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 .

This is a note I prepared for my fourth Trivial Notions talk at Harvard, Spring 2015. Our main sources are [1], [2], [3], [4], [5] and [6].

[-] Contents

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:

Theorem 1 (Class number formula)
The residue of at is equal to Here

- and are the number of real and complex places of .
- is the discriminant of .
- is the class number.
- is the number of roots of unity.
- is the regulator, where .

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".

Remark 2
A general -function can be defined as a sum But if you put random there it is hopeless to get you hands on. -functions people studied a lot involves two major sources of : coming form algebraic varieties/Galois representations (Dedekind, Artin, Hasse-Weil, ...) or coming from modular forms/automorphic representations (Dirichlet, Hecke, ...). It is hard to define precisely what a general -function is, but you know one when you see one: in particular it should have an Euler product and should satisfy a certain functional equation. The first class of -functions (*motivic* -functions) has natural definition as Euler products but the functional equation and analytic continuation are hard to establish, while the second class of -functions (*automorphic -functions*) are harder to construct but has more accessible analytic property once constructed. One major motivation of the Langlands program is trying to relate these two sources of -functions.

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.

Example 3
Consider the elliptic curve . Counting points over we can tabulate the first few 's.
On the other hand, the modularity theorem ensures for some . It turns out . The unique such form has the classical eta product
One now can have infinite fun checking the coefficient before actually agrees with ! It is far from obvious why this power series expansion should have anything to do with solving the cubic equation over all finite fields.

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 .

Question
What is this number?

The idea is to generalize the logarithm appearing in Dirichlet's formula to the *dilogarithm*.

Definition 1
The *polylogarithm* is defined to be . In particular . It converges when and like the logarithm, it can be analytic continued to obtain a *multi-valued* function on . The dilogarithm jumps by as go around . To obtain a single valued function, we modify the dilogarithm and define the *Bloch-Wigner dilogarithm* It becomes a continuous single-valued function on , real analytic on (with singular type ), vanish on the real line and satisfies .

Definition 2
Suppose , with . Bloch defined the *elliptic dilogarithm* by averaging using the action of on ,

This is a purely complex analytic construction. Interesting things happen when evaluating at rational points of .

Example 5
has complex uniformization , where We have with Under the complex uniformization, corresponds to . We can compute and Now the miracle is which looks like a rational number! In other words, we find a rational point whose elliptic dilogarithm contributes the transcendental part of .

Example 6
Notice only depends on the -isogeny class of . What happens when we choose a different curve in the same isogeny class? Here is an example to illustrate the subtlety. Let . It is 5-isogenous to and thus has the same -function as . It has complex uniformization with We also have , with Again corresponds to . We can compute neither of which looks like a rational number. If we take the linear combination of these two transcendental numbers, the miracle happens again: which should be rational! In other words we find the divisor supported on whose elliptic dilogarithm contributes to the transcendental part 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:

Theorem 2 (Bloch, Beilinson)
Let be an elliptic curve with . There exists a -stable divisor on such that .

Remark 3
This theorem was first proved by Bloch [3] for elliptic curves with complex multiplication and later proved by Beilinson [7] in general.

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.

Remark 4
These are called -groups simply because Grothendieck used the letter to denote his group (now we see the first connection between the two words in the title). Though this time the name had a bit justification: was intended to stand for the German word *Klasse*.

Example 8
For a number field , we have , . So the invariants appearing in Dirichlet's theorem can all be rephrased in terms of pieces of these -groups. In particular, we can interpret the regulator as the determinant of the regulator map

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 .

Remark 5
For the CM case, is a Hecke -series and Bloch's evaluation of the regulator map at functions supported at torsion points mimics the classical computation of Kummer relating Dirichlet -series and logarithm of cyclotomic units, via Fourier transform. For the general case, Beilinson used another construction of the regulator map, which replaces the target by the Deligne cohomology group (which is one dimensional). To be more precise, Beilinson works with modular curves and the value of the regulator map at a modular unit gives an integral of the product of and two Eisenstein series, which evaluates to for some Dirichlet characters , via Rankin's method.

Remark 6
The elements in of modular curves constructed by Beilinson were later used by Kato to construct Euler systems and became the starting point of Kato's proof [8] of the BSD conjecture in the analytic rank zero case and one divisibility in the Iwasawa main conjecture for .

Remark 7
Beilinson formulated more general conjectures for the special values of motivic -functions by constructing regulator maps from motivic cohomology to Deligne cohomolgy, which is further generalized in the framework of mixed motives. For the list of (very few) proved cases, see the end of [9].

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