View source for Shimura correspondence

By a [[Modular form|modular form]] of weight $k$ one understands a function $f$ on the upper half-plane satisfying $f(\gamma z)=\chi(\gamma)(cz+d)^kf(z)$ for some suitable function $\chi:\Gamma\to\textbf{C}^{\times}$ when

\begin{equation}\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\end{equation}

is an element of some congruence subgroup of $\text{SL}(2,\textbf{Z})$ (cf. also [[Modular function|Modular function]]).

If $k$ is an integer, E. Hecke defined operators $T_n$ for every integer $n$, and showed they could be simultaneously diagonalizable (cf. also [[Hecke operator|Hecke operator]]). The $L$-series of a simultaneous eigenfunction (cf. also [[Dirichlet L-function|Dirichlet $L$-function]]) is then an [[Euler product|Euler product]].

Modular forms of half-integral weight arise naturally, for example as [[Theta-series|theta-series]]. A theta-series in $r$ variables is a modular form of weight $r/2$.

If $k$ is a half-integer, $T_n$ can only be defined if $n$ is a square on forms of weight $k$, and there is not enough information in the Hecke eigenvalues to determine the [[Fourier coefficients|Fourier coefficients]]. The coefficients are not multiplicative, so the $L$-series is not an Euler product.

Using the Rankin–Selberg method and a converse theorem, G. Shimura [[#References|[a1]]] showed that if $\widetilde{f}$ is a modular form of weight $k+1/2$, then there is a corresponding modular form of weight $2k$ such that the $T_{n^2}$ Hecke eigenvalue on $\widetilde{f}$ agrees with the $T_n$ Hecke eigenvalue of $f$.

This result was complemented by the important theorem of J.-L. Waldspurger [[#References|[a2]]], showing that the $D$th Fourier coefficient of $\widetilde{f}$ agrees with $L(k/2,f,\chi D)$. Waldspurger also gave interpretations of these special values as periods of $f$ (integrals over over geodesics). W. Kohnen and D. Zagier [[#References|[a3]]] gave a particularly useful treatment of a special case. Also useful is [[#References|[a4]]]. P. Sarnak and S. Katok [[#References|[a5]]] found similar results for Maass forms.

Given Waldspurger's theorem, the case where $k=1$ becomes particularly interesting, since if $f$ is the modular form of weight two associated with an [[Elliptic curve|elliptic curve]], $L(1,f,\chi D)$ has an interpretation in terms of the Birch–Swinnerton-Dyer conjecture. The period interpretation of the special values is then connected with the work of B.H. Gross, Kohnen and Zagier [[#References|[a6]]] on heights of Heegner points. A beautiful application of this connection with the Birch–Swinnerton-Dyer conjecture to the classical problem of computing the set of areas of rational right triangles was given in [[#References|[a7]]].

An interesting approach to the Shimura correspondence and Waldspurger's theorem is offered by the theory of Jacobi forms, in which both $\widetilde{f}$ and its correspondent $f$ may be related to automorphic forms on the Jacobi group. See [[#References|[a8]]] and [[#References|[a9]]]; cf. also [[Automorphic form|Automorphic form]].

A. Weil realized that (Siegel) modular forms, particularly theta-series, should be interpreted as automorphic forms not on $\text{Sp}(2n)$, but on a certain double cover $\widetilde{\text{Sp}}(2n)$, the so-called metaplectic group. If $n=1$, then $\text{Sp}(2n)=\text{SL}(2)$, and this is the proper framework for understanding the classical Shimura correspondence, which can be regarded as a lifting from either $\widetilde{\text{SL}}(2)$ to $\text{PGL}(2)=O(3)$, or from $\widetilde{\text{GL}}(2)$ to $\text{GL}(2)$.

T. Kubota and K. Matsumoto constructed metaplectic covers of more general groups, provided the ground field contains sufficiently many roots of unity. The Shimura correspondence in this context is a lifting from automorphic forms on the covering group to automorphic forms on $G$ or (sometimes) its dual, obtained by reversing the long and short roots and interchanging the fundamental group with the dual of the centre. See [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]] for the Shimura correspondence on higher covers of higher rank groups. Finding analogues of Waldspurger's theorem in this context is an important open problem (as of 2000).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Shimura,   "On modular forms of half integral weight"  ''Ann. of Math.'' , '''97'''  (1973)  pp. 440–481</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.-L. Waldspurger,   "Sur les coefficients de Fourier des formes modulaires de poids demi-entier"  ''J. Math. Pures Appl.'' , '''60'''  (1981)  pp. 375–484</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Kohnen,   D. Zagier,   "Values of $L$-series of modular forms at the center of the critical strip"  ''Invent. Math.'' , '''64'''  (1981)  pp. 175–198</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I. Piatetski–Shapiro,   "Work of Waldspurger" , ''Lie Group Representations II'' , ''Lecture Notes in Mathematics'' , '''1041''' , Springer  (1984)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Sarnak,   S. Katok,   "Heegner points, cycles and Maass forms"  ''Israel J. Math.'' , '''84'''  (1993)  pp. 193–227</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B.H. Gross,   W. Kohnen,   D. Zagier,   "Heegner points and derivatives of $L$-series II"  ''Math. Ann.'' , '''278'''  (1987)  pp. 497–562</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.B. Tunnell,   "A classical Diophantine problem and modular forms of weight $3/2$"  ''Invent. Math.'' , '''72'''  (1983)  pp. 323–334</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Eichler,   D. Zagier,   "Jacobi forms" , Birkhäuser  (1985)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D. Ginzburg,   S. Rallis,   D. Soudry,   "A new construction of the inverse Shimura correspondence"  ''Internat. Math. Res. Notices'' , '''7'''  (1997)  pp. 349–357</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  D. Kazhdan,   S.J. Patterson,   "Towards a generalized Shimura correspondence"  ''Adv. Math.'' , '''60'''  (1986)  pp. 161–234</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  Y.Z. Flicker,   "Automorphic forms on covering groups of $\text{GL}(2)$"  ''Invent. Math.'' , '''57''' :  2  (1980)  pp. 119–182</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  Y.Z. Flicker,   D. Kazhdan,   "Metaplectic correspondence"  ''Publ. Math. IHES'' , '''64'''  (1986)  pp. 53–110</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  D. Bump,   J. Hoffstein,   "On Shimura's correspondence"  ''Duke Math. J.'' , '''55'''  (1987)  pp. 661–691</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  D. Savin,   "Local Shimura correspondence"  ''Math. Ann.'' , '''280'''  (1988)  pp. 185–190</TD></TR></table>