Top shape

Edge points at infinity are the tips of SL (2, Z).

In number theory is a holomorphic modular form to module group (sometimes also defined as a module group) as a peak form (engl .: cusp form ) referred to when the tip (cusp), which means for disappears. ${\displaystyle f}$ ${\displaystyle \Gamma =SL(2,\mathbb {Z} )}$ ${\displaystyle PSL(2,\mathbb {Z} )}$ ${\displaystyle Im\,z\to \infty }$

An equivalent condition is that the constant term in the Fourier expansion ${\displaystyle a_{0}=f(0)}$

${\displaystyle f(z)=\sum a_{n}e^{2\pi inz}=\sum a_{n}q^{n}}$

with , disappears: ${\displaystyle q=e^{2\pi iz}}$

${\displaystyle a_{0}=0}$.

and there are no negative n in the expansion (the modular form is holomorphic). Then disappears in the top . ${\displaystyle f}$ ${\displaystyle q=0}$

One can also consider peak shapes for congruence subgroups of the module group, then there are generally several peaks, parameterized by rational numbers in infinity. This corresponds to the limit value for in the transformation law of the module form, whereby only a finite number of peaks at infinity result as each representative of an orbit. If the quotient space of the upper half-plane is compacted by adding the peaks, the Riemann surfaces of the associated module curves are obtained. ${\displaystyle \Gamma _{*}}$ ${\displaystyle {\frac {a}{c}}}$ ${\displaystyle z\to \infty }$ ${\displaystyle z\to {\frac {az+b}{cz+d}}}$ ${\displaystyle \mathbb {H} \backslash \Gamma _{*}}$

Tip shapes with a given weight

In the following, the tip shapes for the full module group are considered. It follows from the definition that there are no non-zero point shapes for odd weights . The dimension of the space of the tip shapes with a given weight can be calculated with the Riemann-Roch theorem . The smallest weights for which nontrivial tip shapes exist are ${\displaystyle k\in \mathbb {Z} }$

${\displaystyle k=12,16,18,20,22,26}$,

In all of these cases the space of the tip shapes is 1-dimensional, so there is a tip shape that is unique apart from multiplication with complex numbers for these weights. In general, the dimension of the vector space of tip shapes is to weight the same case and the same otherwise. ${\displaystyle k=2m}$ ${\displaystyle \left[{\frac {m}{6}}\right]-1}$ ${\displaystyle m\equiv 1\ mod\ 6}$ ${\displaystyle \left[{\frac {m}{6}}\right]}$

For example, except for multiplication with complex numbers, the peak shape for weight 12 is the discriminant

${\displaystyle \Delta (z)=(2\pi )^{12}\sum _{n=1}^{\infty }\tau (n)\,{\mathrm {e} }^{2\pi inz}}$,

whose Fourier coefficients define the Ramanujan tau function . ${\displaystyle \tau (n)}$

The Fourier coefficients of a tip shape to weight vanish in order ${\displaystyle k=2m}$ ${\displaystyle 0}$ ${\displaystyle m}$

${\displaystyle a_{n}=O(n^{k})}$.

The Petersson dot product on the space of the tip shapes is defined by

${\displaystyle \langle f,g\rangle :=\int _{\mathrm {F} }f(\tau ){\overline {g(z)}}(\operatorname {Im} z)^{k}{\rm {d}}\nu (z)}$,

where is the fundamental domain of the module group and with is the hyperbolic volume element. ${\displaystyle \mathrm {F} =\{z\in \mathrm {H} |\left|\operatorname {Re} z\right|\leq {\frac {1}{2}},\left|z\right|\geq 1\}}$ ${\displaystyle \Gamma }$ ${\displaystyle {\rm {d}}\nu (z)=y^{-2}{\rm {d}}x{\rm {d}}y}$ ${\displaystyle z=x+iy}$

literature

• Tom Apostol : Modular functions and Dirichlet series in number theory. Second edition. Graduate Texts in Mathematics, 41st Springer-Verlag, New York, 1990. ISBN 0-387-97127-0

Web links

• Cusp Shape (MathWorld)
• The L-functions and modular forms data base