i$$ many, different inverse roots of $$P_1(T)$$. Hence, all coefficients of the polynomials $$P_i(T)$$ can be expressed as polynomial functions of the parameters $$c_1=-9$$, $$c_2=71$$ and $$q=41$$ appearing in $$P_1(T)=1+c_1 T + c_2 T^2 + q c_1 T^3 + q^2 T^4.$$ Calculating these polynomial functions for the coefficients of the $$P_i(T)$$ shows that
 * $$\begin{alignat}{2}

P_0(T) &= 1 - T\\ P_1(T) &= 1 - 3^2\cdot T + 71\cdot T^2 - 3^2\cdot 41\cdot T^3 + 41^2\cdot T^4\\ P_2(T) &=(1 - 41\cdot T)^2 \cdot (1 + 11\cdot T + 3\cdot 7\cdot 41\cdot T^2 + 11\cdot 41^2\cdot T^3 + 41^4\cdot T^4)\\ P_3(T) &=1 - 3^2\cdot 41\cdot T + 71\cdot 41^2\cdot T^2 - 3^2\cdot 41^4\cdot T^3 + 41^6\cdot T^4\\ P_4(T) &=1 - 41^2\cdot T \end{alignat}$$ Polynomial $$P_1$$ allows for calculating the numbers of elements of the Jacobian variety $$\text{Jac}(C)$$ over the finite field $$F_{41}$$ and its field extension $$F_{41^2}$$:


 * $$\begin{alignat}{2}

M_1 &\;\overset{\underset{\mathrm{def}}{}}{=}\; |\text{Jac}(C/F_{41})|=P_1(1)=\prod_{j=1}^4 [1-\alpha_j T]_{T=1}\\ &= [1 - 9\cdot T + 71\cdot T^2 - 9\cdot 41\cdot T^3 + 41^2\cdot T^4]_{T=1} = 1 - 9 + 71 - 9\cdot 41 + 41^2=1375=5^3\cdot 11\text{, and}\\ M_2 &\;\overset{\underset{\mathrm{def}}{}}{=}\; |\text{Jac}(C/F_{41^2})|=\prod_{j=1}^4 [1-\alpha_j^2 T]_{T=1}\\ &= [1 + 61\cdot T + 3\cdot 587\cdot T^2 + 61\cdot 41^2\cdot T^3 + 41^4\cdot T^4]_{T=1} = 2930125 = 5^3\cdot 11\cdot 2131. \end{alignat}$$

The inverses $$\alpha_{i,j}$$ of the zeros of $$P_i(T)$$ do have the expected absolute value of $$41^{i/2}$$ (Riemann hypothesis). Moreover, the maps $$\alpha_{i,j}\longmapsto 41^2/\alpha_{i,j},$$ $$j=1,\ldots,\deg P_i,$$ correlate the inverses of the zeros of $$P_i(T)$$ and the inverses of the zeros of $$P_{4-i}(T)$$. A non-singular, complex, projective, algebraic variety $$Y$$ with good reduction at the prime 41 to $$X=\text{Jac}(C/F_{41})$$ must necessarily have Betti numbers $$B_0=B_4=1, B_1=B_3=4, B_2=6$$, since these are the degrees of the polynomials $$P_i(T).$$ The Euler characteristic $$E$$ of $$X$$ is given by the alternating sum of these degrees/Betti numbers: $$E=1-4+6-4+1=0$$.

By taking the logarithm of



\zeta(\text{Jac}(C/F_{41}), s)\,=\, \exp\left(\sum_{m = 1}^\infty \frac{M_m}{m} (41^{-s})^m\right)\,=\,\prod_{i=0}^{4} \, P_i (41^{-s})^{(-1)^{i+1}} =\frac{P_1(T)\cdot P_3(T)}{P_0(T)\cdot P_2(T)\cdot P_4(T)}, $$ it follows that
 * $$\begin{alignat}{2}

\sum_{m = 1}^\infty & \frac{M_m}{m} (41^{-s})^m\,=\,\log\left(\frac{P_1(T)\cdot P_3(T)}{P_0(T)\cdot P_2(T)\cdot P_4(T)}\right)\\ &=1375\cdot T + 2930125/2\cdot T^2 + 4755796375/3\cdot T^3 + 7984359145125/4 \cdot T^4 + 13426146538750000/5\cdot T^5 + O(T^6). \end{alignat}$$ Aside from the values $$M_1$$ and $$M_2$$ already known, you can read off from this Taylor series all other numbers $$M_m$$, $$m\in\mathbb{N}$$, of $$F_{41^m}$$-rational elements of the Jacobian variety, defined over $$F_{41}$$, of the curve $$C/F_{41}$$: for instance, $$M_3=4755796375=5^3\cdot 11\cdot 61\cdot 56701$$ and $$M_4=7984359145125=3^4\cdot 5^3\cdot 11\cdot 2131\cdot 33641$$. In doing so, $$m_1|m_2$$ always implies $$M_{m_1}|M_{m_2}$$ since then, $$\text{Jac}(C/F_{41^{m_1}})$$ is a subgroup of $$\text{Jac}(C/F_{41^{m_2}})$$.