Euler in 1735 discovered that and Dirichlet in 1839 proved that We begin by re-interpreting these sums as special values of -functions of number fields. The notion of -functions has been vastly generalized and their special values are the subject of the celebrated conjectures of Birch-Swinnerton-Dyer, Beilinson and many others. We then focus on the next simplest (but incredibly rich) case: the -function of an elliptic curve . In 1970s, Bloch discovered a beautiful formula for in terms of Wigner's dilogarithm function (a generalization of the usual logarithm). We illustrate this formula with explicit examples and explain how Quillen's -group plays a surprising role in calculating .
Euler's identity and other similar identities like can be viewed as the special value at , of the Riemann zeta function This sum converges when and the basic observation due to Riemann is that can be extended as a meromorphic function on on via analytic continuation. It is holomorphic everywhere except a simple pole at with residue .
Here are two key properties of . It has an Euler product when , essentially due to the fundamental theorem of arithmetics in : It satisfies a functional equation, also established by Riemann, relating and : The analytic continuation endows the famous formula a mathematical meaning by interpreting the left hand side as , which can be easily computed by the functional equation and the value .
By replacing with arbitrary number field , we obtain the Dedekind zeta function, which has a similar Euler product by replacing prime numbers by prime ideals of . It also has analytic continuation and a functional equation. If you ever wonder what analytic continuation does any good for you besides computing the sum of all natural numbers:
To summarize: you first collect all local information about (at all primes of ) and form the Euler product, then running the mysterious process of analytic continuation pops out deep global information about for you!
Analogously, Dirichlet's identity can be viewed as the special value at of the Dirichlet -function where is a group homomorphism with
The sum converges when and can be analytic continued to a holomorphic function on . The multiplicativity of also gives an Euler product The key observation is that one can also view as the the Legendre symbol whose value dictates the splitting behavior of in . Therefore Now comparing the residues at on both sides we obtain Dirichlet's formula that in the beginning! To summarize: Dirichlet's formula is simply the consequence of the class number formula together with the key that is linked to "arithmetic".
Now let be an elliptic curve over . To define its -function, we would like to first collect the local information about . The natural choice is to look at the number of its -rational points (we will systematically ignore bad reduction in this talk). It is a theorem of Hasse that . Let be the error term. The Hasse-Weil -function of is then defined to be The sum converges when due to Hasse's bound. The definition looks familiar except that the denominator becomes a quadratic polynomial rather than a linear polynomial in (since we are looking at a motive of rank 2).
The most interesting global information about is its rational points , which is a finitely generated abelian group by Mordell-Weil. Does give any hint about it? Let us try to plug in formally: Since each point in reduces to a point in , when has large rank tends to be small. Moreover, the rate of converging to zero should be related to ! In 1960s, Birch and Swinnerton-Dyer did numerical experiments on EDSAC and suggested the heuristic which leads to the famous conjecture that
Now it is a theorem that has analytic continuation to (and satisfies a functional equation relating and ), so the BSD conjecture actually makes sense. The proof of this hard theorem is via the famous modularity theorem due to Wiles, Taylor, Breuil, Conrad and Diamond: there exists (weight 2 cusp newform of level ) such that . The following example may illustrate how nontrivial this modularity theorem is.
Though important progresses toward it have been made, the BSD conjecture is still widely open. In some sense the value of at is more accessible than since it does not involve the process of analytic continuation. Our next goal is to illustrate Bloch and Beilinson's theorem on the special value .
The idea is to generalize the logarithm appearing in Dirichlet's formula to the dilogarithm.
This is a purely complex analytic construction. Interesting things happen when evaluating at rational points of .
In general could be trivial and such a divisor supported on related to may not always exist. The next best thing one can hope:
We finish by explaining the role of algebraic -theory in the proof. Algebraic -theory is a sequence of functors which, roughly speaking, extracts abelian invariants from "linear algebraic construction" over the scheme . For example, are abelian invariants of "vector spaces" over : it is the Grothendieck group on the isomorphism classes of vector bundles over . Similarly, , can be thought of as abelian invariants coming from "matrices" over and "relations" between elementary matrices.
The generalization for the logarithm we seek now can be thought as a regulator map To construct we need to understand what is. Unfortunately Quillen's -group are highly nonconstructive and are very difficult to compute. We do not explain Quillen's construction here since we will not need it. The key fact we will use is that for a field , can be identified as Milnor's -group (defined for fields) This explains the following construction and the importance of the function .
Thus descends to a map Now composing the functorial map, we obtain Bloch's regulator map where denotes the Neron model of over .
This is quite elegant but the difficulty dramatically increases compared to Dirichlet's theorem. For example it is not even known that is finitely generated! Notice, for a field , by the localization exact sequence, the elements in can be represented by elements of the form which maps trivially into for each point . What Bloch and Beilinson did is something weaker than the conjecture: they were able to construct explicit elements (represented as above) in and relate their regulators to . In other words, they prove that there is a subspace of whose image under is a rational multiple of .
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