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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.

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.

One of the motivations of studying the Langlands Conjectures is the question

What are the finite extensions of the rational numbers?

The local Langlands Conjectures give a dictionary between the Galois Theory of local fields and the representation theory of locally compact reductive groups.

The following videos are of the first lecture by Dick Gross in a series of lectures given at Columbia University in Fall 2011 about the Langlands Conjectures where he explains this in more detail.

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