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By a [[Modular form|modular form]] of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s1302801.png" /> one understands a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s1302802.png" /> on the upper half-plane satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s1302803.png" /> for some suitable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s1302804.png" /> when
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s1302805.png" /></td> </tr></table>
+
\begin{equation}\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\end{equation}
  
is an element of some congruence subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s1302806.png" /> (cf. also [[Modular function|Modular function]]).
+
is an element of some congruence subgroup of $\text{SL}(2,\textbf{Z})$ (cf. also [[Modular function|Modular function]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s1302807.png" /> is an integer, E. Hecke defined operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s1302808.png" /> for every integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s1302809.png" />, and showed they could be simultaneously diagonalizable (cf. also [[Hecke operator|Hecke operator]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028010.png" />-series of a simultaneous eigenfunction (cf. also [[Dirichlet-L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028011.png" />-function]]) is then an [[Euler product|Euler product]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028012.png" /> variables is a modular form of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028013.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028014.png" /> is a half-integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028015.png" /> can only be defined if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028016.png" /> is a square on forms of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028017.png" />, and there is not enough information in the Hecke eigenvalues to determine the [[Fourier coefficients|Fourier coefficients]]. The coefficients are not multiplicative, so the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028018.png" />-series is not an Euler product.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028019.png" /> is a modular form of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028020.png" />, then there is a corresponding modular form of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028021.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028022.png" /> Hecke eigenvalue on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028023.png" /> agrees with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028024.png" /> Hecke eigenvalue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028025.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028026.png" />th Fourier coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028027.png" /> agrees with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028028.png" />. Waldspurger also gave interpretations of these special values as periods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028029.png" /> (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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028030.png" /> becomes particularly interesting, since if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028031.png" /> is the modular form of weight two associated with an [[Elliptic curve|elliptic curve]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028032.png" /> 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]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028033.png" /> and its correspondent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028034.png" /> may be related to automorphic forms on the Jacobi group. See [[#References|[a8]]] and [[#References|[a9]]]; cf. also [[Automorphic form|Automorphic form]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028035.png" />, but on a certain double cover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028036.png" />, the so-called metaplectic group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028038.png" />, and this is the proper framework for understanding the classical Shimura correspondence, which can be regarded as a lifting from either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028039.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028040.png" />, or from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028041.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028042.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028043.png" /> 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).
+
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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028044.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028045.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028046.png" />"  ''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130280/s13028047.png" />"  ''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>
+
<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>

Latest revision as of 13:41, 18 January 2021

By a 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).

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). The $L$-series of a simultaneous eigenfunction (cf. also Dirichlet $L$-function) is then an Euler product.

Modular forms of half-integral weight arise naturally, for example as 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. 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 [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 [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 [a3] gave a particularly useful treatment of a special case. Also useful is [a4]. P. Sarnak and S. Katok [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, $L(1,f,\chi D)$ 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 [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 [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 [a8] and [a9]; cf. also 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 [a10], [a11], [a12], [a13], [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

[a1] G. Shimura, "On modular forms of half integral weight" Ann. of Math. , 97 (1973) pp. 440–481
[a2] J.-L. Waldspurger, "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" J. Math. Pures Appl. , 60 (1981) pp. 375–484
[a3] 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
[a4] I. Piatetski–Shapiro, "Work of Waldspurger" , Lie Group Representations II , Lecture Notes in Mathematics , 1041 , Springer (1984)
[a5] P. Sarnak, S. Katok, "Heegner points, cycles and Maass forms" Israel J. Math. , 84 (1993) pp. 193–227
[a6] B.H. Gross, W. Kohnen, D. Zagier, "Heegner points and derivatives of $L$-series II" Math. Ann. , 278 (1987) pp. 497–562
[a7] J.B. Tunnell, "A classical Diophantine problem and modular forms of weight $3/2$" Invent. Math. , 72 (1983) pp. 323–334
[a8] M. Eichler, D. Zagier, "Jacobi forms" , Birkhäuser (1985)
[a9] D. Ginzburg, S. Rallis, D. Soudry, "A new construction of the inverse Shimura correspondence" Internat. Math. Res. Notices , 7 (1997) pp. 349–357
[a10] D. Kazhdan, S.J. Patterson, "Towards a generalized Shimura correspondence" Adv. Math. , 60 (1986) pp. 161–234
[a11] Y.Z. Flicker, "Automorphic forms on covering groups of $\text{GL}(2)$" Invent. Math. , 57 : 2 (1980) pp. 119–182
[a12] Y.Z. Flicker, D. Kazhdan, "Metaplectic correspondence" Publ. Math. IHES , 64 (1986) pp. 53–110
[a13] D. Bump, J. Hoffstein, "On Shimura's correspondence" Duke Math. J. , 55 (1987) pp. 661–691
[a14] D. Savin, "Local Shimura correspondence" Math. Ann. , 280 (1988) pp. 185–190
How to Cite This Entry:
Shimura correspondence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shimura_correspondence&oldid=15178
This article was adapted from an original article by D. Bump (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article