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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.

Jacquet and Langlands have shown how to associate classical holomorphic modular forms f to automorphic representations $\pi_f$ of ${\rm GL}(2,\mathbb{A})$. These representations can be written in terms of local components. In this case, the local components, which are infinite dimensional, are complex representations of ${\rm GL}(2,\mathbb{Q}_v)$. Also, the representation is realized in the action of ${\rm GL}(2,\mathbb{A})$ by right translation on a certain space of functions on ${\rm GL}(2,\mathbb{Q})\backslash{\rm GL}(2,\mathbb{A})$.

The classical theory of the passage of modular forms to automorphic representations suggests how it may be extended to Siegel modular forms of higher degree. Spaces of cuspidal Siegel modular forms of degree n are associated to cuspidal automorphic representations of ${\rm GSp}(2n,\mathbb{A})$. These cuspidal automorphic representations $\pi$ can be written in terms of local components $\pi_v$, where v is a place of $\mathbb{Q}$. Many cusp forms are associated to a single such representation $\pi$, but among them is a unique primitive f known as a newform.

In addition to representation theory, modular forms are also related to certain abelian varieties. The nature of this relationship is made precise by the famous Taniyama-Shimura conjecture, proven in 1999.

Taniyama-Shimura Conjecture: Let $E: y^2=x^3+ax+b$ be an elliptic curve with integral coefficients, conductor N, and L-series

$L(s,E)=\sum_{n=1}^\infty\dfrac{a_n}{n^s}.$

Then there is a cusp form of weight 2 and level N which is a Hecke eigenform with Fourier series

$f(\tau)=\sum_{n=1}^\infty a_nq^n,\quad {\rm where}\, q=e^{2\pi i\tau}$

The Langlands philosophy suggests that there should be abelian varieties associated with degree 2 Siegel modular forms. Brumer and Kramer have recently stated a conjectured extension of the Taniyama-Shimura conjecture to the degree 2 case for paramodular forms.

In the GL(2) case, cuspidal eigenforms determine Galois representations as well as automorphic representations. By a theorem of Deligne and Carayol, Galois representations and automorphic representations both have local components which determine each other. This theorem tells us that a modular elliptic curve has the same conductor as the level of the corresponding cusp form.

We would like to do the same thing for the GSp(4) case, but we need to know the corresponding abelian varieties. Unfortunately, these are unknown in general. But there is evidence supporting a conjectured correspondence for the paramodular group K(p) of GSp(4) due to Brumer and Kramer.

The Paramodular Conjecture: There is a one-to-one correspondence between isogeny classes of rational abelian surfaces A of conductor N with ${\rm End}_\mathbb{Q} A = \mathbb{Z}$ and weight 2 newforms f on K(N) with rational eigenvalues, not in the span of the Gritsenko lifts, such that L(A,s) = L(f,s). The $\ell$-adic representations associated to f should be isomorphic to those of the Tate module of A for any $\ell$ prime to N.

The paramodular conjecture is consistent with known examples from Brumer and Kramer and also in Poor and Yuen’s work on weight 2 Siegel paramodular forms. There are many open problems related to this conjecture, such as translating the different actions one can perform on abelian varieties to actions on spaces of paramodular forms.

More generally, Yoshida conjectured that for any rational abelian surface, there exists a discrete subgroup $\Gamma'$ of ${\rm Sp}(4,\mathbb{Q})$ and a degree 2 Siegel modular form of weight 2, $f\in\mathcal{M}_2(\Gamma')$, with the same L-series.

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

C. Poor and D. Yuen, Paramodular Cusp Forms, arXiv:0912.0049v1 (2009).

H. Yoshida, Siegel modular forms and the arithmetic of quadratic forms, Invent. Math. 60 (1980), no. 3, 193-248.

I have been reading “Number Theory I: Fermat’s Dream” by Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito. In Chapter 1: Rational Points on Elliptic Curves, the authors state two propositions:

Proposition 1.1. There is no integral solution $(x,y,z)$ to $x^4 + y^4 = z^4$ satisfying $xyz \neq 0.$

and

Proposition 1.2. The only rational solutions to $y^2=x^3-x$ are
$(x,y)=(0,0)$ and $(\pm1,0).$

They then show how 1.2 implies 1.1:

Suppose $(x,y,z)$ is an integral solution to $x^4 + y^4 = z^4$ satisfying $xyz \neq 0.$ We rewrite the equation as

$x^4 = z^4 - y^4$

We then multiply each side by $\dfrac{z^2}{y^6}$ to get

$\dfrac{x^4z^2}{y^6} = \dfrac{z^6}{y^6} - \dfrac{z^2}{y^2}$
$\left(\dfrac{x^2z}{y^3}\right)^2=\left(\dfrac{z^2}{y^2}\right)^3-\left(\dfrac{z^2}{y^2}\right).$

Setting $Y=\dfrac{x^2z}{y^3},\, X=\dfrac{z^2}{y^2}$, we have
$Y^2=X^3-X.$

Note that X,Y are rational numbers. The only rational points of this elliptic curve are $(0,0), (\pm1,0)$. In either case, $0=Y=\dfrac{x^2z}{y^3}$ implies either $x=0$ or $z=0$, contradicting our assumption. So there are only trivial integral solutions $(x,y,z)$ to $x^4 + y^4 = z^4$.

We can extend this argument to prove that there are no non-trivial integral solutions to $x^n + y^n = z^n$, for $n\equiv 0\, ({\rm mod}\, 4)$.