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Kimball
Kimball
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For the functoriality aspects, the two key properties of $L$-functions are additivity ($L(s,\rho_1 \oplus \rho_2) = L(s, \rho_1)L(s,\rho_2)$) and inductivity ($L(s,I_{E}^F(\rho)) = L(s, \rho)$), and for this it matters what the precise form of the local factors are. Roughly you want it to be a "characteristic polynomial" of the representation (at least for a finite-dimensional Galois representation---on the automorphic side, you want the analogous thing for Langlands parameter) so that additivity will hold.

Recall that the correspondence between Galois and automorphic representations, is says that the representations should locally correspond at almost all places, so it suffices to look at unramified places, where both sides should reduce to the 1-dimensional case of local class field theory. Then the additivity of $L$-functions says the $L$-functions correspond. Of course the main problem in functoriality is showing that $\pi = \otimes \pi_v$ is (globally) automorphic---constructing the $\pi_v$ for almost all $v$ is easy. Then inductivity is useful to get compatibility with base change.

Note: not all Dirichlet series formed from arithmetic objects should give automorphic $L$-functions. For instance, Koecher-Maass series (constructed with Fourier coefficients of Siegel modular forms) and zeta functions of binary cubic forms don't have Euler products. Also, Zagier's "Naive BSD" paper considers variations of $L$-series of elliptic curves which do have Euler products, but not (apparently) meromorphic continuation to $\mathbb C$. So it's better to look at things where, say $a_n$'s are multiplicative, and the $a_p$'s are traces of representations.

Edit: Upon rereading my answer, I realize I may not have spelled out a couple conclusions explicitly:

  • The above reasoning more-or-less motivates the definition of $L$-functions of Galois representations. The automorphic definition is then motivated by a combination of the result of Hecke that gives $L$-functions of modular forms as integral representations combined with the predictions of the local Langlands correspondence.

  • My point of view is that the reason $L$-functions connect different ares of mathematics is because they reflect things like the global Langlands correspondence. Unfortunately, I don't have a good intuitive explanation for why Langlands' conjectures should be true (one can make the trite remark that it generalizes class field theory, but that doesn't answer why in my mind).

For the functoriality aspects, the two key properties of $L$-functions are additivity ($L(s,\rho_1 \oplus \rho_2) = L(s, \rho_1)L(s,\rho_2)$) and inductivity ($L(s,I_{E}^F(\rho)) = L(s, \rho)$), and for this it matters what the precise form of the local factors are. Roughly you want it to a "characteristic polynomial" of the representation (at least for a finite-dimensional Galois representation---on the automorphic side, you want the analogous thing for Langlands parameter) so that additivity will hold.

Recall that the correspondence Galois and automorphic representations, is that the representations should locally correspond at almost all places, so it suffices to look at unramified places, where both sides should reduce to the 1-dimensional case of local class field theory. Then the additivity of $L$-functions says the $L$-functions correspond. Of course the main problem in functoriality is showing that $\pi = \otimes \pi_v$ is (globally) automorphic---constructing the $\pi_v$ for almost all $v$ is easy. Then inductivity is useful to get compatibility with base change.

Note: not all Dirichlet series formed from arithmetic objects should give automorphic $L$-functions. For instance, Koecher-Maass series (constructed with Fourier coefficients of Siegel modular forms) and zeta functions of binary cubic forms don't have Euler products. Also, Zagier's "Naive BSD" paper considers variations of $L$-series of elliptic curves which do have Euler products, but not (apparently) meromorphic continuation to $\mathbb C$. So it's better to look at things where, say $a_n$'s are multiplicative, and the $a_p$'s are traces of representations.

For the functoriality aspects, the two key properties of $L$-functions are additivity ($L(s,\rho_1 \oplus \rho_2) = L(s, \rho_1)L(s,\rho_2)$) and inductivity ($L(s,I_{E}^F(\rho)) = L(s, \rho)$), and for this it matters what the precise form of the local factors are. Roughly you want it to be a "characteristic polynomial" of the representation (at least for a finite-dimensional Galois representation---on the automorphic side, you want the analogous thing for Langlands parameter) so that additivity will hold.

Recall that the correspondence between Galois and automorphic representations says that the representations should locally correspond at almost all places, so it suffices to look at unramified places, where both sides should reduce to the 1-dimensional case of local class field theory. Then the additivity of $L$-functions says the $L$-functions correspond. Of course the main problem in functoriality is showing that $\pi = \otimes \pi_v$ is (globally) automorphic---constructing the $\pi_v$ for almost all $v$ is easy. Then inductivity is useful to get compatibility with base change.

Note: not all Dirichlet series formed from arithmetic objects should give automorphic $L$-functions. For instance, Koecher-Maass series (constructed with Fourier coefficients of Siegel modular forms) and zeta functions of binary cubic forms don't have Euler products. Also, Zagier's "Naive BSD" paper considers variations of $L$-series of elliptic curves which do have Euler products, but not (apparently) meromorphic continuation to $\mathbb C$. So it's better to look at things where, say $a_n$'s are multiplicative, and the $a_p$'s are traces of representations.

Edit: Upon rereading my answer, I realize I may not have spelled out a couple conclusions explicitly:

  • The above reasoning more-or-less motivates the definition of $L$-functions of Galois representations. The automorphic definition is then motivated by a combination of the result of Hecke that gives $L$-functions of modular forms as integral representations combined with the predictions of the local Langlands correspondence.

  • My point of view is that the reason $L$-functions connect different ares of mathematics is because they reflect things like the global Langlands correspondence. Unfortunately, I don't have a good intuitive explanation for why Langlands' conjectures should be true (one can make the trite remark that it generalizes class field theory, but that doesn't answer why in my mind).

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Kimball
Kimball
  • 6k
  • 32
  • 64

Note: not all Dirichlet series formed from arithmetic objects should give automorphic $L$-functions. For instance, Koecher-Maass series (constructed with Fourier coefficients of Siegel modular forms) and zeta functions of binary cubic forms don't have Euler products. Also, Zagier's "Naive BSD" paper considers variations of $L$-series of elliptic curves which do have Euler products, but not (apparently) meromorphic continuation to $\mathbb C$. So it's better to look at things where, say $a_n$'s are multiplicative, and the $a_p$'s are traces of representations.

Note: not all Dirichlet series formed from arithmetic objects should give automorphic $L$-functions. For instance, Koecher-Maass series (constructed with Fourier coefficients of Siegel modular forms) and zeta functions of binary cubic forms don't have Euler products. Also, Zagier's "Naive BSD" paper considers variations of $L$-series of elliptic curves which do have Euler products, but not (apparently) meromorphic continuation to $\mathbb C$. So it's better to look at things where, say $a_n$'s are multiplicative, and the $a_p$'s are traces of representations.

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Kimball
Kimball
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I'm not really sure what you're looking for, but here are some thoughts, and if you want me to try to expand on some aspect, let me know.

As an analytic gadget

$L$-functions provide a way to use complex analysis to study algebraic objects. Why are they good analytic functions for number theory? Well, suppose you have some nice number theoretic object $X$. Then often you can expect some sort of local-global principle, meaning $X$ is mostly (or completely) determined by associated local objects $X_p$ at all primes $p$ (well, sometimes you should include prime powers). In fact, it often happens (Chebotarev density and the like), that you only need to know $X_p$ for a set of primes of sufficiently large density.

So, if you want an analytic function to take advantage of local-global principles, it makes sense to look for a kind of function that has an "Euler product," so the objects can be studied in terms of corresponding local analytic data. In my mind, this is the first property you want. For convergence of this "Euler product," you need local factors that are close to 1 (at least in some large domain), and something like a polynomial or geometric series is one of the simplest things you could guess. For analytic reasons, you also want to be able to control the zeroes/poles of the local factors, which makes polynomials in $q^{-s}$ useful.

From another point of view, if you have some number theoretic sequence $(a_n)$, and you want to make an analytic function out of it, how can you do this? Well, the most natural thing to do is use a series. What are the options? Power series? Fourier series? Dirichlet series? If $(a_n)$ is a nice arithmetic sequence (multiplicative, polynomial growth) you won't typically have much convergence with a power or fourier series, and more importantly, you don't get the Euler product.

As a tool for functoriality

I don't (entirely) think of $L$-functions as some magical world where everything gets tied together (though they are quite magical). I think of them as a useful tool to compare different objects which have related representations (though some people may take the opposite view). Many interesting algebraic objects naturally have associated representations to them (or already are representations themselves), and the central idea in Langlands' conjectures is that automorphic representations of GL($n$) are the mother of all "arithmetic" (Galois) representations. Meaning, to some arithmetic object $X$, we associate $n$-dimensional representations (sometimes just local ones), and these should correspond to automorphic representations of GL($n$).

For the functoriality aspects, the two key properties of $L$-functions are additivity ($L(s,\rho_1 \oplus \rho_2) = L(s, \rho_1)L(s,\rho_2)$) and inductivity ($L(s,I_{E}^F(\rho)) = L(s, \rho)$), and for this it matters what the precise form of the local factors are. Roughly you want it to a "characteristic polynomial" of the representation (at least for a finite-dimensional Galois representation---on the automorphic side, you want the analogous thing for Langlands parameter) so that additivity will hold.

Recall that the correspondence Galois and automorphic representations, is that the representations should locally correspond at almost all places, so it suffices to look at unramified places, where both sides should reduce to the 1-dimensional case of local class field theory. Then the additivity of $L$-functions says the $L$-functions correspond. Of course the main problem in functoriality is showing that $\pi = \otimes \pi_v$ is (globally) automorphic---constructing the $\pi_v$ for almost all $v$ is easy. Then inductivity is useful to get compatibility with base change.