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I've been studying modular forms this semester and we did a lot of calculations of $L$-functions, e.g. $L$-functions of Dirichlet-characters and $L$-functions of cusp-forms.

But I somehow don't see, why they are considered a big deal. To me it sounds more like a fun fact to say: "You know, the Riemann-$\zeta$ has analytic continuation.", and I don't even know what to say about $L$-functions of cusp-forms.

So why are $L$-functions such a big thing in automorphic forms and analytic number theory?

Thanks!

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    $\begingroup$ For one thing, $L$-functions are the connecting link between modular forms and elliptic curves that led to the formulation of the modularity conjecture. $\endgroup$ – Arthur Jul 13 '16 at 9:39
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    $\begingroup$ Strictly speaking, you can state the modularity conjecture in geometric terms, without L-functions (it's done in Mazur, Number Theory as gadfly, for example). But many different mathematical objects define L-functions, which allows us top compare them, and that may be a part of the answer. But anyhow, I'm waiting eagerly for the experts' answers on this question. $\endgroup$ – PseudoNeo Jul 13 '16 at 10:16
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    $\begingroup$ power series are more than helpful for studying additive functions, while Dirichlet series (with an Euler product = L-function) are more than helpful for studying multiplicative functions. almost everything that is related to the factorization and prime numbers can be represented as multiplicative functions, and multiplicative functions naturally occur in modular forms (and conversely) $\endgroup$ – reuns Jul 24 '16 at 14:58
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    $\begingroup$ Have you tried working through Dirichlet's theorem on primes in arithmetic progressions? This is likely the first place where L-functions can be used to prove something meaningful to a general number theory student. $\endgroup$ – lemiller Jul 25 '16 at 1:49
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There's a lot one could say, but I'll try to be brief. Roughly the idea (just like with the zeta functions) is that L-functions provide a way to analytically study arithmetic objects. Specifically a lot of interesting data is encoded in the location of the zeroes and poles of L-functions, and because L-functions are analytic objects, you can now use analysis to study arithmetic. Here are some examples:

  • The fact that $\zeta(s)$ has a pole at $s=1$ implies the infinitude of primes.

  • (added) The Riemann hypotheses and generalizations, which are about location of nontrivial zeroes of zeta-/L-functions, have lots of implications, such as refined information about distribution of prime numbers.

  • The fact that Dirichlet L-functions do not have a zero at $s=1$ implies there are infinitely many primes in arithmetic progressions. Dirichlet introduced the notion of L-functions to prove this fact.

  • If $E : y^2 = x^3+ax+b$ is an elliptic curve and its $L$-function $L(s,E)$ (which is also the $L$-function of an elliptic curve) has a zero at the central value $s=1$, then $y^2=x^3+ax+b$ has only finitely many rational solutions. This is the known direction of the Birch and Swinnerton-Dyer conjecture.

  • (added) In addition to knowing just locations of zeroes and poles of L-functions, the actual values of L-functions at special points contain further arithmetic information. For instance, if $\chi_K$ is the quadratic Dirichlet character associated to an imaginary quadratic field $K$, then the class number formula says $L(1,\chi_K)$ is essentially the class number of $K$. Similarly, the value of $L(1,E)$ in the previous example is conjecturally expressed in terms of the size of the Tate-Shafarevich group of $E$ and the number of rational points on $E$.

As mentioned in the comments, $L$-functions are also a convenient tool to associate different kinds of objects to each other, e.g., elliptic curves and modular forms, but are not strictly needed to do this.

Nice $L$-functions will have at least meromorphic continuation to $\mathbb C$, Euler products, and certain bounds on their growth. For instance, L-functions of eigencusp forms and Dirichlet L-functions. These properties make $L$-functions nice analytic objects to work with. In particular, the Euler product provides a way to study global objects from local data (one finite set of data for each prime number $p$).

(added) See also this MathOverflow question.

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  • $\begingroup$ In complement to Kimball's statement about how L-functions provide a way to analytically study arithmetic objects, allow me to draw your attention to the significance of their "special values", see e.g. my answer to the question "Significance of the RH to algebraic number theory " posted by Vik78 on April 29. $\endgroup$ – nguyen quang do Jul 21 '16 at 7:51
  • $\begingroup$ @nguyenquangdo It might be more helpful if you provide a link to your answer (if you don't know how to do this, click on "share" at the bottom of your answer; see also "help" on the side of the comment bar). $\endgroup$ – Kimball Jul 21 '16 at 11:13
  • $\begingroup$ Sorry, here's the link math.stackexchange.com/a/1771242/30 $\endgroup$ – nguyen quang do Jul 21 '16 at 14:21

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