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In Mathematical, a modular form is a analytical Fonction on the Demi-plan of Poincaré satisfying a certain kind of functional equation and condition of growth. The theory of the modular forms consequently is in the line of the Analyze complexes but the principal importance of the theory holds in its connections with the Théorie of the numbers.

As a function on the networks

At the simplest level, a modular form can be thought like a function F of the whole of the networks $\backslash \; Lambda\; \backslash ,$ in $\backslash \; mathbb\; \{C\}$, towards the whole of the complex numbers which satisfies the following conditions:

(1) If we consider the network generated by a constant $\backslash \; alpha\; \backslash ,$ and a variable Z , then $F\; (\backslash \; Lambda)\; \backslash ,$ is an analytical function of Z .

(2) If $\backslash \; alpha\; \backslash ,$ is a complex number different from zero and $\backslash \; alpha\; \backslash \; Lambda\; \backslash ,$ is the network obtained by multiplying each element of $\backslash \; Lambda\; \backslash ,$ by $\backslash \; alpha\; \backslash ,$, then $F\; (\backslash \; alpha\; \backslash \; Lambda)\; =\; \backslash \; alpha^\; \{-\; K\}\; F\; (\backslash \; Lambda)\; \backslash ,$ where K is a constant (generally a positive entirety) called the weight of the form.

(3) the absolute value of $F\; (\backslash \; Lambda)\; \backslash ,$ remains limited in a lower position as long as the absolute value of the smallest element different from zero in $\backslash \; Lambda\; \backslash ,$ is far from 0.

When K = 0, condition 2 known as that F depends only on the class of similarity of the network. This is a very important particular case, but the only modular forms of weight 0 are the constants. If we eliminate condition 3 and allow the function to have poles, then the examples of weight 0 exist: they are called modular functions . The situation can be compared with profit with what arrives in the search for functions of the projective Espace $P\; (V)\; \backslash ,$. With these parameters, one would ideally wish functions F on the vector space V which are polynomials of coordinates $v\; \backslash \; 0\; \backslash ,$ in V and which satisfy the equation $F\; (cv)\; =\; F\; (v)\; \backslash ,$ for all C different from zero. Unfortunately, the only functions of this kind are the constants. If we allow the rational functions the place of the polynomials, we can let F be the ratio of two polynomials Homogène S of the same degree. Or we can keep the polynomials and lose the dependence of C , leaving $F\; (cv)\; =\; c^\; \{K\}\; F\; (v)\; \backslash ,$. The solutions are then the homogeneous polynomials of degrees K . On a side, this one form a vector space of size finished for each K , and other, if we let K vary, we can find the numerators and the denominators for the construction of all the rational functions who are the functions of projective space $P\; (V)\; \backslash ,$. One could ask, since the homogeneous polynomials are not really the functions on $P\; (V)\; \backslash ,$, that are they, geometrically speaking? The algebraic geometry answers that they are sections of a sheaf (one can also say a beam of right-hand sides in this case). The situation with the modular forms is precisely similar.

As a function on the elliptic curves

Each network $\backslash \; Lambda\; \backslash ,$ in $\backslash \; mathbb\; \{C\}\; \backslash ,$ determines a elliptic Courbe $\backslash \; mathbb\; \{C\}/\backslash \; Lambda\; \backslash ,$ on $\backslash \; mathbb\; \{C\}$; two networks determine elliptic curves isomorphous if and only if one is obtained from the other by multiplying by some $\backslash \; alpha\; \backslash ,$. The modular functions can be thought like functions on the space of the modules of the isomorphous classes of the complex elliptic curves. For example, the Invariant J of an elliptic curve, looked like a function on the whole of all the elliptic curves, is modular. The modular forms can also be approximate with profit starting from this geometrical direction, like sections of beams of right-hand sides on the space of the modules of the elliptic curves.

To convert a modular form F into a function of complex variable single is easy. Either Z = X + iy , where there > 0, and or F ( Z ) = F (<1, Z >). (We cannot allow there = 0 because then 1 and Z will not generate a lattice, thus we reduce our attention to the positive case there ). Condition 2 on F becomes now the functional equation

$f\; \backslash \; left\; (\{az+b\; \backslash \; over\; cz+d\}\; \backslash \; right)\; =\; (cz+d)^k\; F\; (Z)$

for has , B , C , D whole with $ad\; -\; bc\; =\; 1\; \backslash ,$ (the modular group). For example,

$f\; (-\; 1/z)\; =\; F\; (\backslash \; langle\; 1,\; -\; 1/z\; \backslash \; rangle)\; =\; z^k\; F\; (\backslash \; langle\; Z,\; -\; 1\; \backslash \; rangle)\; =\; z^k\; F\; (\backslash \; langle\; 1,\; Z\; \backslash \; rangle)\; =\; z^k\; F\; (Z).$

The functions which satisfy the modular functional equation for all the matrices in a sub-group finished of $SL\_2\; (\backslash \; mathbb\; \{Z\})\; \backslash ,$ are also counted like modular, usually with a qualifier indicating the group. Thus, modular forms of level NR satisfying the functional equation for the congruence of matrices with the matrix identity modulo NR (often makes some for a greater group given by conditions (MOD NR ) on the entries of the matrix).

General standards

That is to say $N\; \backslash ,$ a positive integer. The modular group $\backslash \; Gamma\_0\; (NR)\; \backslash ,$ is defined by

$\backslash \; Gamma\_0\; (NR)\; =\; \backslash \; left\; \backslash \; \{$

\ begin {pmatrix} has & B \ \ C & D \ end {pmatrix} \ in SL_2 (\ mathbf {Z}): C \ equiv 0 \ pmod {NR} \ right \}

That is to say $k$ a positive integer. A modular form of weight $k$ and level $N\; \backslash ,$ (or of level $\backslash \; Gamma\_0\; (NR)\; groups\; \backslash ,$) holomorphic Fonction $f$ is a on the Demi-plan of Poincaré such as for all

and all $z$ in the Half-plane of Poincaré, we have

$$

F \ left (\ frac {az+b} {cz+d} \ right) = (cz+d)^k F (Z)

and $f$ is Holomorphe with the points.

That is to say $\backslash \; chi\; \backslash ,$ a Character of Dirichlet MOD $N\; \backslash ,$, A modular form of weight $k$, of level $N\; \backslash ,$ (or of level $\backslash \; Gamma\_0\; (NR)\; groups\; \backslash ,$) with character $\backslash \; chi\; \backslash ,$ is a holomorphic Fonction $f$ on the Demi-plan of Poincaré such as pout all

and all $z$ in the Half-plane of Poincaré, we have

$$

F \ left (\ frac {az+b} {cz+d} \ right) = \ chi (d) (cz+d)^k F (Z)

and $f$ is Holomorphe with the points. Certain authors use different convention

$\backslash \; chi^\; \{-\; 1\}\; (d)\; (cz+d)^k\; F\; (Z)\; \backslash ,$

for the right-sided of the equation above.

Examples

The simplest examples for this point of view are the series of Eisenstein : For each Integer even K > 2, we define $E\_k\; (\backslash \; Lambda)\; \backslash ,$ as the sum of $\backslash \; lambda^\; \{-\; K\}\; \backslash ,$ on all the vectors $\backslash \; lambda\; \backslash ,$ different from zero of $\backslash \; Lambda\; \backslash ,$ (the condition K > 2 is necessary for convergence and the condition K even to avoid the cancellation of $\backslash \; lambda^\; \{-\; K\}\; \backslash ,$ with $(-\; \backslash \; lambda)\; ^\; \{-\; K\}\; \backslash ,$ and production of form 0).

A even network unimodulaire L in $\backslash \; mathbb\; \{R\}\; ^n\; \backslash ,$ is a network generated by N vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square length of each vector in L is an even integer. Like consequence of the Formula of summation of Poisson, the function theta

$\backslash \; theta\_L\; (Z)\; =\; \backslash \; sum\_\; \{\backslash \; lambda\; \backslash \; in\; L\}\; e^\; \{\backslash \; pi\; I\; \backslash \; Green\; \backslash \; lambda\; \backslash \; Vert^2\; Z\}$

is a modular form of weight $\backslash \; frac\; \{N\}\; \{2\}\; \backslash ,$. It is not also easy to build even networks unimodulaires, but there exists a manner: either N a divisible integer by 8 and let us consider all the vectors v of $\backslash \; mathbb\; \{R\}\; ^n\; \backslash ,$ such as 2 v has whole coordinates, or all pars, all odd, and such as the sum of the coordinates of v or an even integer. We call this network $L\_n\; \backslash ,$. When N =8, this is the network generated by the roots of the Système of roots called E₈.

Since the two terms of the equation are modular forms of weight 8, and since there exists only one modular form of weight 8 with multiplication by a scalar near, there is

$\backslash \; theta\_\; \{L\_8\; \backslash \; times\; L\_8\}\; (Z)\; =\; \backslash \; theta\_\; \{L\_\; \{16\}\}\; (Z),$

although the networks $L\_8\; \backslash \; times\; L\_8\; \backslash ,$ and $L\_\; \{16\}\; \backslash ,$ are not similar. John Milnor observed that the Tore S of dimension 16 obtained by dividing $\backslash \; mathbb\; \{R\}\; ^\; \{16\}\; \backslash ,$ by these two networks are examples of compact varieties riemanniennes which are Isospectrale S but not Isométrique S.

The Fonction eta of Dedekind is defined by

$\backslash \; eta\; (Z)\; =\; q^\; \{1/24\}\; \backslash \; prod\_\; \{n=1\}\; ^\; \backslash \; infty\; (1-q^n),\; \backslash \; Q\; =\; e^\; \{2\; \backslash \; pi\; I\; Z\}$.

The modular Discriminant $\backslash \; Delta\; (Z)\; =\; \backslash \; eta\; (Z)\; ^\; \{24\}\; \backslash ,$ is then a modular form of weight 12. A famous conjecture of Ramanujan says that, for all Prime number p , the coefficient $\backslash \; tau\_p$ of $q^p\; \backslash ,$ in the development of Δ in powers of Q checks

$|\backslash \; tau\_p|\; \backslash \; Leq\; 2p^\; \{\backslash \; frac\; \{11\}\; \{2\}\}\; \backslash ,$.

This was shown by Pierre Deligne in his work on the Conjectures of Weil.

The second and third examples give certain indices on connection between the modular forms and the traditional questions of the theory of the numbers, such as the representation of the integers by the quadratic forms and the function divide. The crucial conceptual bond between the modular forms and the theory of the numbers is provided by the theory of the operators of Hecke, which gives also the bond between the theory of the modular forms and the Théorie of the representation.

Generalizations

There exist various concepts of modular forms more general than that developed above. The assumption of analycity can be removed; the forms of Maass are clean functions Laplacien but are not analytical. The groups which are not sub-groups of $SL\_2\; (\backslash \; mathbb\; \{Z\})\; \backslash ,$ can be examined. The modular forms of Hilbert are functions with variable N , each one being a complex number of the half-plane of Poincaré, satisfying a modular relation for the matrices 2 X 2 with coefficients in a completely real Corps of numbers. The modular forms of Siegel are associated with the symplectic larger groups in the same way as the forms that we exposed are associated with $SL\_2\; (\backslash \; mathbb\; \{R\})\; \backslash ,$; in other words, they are connected to the abelian varieties in the same direction as our forms (which are some times called elliptic modular forms to accentuate the point) are connected to the elliptic curves. The forms automorphes extend the notion of the modular forms to the groups of Dregs.

References

1. Books

   1. For an elementary introduction to the theory of the modular forms, to see chapter VII of
   2. For a more advanced treatment, to see Goro Shimura: Introduction to the arithmetic theory of the functions automorphes . Princeton University Near, Princeton, N.J., 1971 (in English).
   3. For an introduction to the modular forms starting from the point of view of the theory of the representation, one can consult the work of Stephen Gelbart: Automorphic forms one Adele groups . Annals off Mathematics Studies 83, Princeton University Near, Princeton, N.J., 1975.
2. Course in line
   1. Stein' S notes one Ribet' S race Modular Forms and Hecke Operators

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