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A symplectic restriction problem

Valentin Blomer

Universitat Bonn

Automorphic Forms in Budapest

Valentin Blomer A symplectic restriction problem Automorphic Forms in Budapest 1 / 23

Periods ofautomorphic forms

��L -functions //

??

Arithmetic

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Examples

Example 1: (Riemann)

ζ(s) =∞∑

n=1

1ns =

(2π)s/2

2Γ(s/2)

∫ ∞

0

(θ(iy) − 1

)ys/2 dy

y

θ(z) =∑n∈Z

e(n2z)

analytic properties of ζ ! analytic properties of θ.

Example 2: (Hecke) Let f ∈ Sk be a Hecke eigenform. We have

L(f , s)Γ(s +

k − 12

)(2π)−s =

∫ ∞

0f(iy)ys dy

y


... continued

Example 3: Let X = Γ\G/K ⊇ X0. Let Φ, φ0 be constituents (“harmonics”) inspectral decomposition of L2(X) resp. L2(X0). Is there a connection∣∣∣∣ ∫

X0

Φ(y)φ0(y)dy∣∣∣∣2 ! L − value ?

−! Gross-Prasad conjecture

Application: Apply Parseval for the restriction norm:∫X0

|Φ(y)|2dy =

∫X0

∣∣∣∣ ∫X0

Φ(y)φ0(y)dy∣∣∣∣2dµ(φ0)

Examples 1 and 2: G = SL2(R), G0 ={(∗

∗

)⊆ SL2(R)

}� R∗


Siegel modular forms

Let

G = Sp4(R), G0 ={(∗

∗

)⊆ Sp4(R)

}� GL2(R).

The Siegel upper half space is

H(2) ={Z = X + iY ∈ Sym2(C) | Y > 0

}.

The group G acts on H(2) by Mobius transforms

MZ = (AZ + B)(CZ + D)−1, M =(A BC D

)∈ G.

We define

S(2)k = {Siegel modular forms F for Sp4(Z) of weight k }.

Fourier coefficients of F are indexed by integral positive binary quadratic forms Q .


Spectral decomposition

We write X = Sp4(Z)\H(2) with dim X = 6 and

X0 := SL2(Z)\Pos2(R) � SL2(Z)\H × R>0 =: X∗0 × R>0

Y =

√r

y

( 1 −x−x x2 + y2

) ! (x + iy, r)

with dim X0 = 3. Harmonics on X0 are Maaß forms and powers.

Is ∫X0

F(iY) u(z)rs dx dyy2

drr

an L -value?

Yes, it is a Koecher-Maaß series.


The Koecher-Maaß series

For a Siegel modular form F with Fourier coefficients a(Q), Q = (a, b , c) anintegral positive binary quadratic form, and a Maaß form u define

L(F × u, s) =∑

Q mod SL2(Z)

a(Q)u(HQ)

(det Q)s

where

HQ =−b + i

√|b2 − 4ac |

2a∈ SL2(Z)\H

is the Heegner point associated with Q .

This L -function has

a functional equation,

but no Euler product.


The restriction norm

Let F be a Siegel modular form of weight k . Then

N(F) :=vol(X)

vol(X∗0)

1‖F‖22

∫X0

|F(iY)|2(det Y)k dY(det Y)3/2

=

∫X∗0

∫R

|L(F × u, 1/2 + it)|2|G(F × u, 1/2 + it)|2dt du

where G(s,F × u) are suitable gamma factors.

For large k , the gamma factors decay at t , tu � k 1/2, so this is an average of sizek 3/2, and the L -function squared has conductor k 8.

Lindelof hypothesis: N(F) � k ε.

In absence of an Euler product it is not clear if the Lindelof hypothesis is true.

But we expect more...


The mass equidistribution conjecture

Let f ∈ Sk be a classical Hecke eigenform of weight k , h a test function.

Theorem (Holowinsky-Soundararajan 2009)∫SL2(Z)\H

|f(z)|2yk

‖f‖22h(z)

dx dyy2

k!∞−!

1vol(SL2(Z)\H)

∫SL2(Z)\H

h(z)dx dy

y2 .

Is this true on thin subsets?

Is this true in higher rank?


Thin symplectic mass equidistribution conjecture

N(F) = 4 log k + O(1) , k ! ∞

Why? Recall

X0 := SL2(Z)\Pos2(R) � SL2(Z)\H × R>0 =: X∗0 × R>0

Y =

√r

y

( 1 −x−x x2 + y2

) ! (x + iy, r)

The space X0 has infinite volume, but a Siegel cusp form of weight k decaysquickly as soon as λmin(Y) � 1/k , λmax(Y) � k . We obtain roughly an effectivevolume ∫ k 2

1/k 2

drr

= 4 log k .


Saito-Kurokawa lifts

M+k−1/2 � S2k−2

SK↪−! S(2)

k

dim � k dim � k 3

a(det Q) a(Q) depends only on det Q

Recall

L(F × u, s) =∑

Q mod SL2(Z)

a(Q)u(HQ)

(det Q)s =∑D<0

a(|D |)P(D, u)

|4D |s

where P(D, u) is the Heegner period

P(D, u) =∑z∈HD

u(z),

HD ={−b + i

√|D |

2a| ax2 + bxy + cy2 of disc D

}/SL2(Z)


The main result

Theorem 1: (B-Corbett 2019)Let W be a smooth weight function with support in [1, 2], ω =

∫W(x)x dx. Then

Nav(K) =1ω

12K2

∑k∈2N

W( kK

) ∑f∈B2k−2

N(SK(f)) = 4 log K + O(1).

This is an averaged version of

the thin symplectic mass equidistribution conjecture and

(a strong form of) the Lindelof hypothesis for L(F × u, s).


Steps of the proof...

Apply the Parseval period formula and an approximate functional equation:∑k

∑f∈B2k−2

∑j

∫R

|L(SK(f) × uj , 1/2 + it)|2|G(· · · )|2dt .

separate treatment of the constant function u0 where P(D, u0) = H(D) is theHurwitz class number


Interlude I: Voronoi formula for Hurwitz class numbers

Let c ∈ N, 4 | c, (a, c) = 1. Let φ be a smooth function with compact support in(0,∞). Then∑

D<0

H(D)

|D |1/4e(a |D |

c

)φ(|D |)

=

√2

c1/2

(−ca

)εae

(38

)∑D<0

H(D)

|D |1/2e(−

a |D |c

) ∫ ∞

0sin

(4π√|D |t

c

)φ(t)

dtt1/4

+12

∑n=�

e( an

c

) ∫ ∞

0exp

(−

4π√

ntc

)φ(t)

dtt1/4

+

∫ ∞

0φ(x)

( 14x1/4 −

π

3cx1/4

)dx

.


Steps of the proof... continued

Apply the Parseval period formula and an approximate functional equation.∑k

∑f∈B2k−2

∑j

∫R

|L(SK(f) × uj , 1/2 + it)|2|G(· · · )|2dt


sum over f ∈ B2k−2 by a half-integral weight Kohnen-Petersson formula.

the diagonal term: by Waldspurger/Zhang we have

|P(D, uj)|2 = |D |1/2

L(uj , 1/2)L(uj × χD , 1/2)

4L(sym2uj , 1)


Interlude II: a commutative diagram

P(D, u)Katok-Sarnak

Biro

Duke-Imamoglu-Toth//

Zhang

''

aSh−1(u)(|D |)

Baruch-Mao

��L(u × χD , 1/2)




∑f∈B2k−2

∑j

∫R

|L(SK(f) × uj , 1/2 + it)|2|G(· · · )|2dt




|P(D, uj)|2 = |D |1/2

L(uj , 1/2)L(uj × χD , 1/2)

4L(sym2uj , 1)

sum over uj with the Kuznetsov formula

the diagonal-diagonal term: explicit evaluation


Interlude III: an Euler product

The various integral and half-integral weight Hecke relations lead to the followingEuler product

∏p

(1 −

1p2 −

1p3 +

1p4

)×

∑(dδν,m)=1

∑(f ,δ)=1

∑d1 |fdd2 |fν

∑d3 |(

fdd1, fν

d2)

∑r1r2= f2dν

d1d2d23

µ(dδν)µ2(m)µ(d1)µ(d2)d3

(dν)4δ2f3m3rad(δd1d2r2)

×∏

p|δd1d2r2

(1 +

1p−

1p3

)−1

where rad(n) is the squarefree kernel. It equals

ζ(4)−1




∑f∈B2k−2

∑j

∫R

|L(SK(f) × uj , 1/2 + it)|2|G(· · · )|2dt




|P(D, uj)|2 = |D |1/2

L(uj , 1/2)L(uj × χD , 1/2)

4L(sym2uj , 1)

sum over uj with the Kuznetsov formula

the diagonal-diagonal term: explicit evaluation

the diagonal-offdiagonal term: multiple Poisson summation andHeath-Brown’s large sieve for quadratic characters.


The off-diagonal term

sum over k � K

interpret P(D, uj) as metaplectic Fourier coefficients (by Katok-Sarnak) andapply half-integral Voronoi summation

heart of the proof: need cancellation in∑u

P(D1, u)P(D2, u)︸︷︷︸product of 4 half-integral weight Fourier coefficients

h(tu)

“Kuznetsov formula for toric Fourier coefficients”


A trace formula for pairs of Heegner periodsTheorem 2: Let D1,D2 be two negative fundamental discriminants. Let

F(x, t) = Jit (x) cos(π/4 − πit/2) − J−it (x) cos(π/4 + πit/2),

Wt (n) =1

2πi

∫(2)

Γ( 12 ( 1

2 + s + 2it))Γ( 12 ( 1

2 + s − 2it))

Γ( 14 + it)Γ( 1

4 − it)πses2

n−s dss.

Then

1|D1D2 |1/4

∫X∗0

P(D1; u)P(D2; u)h(tu)du

=3π

H(D1)H(D2)

|D1D2 |1/4h(i/2) − constant function

+

∫ ∞

−∞

∣∣∣∣D1D2

4

∣∣∣∣it/2 Γ(− 14 + it

2 )e(1/2−it)2

√8πΓ( 1

4 + it2 )

L(χD1 , 1/2 + it)L(χD2 , 1/2 + it)ζ(1 + 2it)

h(t)dt4π

polar term

+ δD1=D2

∑m

χD1 (m)

m

∫ ∞

−∞

Wt (m)h(t)t tanh(πt)dt

4π2 − diagonal term

+ e(3/8)∑

n,c,m

K+3/2(|D1 |n2, |D2 |, c)χD1 (m)

n1/2cm

∫ ∞

−∞

F(4πn√|D1D2 |/c, t)

cosh(πt)h(t)Wt (nm)t

dtπ.


Idea of proof

We transform P(D1, u)P(D2, u) as follows:

Katok-Sarnak: av(|D1|)av(1)av(|D2|)av(1), v = Sh−1(u)

Waldspurger: av(|D1|)av(|D2|)L(u, 1/2)

approximate functional equation av(|D1|)av(|D2|)∑

n λ(n)n−1/2

Shimura:∑

n n−1/2av(|D1|n2)av(|D2|)

metaplectic Kuznetsov formula:∑

n,c K(|D1|n2, |D2|, c)


Periods ofautomorphic forms

��L -functions //

??

Arithmetic

__

oo


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