Ramanujan–Petersson conjecture

In

when p is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by Petersson (1930), is a generalization to other modular forms or automorphic forms.

Ramanujan L-function

The Riemann zeta function and the Dirichlet L-function satisfy the Euler product,

${\displaystyle L(s,a)=\prod _{p}\left(1+{\frac {a(p)}{p^{s}}}+{\frac {a(p^{2})}{p^{2s}}}+\dots \right)}$

(1)

and due to their

${\displaystyle L(s,a)=\prod _{p}\left(1-{\frac {a(p)}{p^{s}}}\right)^{-1}.}$

(2)

Are there L-functions other than the Riemann zeta function and the Dirichlet L-functions satisfying the above relations? Indeed, the L-functions of automorphic forms satisfy the Euler product (1) but they do not satisfy (2) because they do not have the completely multiplicative property. However, Ramanujan discovered that the L-function of the modular discriminant satisfies the modified relation

${\displaystyle L(s,\tau )=\prod _{p}\left(1-{\frac {\tau (p)}{p^{s}}}+{\frac {1}{p^{2s-11}}}\right)^{-1},}$

(3)

where τ(p) is Ramanujan's tau function. The term

${\displaystyle {\frac {1}{p^{2s-11}}}}$

is thought of as the difference from the completely multiplicative property. The above L-function is called Ramanujan's L-function.

Ramanujan conjecture

Ramanujan conjectured the following:

1. τ is multiplicative,
2. τ is not completely multiplicative but for prime p and j in N we have: τ(p^j+1) = τ(p)τ(p^j ) − p¹¹τ(p^j−1 ), and
3. |τ(p)| ≤ 2p^11/2.

Ramanujan observed that the quadratic equation of u = p^−s in the denominator of RHS of (3),

${\displaystyle 1-\tau (p)u+p^{11}u^{2}}$

would have always imaginary roots from many examples. The relationship between roots and coefficients of quadratic equations leads the third relation, called Ramanujan's conjecture. Moreover, for the Ramanujan tau function, let the roots of the above quadratic equation be α and β, then

${\displaystyle \operatorname {Re} (\alpha )=\operatorname {Re} (\beta )=p^{11/2},}$

which looks like the

${\displaystyle O\left(n^{11/2+\varepsilon }\right).}$

In 1917, L. Mordell proved the first two relations using techniques from complex analysis, specifically what are now known as Hecke operators. The third statement followed from the proof of the Weil conjectures by Deligne (1974). The formulations required to show that it was a consequence were delicate, and not at all obvious. It was the work of Michio Kuga with contributions also by Mikio Sato, Goro Shimura, and Yasutaka Ihara, followed by Deligne (1971). The existence of the connection inspired some of the deep work in the late 1960s when the consequences of the étale cohomology theory were being worked out.

Ramanujan–Petersson conjecture for modular forms

In 1937, Erich Hecke used Hecke operators to generalize the method of Mordell's proof of the first two conjectures to the automorphic L-function of the discrete subgroups Γ of SL(2, Z). For any modular form

${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}q^{n}\qquad q=e^{2\pi iz},}$

one can form the Dirichlet series

${\displaystyle \varphi (s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}$

For a modular form f (z) of weight k ≥ 2 for Γ, φ(s) absolutely converges in Re(s) > k, because a_n = O(n^k−1+ε). Since f is a modular form of weight k, (s − k)φ(s) turns out to be an entire and R(s) = (2π)^−sΓ(s)φ(s) satisfies the functional equation:

${\displaystyle R(k-s)=(-1)^{k/2}R(s);}$

this was proved by Wilton in 1929. This correspondence between f and φ is one to one (a₀ = (−1)^k/2 Res_s=k R(s)). Let g(x) = f (ix) −a₀ for x > 0, then g(x) is related with R(s) via the Mellin transformation

${\displaystyle R(s)=\int _{0}^{\infty }g(x)x^{s-1}\,dx\Leftrightarrow g(x)={\frac {1}{2\pi i}}\int _{\operatorname {Re} (s)=\sigma _{0}}R(s)x^{-s}\,ds.}$

This correspondence relates the Dirichlet series that satisfy the above functional equation with the automorphic form of a discrete subgroup of SL(2, Z).

In the case k ≥ 3 Hans Petersson introduced a metric on the space of modular forms, called the Petersson metric (also see Weil–Petersson metric). This conjecture was named after him. Under the Petersson metric it is shown that we can define the orthogonality on the space of modular forms as the space of cusp forms and its orthogonal space and they have finite dimensions. Furthermore, we can concretely calculate the dimension of the space of holomorphic modular forms, using the Riemann–Roch theorem (see the dimensions of modular forms).

Deligne (1971) used the Eichler–Shimura isomorphism to reduce the Ramanujan conjecture to the Weil conjectures that he later proved.　The more general Ramanujan–Petersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent (k − 1)/2 where k is the weight of the form. These results also follow from the Weil conjectures, except for the case k = 1, where it is a result of Deligne & Serre (1974).

The Ramanujan–Petersson conjecture for

Ramanujan–Petersson conjecture for automorphic forms

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