Modular group

of one complex variable, elliptic modular form

A function $ f $ on the upper half-plane $ H = \{ {z \in \mathbf C } : { \mathop{\rm Im} z > 0 } \} $ satisfying for some fixed $ k $ the automorphicity condition

$$ \tag{1 } f \left ( \frac{a z + b }{c z + d } \right ) = \ ( c z + d ) ^ {k} f ( z) $$

for any element

$$ \left ( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right ) \ \in \mathop{\rm SL} _ {2} ( \mathbf Z ) $$

( $ \mathop{\rm SL} _ {2} ( \mathbf Z ) $ is the group of integer-valued matrices with determinant $ a d - b c = 1 $), and such that

$$ f ( z) = \ \sum _ { n= } 0 ^ \infty a _ {n} q ^ {n} , $$

where $ q = \mathop{\rm exp} ( 2 \pi i z ) $, $ z \in H $, $ a _ {n} \in \mathbf C $. The integer $ k \geq 0 $ is called the weight of the modular form $ f $. If $ a _ {0} = 0 $, then $ f $ is called a parabolic modular form. There is also [8] a definition of modular forms for all real values of $ k $.

An example of a modular form of weight $ k \geq 4 $ is given by the Eisenstein series (see [4])

$$ G _ {k} ( z) = \ \sum _ {m _ {1} , m _ {2} \in \mathbf Z } {} ^ {*} ( m _ {1} + m _ {2} z ) ^ {-} k , $$

where the asterisk means that the pair $ ( m _ {1} , m _ {2} ) = ( 0 , 0 ) $ is excluded from summation. Here $ G _ {k} ( z) \equiv 0 $ for odd $ k $ and

$$ G _ {k} ( z) = \ \frac{2 ( 2 \pi i ) ^ {k} }{( k - 1 ) ! } \left [ - \frac{B _ {k} }{2 k } + \sum _ { n= } 1 ^ \infty \sigma _ {k-} 1 ( n) q ^ {n} \right ] , $$

where $ \sigma _ {k-} 1 ( n) = \sum _ {d \mid n } d ^ {k-} 1 $ and $ B _ {k} $ is the $ k $- th Bernoulli number (cf. Bernoulli numbers).

The set of modular forms of weight $ k $ is a complex vector space, denoted by $ M _ {k} $; in this connection, $ M _ {k} M _ {l} \subset M _ {k+} l $. The direct sum $ \oplus _ {k = 0 } ^ \infty M _ {k} $ forms a graded algebra isomorphic to the ring of polynomials in the independent variables $ G _ {4} $ and $ G _ {6} $( see [3]).

For each $ z \in H $ the complex torus $ \mathbf C / ( \mathbf Z + \mathbf Z z ) $ is analytically isomorphic to the elliptic curve given by the equation

$$ \tag{2 } y ^ {2} = 4 x ^ {3} - g _ {2} ( z) x - g _ {3} ( z) , $$

where $ g _ {2} ( z) = 60 G _ {4} ( z) $, $ g _ {3} ( z) = 140 G _ {6} ( z) $. The discriminant of the cubic polynomial on the right-hand side of (2) is a parabolic modular form of weight 12:

$$ \frac{1}{2 ^ {4} } ( g _ {2} ^ {3} - 27 g _ {3} ^ {2} ) = \ \frac{( 2 \pi ) ^ {12} }{2 ^ {4} } q \prod _ { m= } 1 ^ \infty ( 1 - q ^ {m} ) ^ {2k} = \ \frac{( 2 \pi ) ^ {12} }{2 ^ {4} } \sum _ { n= } 1 ^ \infty \tau ( n) q ^ {n} , $$

where $ \tau ( n) $ is the Ramanujan function (see [1]).

For each integer $ N \geq 1 $ modular forms of higher level $ N $ have been introduced, satisfying (1) only for elements

$$ \left ( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right ) $$

of a congruence subgroup $ \widetilde \Gamma $ of level $ N $ of the modular group. In this case, related to the modular form $ f $ is the holomorphic differential $ f ( z) ( d z ) ^ {k/2} $ on the modular curve $ X _ {\widetilde \Gamma } $. A well-known example of a modular form of higher level is the theta-series $ f ( z) $ associated to an integer-valued positive-definite quadratic form $ F ( x _ {1} \dots x _ {m} ) $:

$$ f ( z) = \ \sum _ {x _ {1} , \dots, x _ {m} \in \mathbf Z } \mathop{\rm exp} ( 2 \pi i F ( x _ {1} \dots x _ {m} ) ) , $$

which is a modular form of higher level and of weight $ k = m / 2 $. In this example $ a _ {n} $ is the integer equal to the number of solutions of the Diophantine equation $ F ( x _ {1} \dots x _ {m} ) = n $.

The theory of modular forms allows one to obtain an estimate, and sometimes a precise formula, for numbers of the type $ a _ {n} $( and congruences, such as the Ramanujan congruence $ \tau ( n) \equiv \sum _ {d \mid n } d ^ {11} $( $ \mathop{\rm mod} 691 $)), and also to investigate their divisibility properties (see [7]). Best estimates for numbers of the type $ a _ {n} $ have been obtained (see [2]).

Important arithmetic applications of modular forms are related to the Dirichlet series

$$ L _ {f} ( s) = \ \sum _ { n= } 1 ^ \infty a _ {n} n ^ {-} s , $$

i.e. the Mellin transform of $ f $. Such Dirichlet series have been the subject of detailed study (estimates of coefficients, analyticity properties, the functional equation, Euler product expansion) in view of the presence of a non-trivial ring of correspondences $ R $ on a modular curve. For a curve $ X _ \Gamma $ this ring is generated by the correspondence $ T _ {n} ( z) = \sum _ \gamma \gamma ( z) $, where $ \gamma $ runs through the set of all representatives of the elements of the quotient set

$$ \mathop{\rm SL} ( 2 , \mathbf Z ) \setminus \{ {A \in M _ {2} ( \mathbf Z ) } : { \mathop{\rm det} A = n } \} . $$

The correspondences induce linear operators (Hecke operators) acting on the space of modular forms. They are self-adjoint relative to the Peterson scalar product (see [3], [7]). Modular forms which are eigen functions of the Hecke operators are characterized by the fact that their Mellin transforms have Euler product expansions.

Another direction in the theory of modular forms is related to the study of modular curves and the associated fibrations, the Kuga varieties (cf. Modular curve), and also to the theory of infinite-dimensional representations of algebraic adèle groups. Here the theory of modular forms of one variable was successfully transferred to the case of several variables (see [6]). A survey of the number-theoretic applications of modular forms is given in [5].

References

[1]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) pp. Chapt.8
[2]	P. Deligne, "La conjecture de Weil I" Publ. Math. IHES , 43 (1974) pp. 273–307
[3]	S. Lang, "Introduction to modular forms" , Springer (1976)
[4]	J.-P. Serre, "A course in arithmetic" , Springer (1973) (Translated from French)
[5]	O.M. Fenko, "Applications of the theory of modular forms to number theory" J. Soviet Math. , 14 : 4 (1977) pp. 1307–1362 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 15 (1977) pp. 5–91
[6]	J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.) , Modular functions of one variable. 1–6 , Lect. notes in math. , 320; 349; 350; 476; 601; 627 , Springer (1973–1977)
[7]	A. Ogg, "Modular forms and Dirichlet series" , Benjamin (1969)
[8]	R.A. Rankin, "Modular forms and functions" , Cambridge Univ. Press (1977)

Comments

A parabolic modular form is also called a cusp form.