An important theme in both modern and classical number theory has been to study analytic functions associated to number theoretic objects. A famous example is the Riemann zeta function for Re(s) > 1:
ζ(s) = ∞ X n=1 1 ns
and its functional equation given by Riemann in his famous 1859 paper: ξ(s) = ξ(1 − s) where ξ(s) = 1 2π −s 2s(s − 1)Γ s 2 ζ(s).
This functional equation allows ζ(s) to be analytically continued to C\{1} (once one continues ζ(s) to 0 < Re(s) < 1).
Hidden within ζ(s) are many interesting results on the distribution of prime numbers. A small justification for this lies in the Euler product expansion of ζ(s) for Re(s) > 1: ζ(s) =Y p 1 − 1 ps −1 ,
this being an analytic way of expressing unique factorisation.
Riemann showed how the analytic continuation of ζ(s) allows us to find families of good approximations to the prime counting function π(x), given
1.1. Elliptic modular forms
that the zeros of ζ(s) are well behaved. This is the subject of the Riemann Hypothesis, still an unsolved problem. Included in his approximations is the Prime Number Theorem:
lim
x→∞
π(x) x/ln(x)= 1.
There is also merit to studying the special values of ζ(s) at integers. Such values appear in the coefficients of Eisenstein series but it is also worth studying them for their own right to uncover mysterious identities. For example Euler was able to justify that:
ζ(2) = ∞ X n=1 1 n2 = π2 6
and went on to find the general formula for ζ(2k) (given after Theorem 1.1.16). Note that ζ(2k) = α2kπ2k for some α2k ∈ Q so that these zeta values split
neatly into rational and transcendental parts.
In general one can attach to any arithmetic function g : N → C a Dirichlet series: L(g, s) = ∞ X n=1 g(n) ns .
Whenever g has nice enough properties such as multiplicativity there is a well known theory of Euler products.
It is hoped that analytic properties of L(g, s) encode arithmetic properties of g amongst other things of number theoretic significance, e.g. distribution of primes.
Indeed the study of the functions L(χ, s) for lifts of characters χ of (Z/mZ)× allowed Dirichlet to prove his famous theorem on primes in arithmetic progres- sions, and to give information on densities of primes lying in given classes mod m.
Classically other notable Dirichlet series are ones attached to number fields, known as Dedekind zeta functions. Studying these gives precise information on class numbers, embeddings, fundamental units, discriminants etc. This is the basis of the Dirichlet class number formula.
In a modern setting we attach such “L-functions” and “zeta functions” to varieties, modular forms, Galois representations and many other interesting objects. We still appear to be finding deep number theoretic results encoded in such functions. For example the famous Birch Swinnerton-Dyer conjecture tells us how studying L(E, s) for a rational elliptic curve gives lots of information of interest about E, such as the rank of the Mordell-Weil group E(Q), the number of torsion points, the size of the Shafarevich-Tate group of E etc.
to newforms for Γ0(N ). Of course we already have an arithmetic function to
hand given by the Fourier coefficients of the form.
Definition 1.1.42. Let f ∈ Mk(Γ0(N )) have q-expansion f (z) =P ∞ n=0anqn.
The L-function associated to f is:
L(f, s) = ∞ X n=1 an ns.
It is known that this L-function converges absolutely for Re(s) > k+12 if f is a cusp form and for Re(s) > k otherwise. This follows from known upper bounds for |an|.
First one asks whether such L-functions have Euler products. It turns out that in some cases they do but they have quadratic Euler factors in p−s rather than linear ones.
Theorem 1.1.43. The form f is a normalized eigenform for all Hecke operators if and only if L(f, s) has an Euler product expansion of the form:
L(f, s) =Y
p
(1 − app−s+ pk−1−2s)−1.
This again highlights the historical significance of eigenforms. In general L(f, s) will not have such an expansion.
Of course we have seen that it is not always possible to construct normalized eigenforms for all Hecke operators, but that you may always find them for index coprime to the level. In such cases L(f, s) will still have an Euler product but it will be of the form:
L(f, s) =Y p|N (1 − app−s)−1 Y p-N (1 − app−s+ pk−1−2s)−1.
Secondly we ask whether such L-functions have analytic continuation to the whole complex plane. Indeed they do but it is first convenient to define the completed L-function:
Λ(f, s) = Ns2(2π)−sΓ(s)L(f, s).
This function is the analogue of Riemann’s ξ function mentioned above. One can view Λ(f, s) as a Mellin transform as follows:
Λ(f, s) = Ns2
Z ∞
0
1.1. Elliptic modular forms
Using this one proves a functional equation:
Λ(f, s) = ikΛ(WN(f ), k − s). Here WN(f ) = N1− k 2f |k 0 −1 N 0
∈ Sk(Γ0(N )) is the Atkin-Lehner invo-
lute of f . Whenever WN(f ) = ±f (i.e. f is an eigenvector for WN) we have:
Λ(f, s) = ±ikΛ(f, k − s).
Finally we address the question of finding special values of such L-functions. It seems natural to want to evaluate L(f, s) at integer values of s and get nice formulae. However this is very ambitious.
To put this in perspective consider the same question for the Riemann zeta function. We know next to nothing about closed formulae for ζ(2k + 1) for positive integers k. We can only really compute these values numerically. It was only around 1979 that ζ(3) was proved to be irrational by Ap´ery [1].
However if we consider instead the values ζ(2k) then much is known. We happen to know that ζ(2k)π2k ∈ Q and as seen earlier we even have a formula for
the “rational part”, involving Bernoulli numbers. Of course one can use the functional equation to deal with negative integer inputs.
It turns out that for L-functions of modular forms the integer inputs that we know the most about are s = 1, 2, ..., k − 1. These are called the critical values of f . Note that by the symmetry in the functional equation it suffices to evaluate at the values s = k2,k2 + 1, ..., k − 1.
As a brief remark the notion of critical value has been made precise in a paper of Deligne [17]. In this paper he defines critical values of L-functions L(M, s) attached to motives (of which all previously stated L-functions are examples). For ζ(s) the critical values are, as expected, the even positive integers and the odd negative ones. For L(f, s) they are exactly the critical values mentioned above.
Deligne even goes on to conjecture the existence of a “period” c+
(M ) ∈ C such that L(M,0)c+(M ) ∈ Q. This is the natural generalisation of the statement
ζ(2k)
π2k ∈ Q. One might think of this period as being the “transcendental part”
of the L-value, that once divided out leaves something algebraic.
For the case of modular forms Deligne’s conjecture has been proved. However a surprise occurs in that the period we must divide by is only dependent on the parity of the critical value.
Theorem 1.1.44. (Manin/Vishik) Let m be a critical value for f . There exist constants Ω+, Ω− ∈ C such that Λ(f,m)
Ω+ ∈ Qf if m is even and Λ(f,m)
Ω− ∈ Qf if
The constants Ω± are not unique but determined up to scalar multiples in Q×f. Often it is possible to pin it down to a scalar multiple in O
×
Qf (so as to make
it almost canonical). By making this normalization we can study divisibility of such values by primes without ambiguity.
The proof of the above result is beautiful and is found in Manin’s paper [49]. The idea is to define for m = 0, 1, 2, ..., k − 2 the periods of f ∈ Sk(Γ0(N )):
rm(f ) =
Z i∞
0
f (z)zmdz.
Note that Λ(f, m + 1) = Nm+12 rm(f ) so that once one knows the periods of f
it is possible to extract critical Λ-values.
Manin exhibits an infinite system of homogeneous linear equations for the periods of f by using certain actions of Hecke operators. The coefficients of these equations lie in Qf. He then manages to prove that these equations naturally
break up into two sets of equations, one for the even index periods and one for the odd ones. Finally he is able to prove that the two sets of equations each give a 1-dimensional space of solutions, explaining the existence of Ω±. This gives an algebraic method for finding ratios of periods, at no point do we need to find the above integrals.
When a choice of Ω± is fixed and m is critical we will write Λ
alg(f, m) = Λ(f,m)
Ω± (where the choice of sign depends on the parity of m). We will be
interested in special primes dividing these values since these will eventually be the moduli of our congruences (in direct analogue with the 691 in Ramanujan’s congruence).