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 $k\geq 1$; and
3. $|\alpha _{j}|={\sqrt {41}}$ 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 (Kahn 2020):

$P_{i}(T)=\prod _{1\leq j_{1}<j_{2}<\ldots <j_{i-1}<j_{i}\leq 4}(1-\alpha _{j_{1}}\cdot \ldots \cdot \alpha _{j_{i}}T)$

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 $P_{1}(T)=1+c_{1}T+c_{2}T^{2}+qc_{1}T^{3}+q^{2}T^{4}.$ Calculating these polynomial functions for the coefficients of the $P_{i}(T)$ shows that

${\begin{alignedat}{2}P_{0}(T)&=1-T\\P_{1}(T)&=1-3^{2}\cdot T+71\cdot T^{2}-3^{2}\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}\\P_{2}(T)&=(1-41\cdot T)^{2}\cdot (1+11\cdot T+3\cdot 7\cdot 41\cdot T^{2}+11\cdot 41^{2}\cdot T^{3}+41^{4}\cdot T^{4})\\P_{3}(T)&=1-3^{2}\cdot 41\cdot T+71\cdot 41^{2}\cdot T^{2}-3^{2}\cdot 41^{4}\cdot T^{3}+41^{6}\cdot T^{4}\\P_{4}(T)&=1-41^{2}\cdot T\end{alignedat}}$

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 $F_{41^{2}}$:

${\begin{alignedat}{2}M_{1}&\;{\overset {\underset {\mathrm {def} }{}}{=}}\;|{\text{Jac}}(C/F_{41})|=P_{1}(1)=\prod _{j=1}^{4}[1-\alpha _{j}T]_{T=1}\\&=[1-9\cdot T+71\cdot T^{2}-9\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}]_{T=1}=1-9+71-9\cdot 41+41^{2}=1375=5^{3}\cdot 11{\text{, and}}\\M_{2}&\;{\overset {\underset {\mathrm {def} }{}}{=}}\;|{\text{Jac}}(C/F_{41^{2}})|=\prod _{j=1}^{4}[1-\alpha _{j}^{2}T]_{T=1}\\&=[1+61\cdot T+3\cdot 587\cdot T^{2}+61\cdot 41^{2}\cdot T^{3}+41^{4}\cdot T^{4}]_{T=1}=2930125=5^{3}\cdot 11\cdot 2131.\end{alignedat}}$

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

${\begin{alignedat}{2}\sum _{m=1}^{\infty }&{\frac {M_{m}}{m}}(41^{-s})^{m}\,=\,\log \left({\frac {P_{1}(T)\cdot P_{3}(T)}{P_{0}(T)\cdot P_{2}(T)\cdot P_{4}(T)}}\right)\\&=1375\cdot T+2930125/2\cdot T^{2}+4755796375/3\cdot T^{3}+7984359145125/4\cdot T^{4}+13426146538750000/5\cdot T^{5}+O(T^{6}).\end{alignedat}}$

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 ${\text{Jac}}(C/F_{41^{m_{2}}})$.

References

• Kahn, Bruno (2020), "The zeta function of an abelian variety", Zeta and L-Functions of Varieties and Motives, London Mathematical Society Lecture Note Series, Cambridge: Cambridge University Press, pp. 52–53, doi:10.1017/9781108691536, ISBN 978-1-108-70339-0
• Koblitz, Neal (1998), "Elliptic and Hyperelliptic Cryptosystems", Algebraic Aspects of Cryptography, Algorithms and Computation in Mathematics, vol. 3, Berlin, Heidelberg: Springer-Verlag, pp. 146–147, doi:10.1007/978-3-662-03642-6, ISBN 978-3-662-03642-6
• Milne, James (1986), "Abelian Varieties", Arithmetic Geometry, New York: Springer-Verlag, pp. 103–150, doi:10.1007/978-1-4613-8655-1, ISBN 978-1-4613-8655-1