I’ve just sat through several lectures that proved most of the results in Tate’s thesis: the self-duality of the adeles, the construction of “zeta functions” by integration, and the proof of the functional equation. However, while I was able to follow at least some of the arguments in the individual steps, I understand almost nothing about the big picture. My impression so far is that Tate invented a new and fancier way of proving the functional equation that the Hecke analytic approach. But is there more to the story than “this is a neat way of proving something already known”?
I’m under the impression that Tate’s thesis laid the foundations for the Langlands program, but I don’t understand this properly yet.
Can someone explain to me what’s the real significance and meaning of Tate’s thesis?
Tate’s thesis introduces the concept, ubiquitous now, of doing analysis, and especially Fourier analysis, on the locally compact ring of adeles. In this setting, the discrete subgroup $\mathbb Z \subset \mathbb R$ is replaced
by the discrete subgroup $\mathbb Q \subset \mathbb A$.
This has a number of implications, some of which are:
$\mathbb Q$ is a field, and $\mathbb A$ is essentially a product of fields.
It is technically almost always easier to work with fields rather than more general rings (such as $\mathbb Z$). The adelic formalism allows one to have one’s cake and eat it too (in some sense): one is working with the field $\mathbb Q$, not the ring $\mathbb Z$, but the primes are still present, in the factorization of $\mathbb A$ as a product. (And this product structure of
$\mathbb A$, which is formally very simple, captures in some subtle way the
deeper sense of “product” in the statement of the fundamental theorem of arithmetic, i.e. that any natural number is a unique product of prime powers.)
Tate writes zeta-functions, or more generally, Hecke $L$-series, as integrals over $\mathbb A^{\times}$. The Euler product structure of the $L$-series then becomes simply a factorization of this integral according to the product structure of $\mathbb A^{\times}$. (This is a manifestation of the parenthetical remark at the end of point (1).)
The proof of the functional equation becomes (more-or-less) just an application of Poisson summation (in the adelic context).
It is worth comparing this with the classical proof (which one can read in Lang’s book, among other places, if memory serves). Classically, one takes
the sum over ideals representation of the $L$-function, and decomposes it first
into a finite collection of sums, indexed by the ideal class group, each sum taking place over all the integral ideals in a given ideal class. These individual series are then described as Mellin transforms of theta series,
and the functional equation is derived from the transformation properties of the
theta series, the latter being proved by an application of Poisson summation in the classical setting.
Once one unpacks all the details, Tate’s proof and Hecke’s proof don’t look so different; but the difference in packaging is enormous! In Tate’s approach there is no need to unpack everything (for example, the ideal class group is just lurking around in the background implicitly, and there is no need to bring it out explicitly), while in the classical arguments such unpacking is key to the whole thing.
As another example of the conceptual clarity and simplification that Tate’s approach gives, you might consider the way he derives the formula for the residue at $s = 1$ of the zeta function of a number field (i.e. the general class number formula) and compare it with the classical derivation.
Working in the case of a function field over a finite field, Tate derives the Riemann–Roch formula (in the form $\dim H^0(C,\mathcal O(D)) – \dim
H^0(C,\mathcal O(K – D)) = 1 + \deg D – g$) as a straightforward consequence of Poisson summation. Among other things, this provides a rather striking unification of (what we now call) Serre duality and Fourier duality. (Although I don’t know the precise history, this probably has antecedents in the literature: the original proof of the functional equation of the $\zeta$-function for a curve over a finite field, by Schmidt, proceeded by applying Riemann–Roch; so Tate is essentially reversing this argument.)
Tate’s explication of the functional equation of $L$-series in terms of local functional equations shows that the global root number — i.e. the constant that appears in the functional equation — is a product of local numbers. As far as I understand, this wasn’t known (and perhaps not even suspected) prior to Tate’s proof.
This may seem slightly esoteric, but experience shows that one should regard global root numbers, and their factorization into a product of local root numbers (or $\epsilon$-factors), to be of essentially equal importance to global $L$-series, and their (Euler product) factorization into local $L$-factors.
Summary/Conclusion: The aim of the above list is just to highlight some of the points to watch out for while studying Tate’s thesis. Let me now make some remarks at a more general level.
In the classical theory of zeta and $L$-functions, there is a tension between the analytic tools, which are essentially additive Fourier theory (e.g. Poissson summation) and the multiplicative aspects of the theory (exemplified by the Euler product). Tate’s thesis resolves these tensions by moving to the adelic context.
In the general theory of automorphic forms (say on a quotient $\Gamma
\backslash G(\mathbb R)$) for some congruence subgroup $\Gamma$ of the integral points $G(\mathbb Z)$ of a semi-simple or reductive Lie group $G(\mathbb R)$) there is the same tension between the harmonic analysis and Lie theory (which $\Gamma \backslash G(\mathbb R)$ is well set-up to accommodate) and the theory of Hecke operators (which pertain to the finite primes, which are not particular visible in this classical description), which is resolved by moving to the adelic picture $G(\mathbb Q)\backslash G(\mathbb A)$.
Another thing to bear in mind is that the theories of $L$-series and of automorphic forms are quite technical in nature, and so conceptual and aesthetic simplifications (as in Tate’s thesis) go hand in hand with technical simplifications. (See e.g. points (1) and (3) above.) One instance of this in the automorphic forms context is that conjugacy classes in $G(\mathbb Q)$ are much easier to understand than in a congruence subgroup $\Gamma$ of $G(\mathbb Z)$. (Another instance of the technical superiority of fields over more general rings.) One might also consider the Tamagawa number one theorem, which gives an elegant reformulation and generalization of a myriad of classical results.
So, to finish, Tate’s thesis is significant because it improves the classical point of view in a number of ways, achieving conceptual, technical, and aesthetic simplifications. At the same time, it suggests a way of unifying harmonic analytic and arithmetic considerations in the general context of automorphic forms, by working in the adelic context.
Finally, I strongly suggest working through the details of Tate’s thesis in the particular case of the Riemann zeta function, and seeing how his arguments and construction compare with the classical ones. If you haven’t already done this, it should be quite enlightening. (In particular, it will illuminate points (1), (2), and (3) above.)