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Let $C$ be a smooth projective variety of genus $g\leq3$ over $\mathbb{Q}$. Arithmetic information about the curve is encoded in its L-function $L(C,s)$. The conjectures of Birch and Swinnerton-Dyer about elliptic curves over $\mathbb{Q}$ were generalized to arbitrary abelian varieties over number fields by John Tate.

In the case of hyperelliptic curves (genus 2 curves) over $\mathbb{Q}$, the first conjecture is that the order of vanishing of the L-function of the Jacobian at s=1 (the analytic rank) is equal to the Mordell-Weil rank of the Jacobian. The second is

$\lim_{s\rightarrow1}(s-1)^{-r}L(J,s)=\Omega\cdot{\rm Reg}\cdot\prod_{p}c_p\cdot\#$ Ш $(J,\mathbb{Q})\cdot(\# J(\mathbb{Q})_{\rm tors})^{-2}$

where $L(J,s)$ is the L-series of J and r is its analytic rank. $\Omega$ denotes the integral over $J(\mathbb{R})$ of a certain differential 2-form, Reg is the regulator of $J(\mathbb{Q})$, $c_p = \# J(\mathbb{Q}_p)/J^0(\mathbb{Q})$ is the Tamagawa number, Ш$(J,\mathbb{Q})$ is the Tate-Shafarevich group of J over $\mathbb{Q}$, and $J(\mathbb{Q})_{tors}$ is the torsion subgroup of $J(\mathbb{Q})$. Here, $J^0(\mathbb{Q})$ is the subgroup of the Jacobian isomorphic to $\mathcal{J}^0(\mathbb{Z}_p)$, where $\mathcal{J}^0$ is the open subgroup scheme of the closed fiber of the Néron model of J over $\mathbb{Z}_p$

The L-series of the curve C is given as both an Euler product and a Dirichlet series.

$L(C,s)=\prod_p L_p(p^{-s})^{-1}=\sum_{n=1}^\infty a_n n^{-s}.$

If C has good reduction at the prime p, the factor in the Euler product at this prime is determined by a polynomial $L_p(T)$ of degree 4. It appears in the local zeta function of the curve over the finite field $\mathbb{F}_p$ of order p.

$Z(C/\mathbb{F}_p;T)=exp\left(\sum_{k=1}^\infty N_kT^k/k\right)=\dfrac{L_p(T)}{(1-T)(1-pT)},$

where $N_k$ is the number of $\mathbb{F}_{q^k}$-points on C.

By a theorem of Weil, the polynomial $L_p(T)=\sum_{n=0}^4 a_nT^n$, can be determined by counting points on C only over $\mathbb{F}_p$ and $\mathbb{F}_{p^2}$ since the coefficients must satisfy $a_0=1, a_3=pa_1,$ and $a_4=p^2$. We consider five special hyperelliptic curves (those associated to paramodular forms taken from Brumer and Kramer’s paper) and compute their discriminant and local L-factor at p=7.

Curve: $y^2=x^6 + 4x^5 + 4x^4 + 2x^3 + 1$
Conductor: $249$
Discriminant: $261095424 = 2^{20} \cdot 3 \cdot 83$
Local L-factor at p=7: $L_7(7^{-s})^{-1}=\dfrac{1}{7^{2-4 s}+8\cdot 7^{1-3 s}+44\cdot 7^{-2 s}+8\cdot 7^{-s}+1}$

Curve: $y^2=x^6 + 2x^5 + 3x^4 + 4x^3 - x^2 - 2x + 1$
Conductor: $277$
Discriminant: $290455552 = 2^{20} \cdot 277$
Local L-factor at p=7: $L_7(7^{-s})^{-1}=\dfrac{1}{7^{2-4 s}+6\cdot 7^{1-3 s}+54\cdot 7^{-2 s}+6\cdot 7^{-s}+1}$

Curve: $y^2=x^6 - 2x^3 - 4x^2 + 1$
Conductor: $295$
Discriminant: $309329920 = 2^{20} \cdot 5 \cdot 59$
Local L-factor at p=7: $L_7(7^{-s})^{-1}=\dfrac{1}{7^{2-4 s}+6\cdot 7^{1-3 s}+48\cdot 7^{-2 s}+6\cdot 7^{-s}+1}$

Curve: $y^2=x^6 - 2x^5 + 3x^4 - x^2 - 2x + 1$
Conductor: $349$
Discriminant: $365953024 = 2^{20} \cdot 349$
Local L-factor at p=7: $L_7(7^{-s})^{-1}=\dfrac{1}{7^{2-4 s}+9\cdot 7^{1-3 s}+69\cdot 7^{-2 s}+9\cdot 7^{-s}+1}$

Curve: $y^2=x^6 + 2x^5 + 5x^4 + 2x^3 + 2x^2 + 1$
Conductor: $353$
Discriminant: $370147328 = 2^{20} \cdot 353$
Local L-factor at p=7: $L_7(7^{-s})^{-1}=\dfrac{1}{7^{2-4 s}+7^{2-3 s}+7^{1-s}+37\cdot 7^{-2 s}+1}$

If one were to attempt to find an automorphic representation $\pi=\otimes\pi_p$ that could be associated to these varieties, one could use this information about the local L-factors to rule out possible local components. For example, Schmidt determined the local L-factors of the representations $\Pi({\rm St}\otimes 1)$, $\Pi(\xi{\rm St}\otimes 1)$, $\Pi({\rm St}\otimes{\rm St})$:

$\Pi({\rm St}\otimes 1)$ : $L_p(s,\Pi_p)^{-1} = (1-p^{-s-1/2})^2(1-p^{-s+1/2}),$
$\Pi(\xi{\rm St}\otimes 1)$ : $L_p(s,\Pi_p)^{-1} = (1-p^{-s-1/2})(1-p^{-s+1/2})(1+p^{-s-1/2}),$
$\Pi({\rm St}\otimes{\rm St})$ : $L_p(s,\Pi_p)^{-1} = (1-p^{-s-1/2})^2.$

Comparing these with the L-factors of the curves we found at p=7, we can say that the local component at p=7 of an associated automorphic representation is not one of these.

References
A. Brumer and K. Kramer, Paramodular abelian varieties of odd conductor, arXiv:1004.4699v2 (2010).

E. Flynn, F. Leprévost, et. al., Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comp. 70 (2001), no. 236, 1675-1697

K. Kedlaya and A. Sutherland, Computing L-series of hyperelliptic curves, Algorithmic number theory, 312–326, Lecture Notes in Comput. Sci., 5011, Springer, Berlin, 2008.

R. Schmidt, On classical Saito-Kurokawa liftings, J. Reine Angew. Math. 604 (2007), 211-236.

J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, 306 1965/1966.

A. Weil, Number of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55, (1949), 497-508.