Waldspurger formula explained

In representation theory of mathematics, the Waldspurger formula relates the special values of two L-functions of two related admissible irreducible representations. Let be the base field, be an automorphic form over, be the representation associated via the Jacquet–Langlands correspondence with . Goro Shimura (1976) proved this formula, when

k = Q and is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper. Marie-France Vignéras (1980) proved this formula, when k = Q and is a newform. Jean-Loup Waldspurger, for whom the formula is named, reproved and generalized the result of Vignéras in 1985 via a totally different method which was widely used thereafter by mathematicians to prove similar formulas.

Statement

Let

k be a number field, A be its adele ring, k ^{x} be the subgroup of invertible elements of k , A ^{x} be the subgroup of the invertible elements of A , \chi, \chi _{1,} \chi _{2} be three quadratic characters over A ^{x /k} ^{x} , G = SL _{2(k)} , l{A}(G) be the space of all cusp forms over G(k)\backslash G( A ) , l{H} be the Hecke algebra of G( A ) . Assume that, \pi is an admissible irreducible representation from G( A ) to l{A}(G) , the central character of π is trivial, \pi _{\nu} \sim \pi[h _{\nu]} when \nu is an archimedean place, {A} is a subspace of { l{A}(G)} such that \pi| _{l{H}:} l{H} \to A . We suppose further that, \varepsilon(\pi ⊗ \chi, 1 / 2 ) is the Langlands \varepsilon -constant [{{harv | Langlands | 1970 }}; {{harv | Deligne | 1972 }} ] associated to \pi and \chi at s = 1 / 2 . There is a {\gamma \in k ^{x }} such that k(\chi) = k( \sqrt{\gamma} ) .\left( (\chi)/(\pi) \right) = \varepsilon(\pi ⊗ \chi, 1 / 2 ) ⋅ \varepsilon(\pi, 1 / 2 ) ⋅ \chi( -1 ).

• Comment. Because all the terms in the right either have value +1, or have value -1, the term in the left can only take value in the set .

Definition 2. Let

{D _{\chi}} be the discriminant of \chi . $$p(\backslash chi)\; =\; D\_\backslash chi^\; \backslash sum\_\; \backslash left\backslash vert\; \backslash gamma\_\backslash nu\; \backslash right\backslash vert\_\backslash nu^.$$

Definition 3. Let

f _{0,} f _{1} \in A . b(f _{0,} f _{1)} = \int _{x\in k ^{x}} f _{0(x)} ⋅ \overline{f _{1(x)}} dx.

Definition 4. Let

{T} be a maximal torus of {G} , {Z} be the center of {G} , \varphi \in A . $$\backslash beta\; (\backslash varphi,\; T)\; =\; \backslash int\_\; b(\backslash pi\; (t)\backslash varphi,\; \backslash varphi)\; \backslash ,\; dt\; .$$

• Comment. It is not obvious though, that the function

\beta is a generalization of the Gauss sum.

Let

K be a field such that k(\pi)\subset K\subset C . One can choose a K-subspace{A ^{0}} of A such that (i) A = A ^{0} ⊗ _{KC} ; (ii) (A ^{0)} ^{\pi(G)} = A ^{0} . De facto, there is only one such A ^{0} modulo homothety. Let T _{1,} T _{2} be two maximal tori of G such that \chi _{T _{1}} = \chi _{1} and \chi _{T _{2}} = \chi _{2} . We can choose two elements \varphi _{1,} \varphi _{2} of A ^{0} such that \beta(\varphi _{1,} T _{1)} ≠ 0 and \beta(\varphi _{2,} T _{2)} ≠ 0 .

Definition 5. Let

D _{1,} D _{2} be the discriminants of \chi _{1,} \chi _{2} .p(\pi, \chi _{1,} \chi _{2)} = D ^{-1 / 2} _{1} D ^{1 / 2} _{2} L(\chi _{1,} 1 ) ^{-1} L(\chi _{2,} 1 ) L(\pi ⊗ \chi _{1,} 1 / 2 ) L(\pi ⊗ \chi _{2,} 1 / 2 ) ^{-1} \beta(\varphi _{1,} T ^{-1} _{1 )} \beta(\varphi _{2,} T _{2).}

• Comment. When the

\chi _{1} = \chi _{2} , the right hand side of Definition 5 becomes trivial.

We take

\Sigma _{f} to be the set, {\Sigma _{s}} to be the set of (all k -places \nu \mid \nu is real, or finite and special).

Comments:

The case when and is a metaplectic cusp form

Let p be prime number,

F _{p} be the field with p elements, R = F _{p[T],} k = F _{p(T),} k _{infty} = F ^{-1} _{p((T} )), o _{infty} be the integer ring of k _{infty,} l{H} = PGL _{2(k} _{infty)/PGL} _{2(o} _{infty),} \Gamma = PGL _{2(R)} . Assume that, N, D\in R , D is squarefree of even degree and coprime to N, the prime factorization of N is $\backslash prod\_\backslash ell\; \backslash ell^$. We take \Gamma _{0(N)} to the set $\backslash left\backslash ,$ S _{0(\Gamma} _{0(N))} to be the set of all cusp forms of level N and depth 0. Suppose that, \varphi, \varphi _{1,} \varphi _{2} \in S _{0(\Gamma} _{0(N))} .

Definition 1. Let

\left ( (c)/(d) \right ) be the Legendre symbol of c modulo d, \widetilde{SL} _{2(k} _{infty)} = Mp _{2(k} _{infty)} . Metaplectic morphism

Definition 2. Let

z = x + iy \in l{H}, d\mu = (dxdy)/(\left \vert y \right \vert ^{2}) . Petersson inner product $$\backslash langle\; \backslash varphi\_1,\; \backslash varphi\_2\backslash rangle\; =\; [\backslash Gamma\; :\; \backslash Gamma\_0(N)]^\; \backslash int\_\; \backslash varphi\_1(z)\; \backslash overline\; \backslash ,\; d\backslash mu.$$

Definition 3. Let

n, P \in R . Gauss sum $$G\_n(P)\; =\; \backslash sum\_\; \backslash left\; (\backslash frac\; \backslash right)\; e(rnT^2).$$

Let

λ _{infty,} be the Laplace eigenvalue of \varphi . There is a constant \theta \in R such that λ _{infty,} = (e ^{-i\theta} + e ^{i\theta})/(\sqrt{p) }.

Definition 4. Assume that

v _{infty(a/b)} = \deg(a) - \deg(b), \nu = v _{infty(y)} . Whittaker function

Definition 5. Fourier–Whittaker expansion $$\backslash varphi(z)\; =\; \backslash sum\_\; \backslash omega\_\backslash varphi(r)\; e(rxT^2)\; W\_(y).$$ One calls

\omega _{\varphi(r)} the Fourier–Whittaker coefficients of \varphi .

Definition 6. Atkin–Lehner operator with

\ell ^{2 \alpha _{\ell}} d - bN = \ell ^{\alpha _{\ell}} .

Definition 7. Assume that,

\varphi is a Hecke eigenform. Atkin–Lehner eigenvalue $$w\_\; =\; \backslash frac$$ with w _{\alpha _{\ell,} \varphi} = \pm 1 .

Definition 8. $$L(\backslash varphi,\; s)\; =\; \backslash sum\_\; \backslash frac\; .$$

Let

\widetilde{S} _{0(\widetilde{\Gamma}} _{0(N))} be the metaplectic version of S _{0(\Gamma} _{0(N))} , \{ E _{1,} \ldots, E _{d} \} be a nice Hecke eigenbasis for \widetilde{S} _{0(\widetilde{\Gamma}} _{0(N))} with respect to the Petersson inner product. We note the Shimura correspondence by \operatorname{Sh}.

Theorem [{{harv | Altug | Tsimerman | 2010 }}, Thm 5.1, p. 60 ]. Suppose that $K\_\backslash varphi\; =\; \backslash frac\; 1$,

\chi _{D} is a quadratic character with \Delta(\chi _{D)} = D . Then$$\backslash sum\_\; \backslash left\; \backslash vert\; \backslash omega\_(D)\; \backslash right\; \backslash vert\_p^2\; =\; \backslash frac\; L(\backslash varphi\; \backslash otimes\; \backslash chi\_D,\; 1/2)\; \backslash prod\_\backslash ell\; \backslash left(1\; +\; \backslash left\; (\backslash frac\; D\; \backslash right)\; w\_\; \backslash right).$$

References

• • • • Altug . Salim Ali . Tsimerman . Jacob . Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms . International Mathematics Research Notices . 1008.0430 . 2010 . 10.1093/imrn/rnt047 . 119121964 .
• Book: Langlands, Robert . On the Functional Equation of the Artin L-Functions . 1970. 1-287 .
• Deligne . Pierre . Les constantes des équations fonctionelle des fonctions L . Modular Functions of One Variable II . 501–597 . International Summer School on Modular functions . Antwerp . 1972 .