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Let be a smooth projective variety of genus over . Arithmetic information about the curve is encoded in its *L*-function . The conjectures of Birch and Swinnerton-Dyer about elliptic curves over were generalized to arbitrary abelian varieties over number fields by John Tate.

In the case of hyperelliptic curves (genus 2 curves) over , 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

Ш

where is the *L*-series of *J* and *r* is its analytic rank. denotes the integral over of a certain differential 2-form, Reg is the regulator of , is the Tamagawa number, Ш is the Tate-Shafarevich group of *J* over , and is the torsion subgroup of . Here, is the subgroup of the Jacobian isomorphic to , where is the open subgroup scheme of the closed fiber of the Néron model of *J* over

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

If *C* has good reduction at the prime *p*, the factor in the Euler product at this prime is determined by a polynomial of degree 4. It appears in the local zeta function of the curve over the finite field of order *p*.

where is the number of -points on *C*.

By a theorem of Weil, the polynomial , can be determined by counting points on *C* only over and since the coefficients must satisfy and . 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:

Conductor:

Discriminant:

Local *L*-factor at *p*=7:

Curve:

Conductor:

Discriminant:

Local *L*-factor at *p*=7:

Curve:

Conductor:

Discriminant:

Local *L*-factor at *p*=7:

Curve:

Conductor:

Discriminant:

Local *L*-factor at *p*=7:

Curve:

Conductor:

Discriminant:

Local *L*-factor at *p*=7:

If one were to attempt to find an automorphic representation 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 , , :

:

:

:

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.

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.