Prime Solutions to Quadratic Equations
                                                        KAIST, Jan.  23, 2008
              Jianya Liu
         Shandong University
Main part:  joint work with Peter Sarnak

      1. Classical theorems and conjectures
□ Hua (1938):

\begin{matrix}n ≡ 5(mod24) \\ n ≥ n_{0} \end{matrix} \} ⇒ n = p_{1}^{2} + … + p_{5}^{2}.

□ Here n ≡ 5(mod24) is necessary, since

p ≥ 5 ⇒ p^{2} ≡ 1(mod24).

□ Proof by the circle method.  Need
 • the distribution of primes in arithmetic progressions,
 • estimates for exponential sums over primes.

■ Reformulation:
Let k ≥ 5, and

q(x) = x_{1}^{2} ± x_{2}^{2} ± … ± x_{k}^{2}.

Put

V = {x : q(x) = t}.

Then

Zcl(V ∩ 𝔓^{k}) = Zcl(V),

iff there is no local obstruction, i.e

t ≡ 1 ± 1 ± … ± 1(mod24). 3

□ Hua (1938):  Conj H

\begin{matrix}n ≡ 3(mod24) \\ 5 ∤ n \\ n ≥ n_{0} \end{matrix} \} ⇒ n = p_{1}^{2} + p_{2}^{2} + p_{3}^{2}.

He proved that H is true for almost all integers satisfying
necessary conditions, i.e.

E(x) ≪ \frac{x}{log^{A} x}.

□ Conj H seems more difficult then binary Goldbach/twin
prime conj, since by Rieger,

∑_{{m≤x \atop m=p21+p22}} 1 ∼ \frac{π}{2} \frac{x}{log^{2} x}.

Compare with PNT

∑_{p≤x} 1 ∼ \frac{x}{log x}.

□ Zhan + L (2001):

E(x) ≪ x^{11/12+ε}.

□ Blomer + Brüdern (2005):

n = m_{1}^{2} + m_{2}^{2} + m_{3}^{2} with m_{j} ∈ 𝔓_{521}.

       2. Schinzel and Nonabelian Schinzel
Now we are concerned with prime values of polynomials.
■ Schinzel's Hypothesis:
Let f_{1}, ..., f_{t} ∈ ℤ[x] be distinct irr polynomials.  There
are infinitely many x ∈ ℤ such that f_{j}(x) is prime
for each j iff there is no local congruence obstruction
[i.e.

f_{1}(x)f_{2}(x) ⋯ f_{t}(x)

is not identically congruent to 0 mod any prime p].
□ E.g.  OK with

f_{1}(x) = x, f_{2}(x) = x + 2,

Shinzel ⇒ Twin Prime Conjecture.

□ E.g.  But for

f_{1}(x) = x, f_{2}(x) = x + 2, f_{3}(x) = x + 4,

one has

3|f_{1}f_{2}f_{3},

and therefore f_{1}, f_{2}, f_{3} cannot take prime value infinitely
times.
□ E.g.  OK with

f_{1}(x) = x, f_{2}(x) = x + 2, f_{3}(x) = x + 6.

Therefore, conjecturally, f_{1}, f_{2}, f_{3} should take prime
value infinitely times.

■ Nonabelian Schinzel (Sarnak's conj.):
Let L be a subgroup of GL_{n}(ℤ) with Zcl(L) semisimple,
connected and simply connected.  Let V = Lv be an
orbit of L, with v ∈ ℤ^{n}. Fix

f_{1}, f_{2}, ..., f_{t} ∈ ℤ[x_{1}, ..., x_{n}]

[and other conditions!].  Let

P = {x ∈ V : f_{j}(x) ∈ 𝔓}.

Then

Zcl(P) = Zcl(V)

iff there are no local obstructions.

             3. Primes on k-quadrics
■ Connection between non-ablelian Schinzel and primes
on k-quadrics.  Let k ≥ 3, and

q(x) = q(x_{1}, …, x_{k})

be an integral quadratic form.  For t ∈ ℤ, let V
denote the quadrics

V = {x : q(x) = t}.

Let

V (ℤ) := {x ∈ ℤ : q(x) = t}.

Throughout we assume that

V (ℤ) ≠ ∅. 9

□ Let

L = SO(q, ℝ)

be the real k×k orthogonal group of motions preserving
q, and choose a v ∈ V (ℤ).  Then

V = Lv.

Let

\{ \begin{matrix}f_{1} & = & x_{1}, \\ f_{2} & = & x_{2}, \\ & & …, \\ f_{k} & = & x_{k}. \end{matrix}

Then non-ablelian Schinzel for these f_{1}, . . . , f_{k} predicts
that

Zcl(V ∩ 𝔓^{k}) = Zcl(V),

iff there is no local obstruction.

□ Non-ablelian Schinzel ⇒ When k ≥ 3 and q is indefinite,
the k-quadrics

q(x) = t

has infinitely many solutions with all the co-ordinates
prime, iff there is no local obstruction.

         4. Almost primes on 3-quadrics
■ Distinguish definite and indefinite.
 • Definite case:  circle method, sieves; Introduction.
 • Indefinite case:  this section.
□ Let

q(x) = q(x_{1}, x_{2}, x_{3})

be an integral \underline{indefinite} quadratic form with square-free
det.  For t ∈ ℤ with t ≠ 0, recall

V = {x : q(x) = t}.

and

V (ℤ) := {x ∈ ℤ : q(x) = t}. 12

Throughout we assume that V (ℤ) ≠ ∅.
■ Easier case:
q \underline{isotropic} over ℚ, i.e.  there are non-constant polynomial
parameterizations of points in V (ℤ).  That is, there
are

T = (T_{1}, T_{2}, T_{3})

with T_{j} an integral polynomial in one variable and
for which T (y) ∈ V (ℤ) if y ∈ ℤ.  With this one can
produce almost primes on such curves in V using sieve
methods.
□ Diamond + Halberstam:  for the conic

q(x_{1}, x_{2}, x_{3}) = x_{1}^{2} + x_{2}^{2} − x_{3}^{2} = 0, 13

there are infinitely many such Pythagorean triples
for which the product

x_{1}x_{2}x_{3} ∈ 𝔓_{17}.

But such triples are \underline{not} Zariski dense in V.
□ Idea:  Parameterized solutions

x_{1} = 2mn, x_{2} = m^{2} − n^{2}, x_{3} = m^{2} + n^{2}.

Apply sieve methods to the sequence

m(m + 1)(m − 1)(m^{2} + 1).

□ Using similar arguments, one can do the same for
V when q is isotropic.

■ Harder case:
q \underline{anisotropic} over ℚ.  If q is anisotropic, there
are no non-constant integral polynomial maps of the
line into V.  This follows by considering the highest
degree coeff of a potential such map.
□ \underline{Theorem 1} (Sarnak + L). Let q(x_{1}, x_{2}, x_{3}) be an integral
indefinite anisotropic quadratic form with square-free
determinant.  Let t ∈ ℤ and t ≠ 0.  Then the set
of solutions to the eqn

q(x_{1}, x_{2}, x_{3}) = t

with

x_{1}x_{2}x_{3} ∈ 𝔓_{26} 15

is Zariski dense in V.  Under Selberg's Eigenvalue
Conjecture,

26 → 22.

■ \underline{Theorem 2}.
For T ≥ 2, construct a weight function F_{T} : [0, ∞)^{3} →
ℝ.  For 1 ≤ n ≪ T^{3}, set

a_{n}(T) = ∑_{{x∈V (ℤ) \atop x1x2x3=±n}} F_{T} (x).

and

X = ∑_{n≥1} a_{n}(T).

Then, for 1 ≤ d ≤ T,

∑_{n≡0( mod d)} a_{n}(T) = \frac{|V^{0}(ℤ/dℤ)|}{|V (ℤ/dℤ)|}X + O_{ε}(d^{1+ε}T^{1/2+θ}).

□ \underline{Remark 1}:  This is \underline{not} a large sieve inequality.
□ \underline{Remark 2}:  Here

V (ℤ/dℤ) = {x ∈ (ℤ/dℤ)^{3} : q(x) ≡ t(modd)},

and

V^{0}(ℤ/dℤ) = {x ∈ V (ℤ/dℤ) : x_{1}x_{2}x_{3} ≡ 0(modd)}.

To see the level of equi-distribution,

\frac{|V^{0}(ℤ/dℤ)|}{|V (ℤ/dℤ)|} ≍ \frac{1}{d},

X ≍ T.

Therefore,

\frac{T}{d} ≫ d^{1+ε}T^{1/2+θ} ⇐⇒ d ≪ T^{1/4−θ/2−ε},

i.e.

      level of equi-distribution = \frac{1}{4} − \frac{θ}{2} − ε.

□ \underline{Remark 3}:  Here θ is as in the bound towards the
Ramanujan conjecture:

     |α_{π}^{(j)}(p)| ≤ p^{θ} if π is unramified at p,
     |ℜμ_{π}^{(j)}(∞)| ≤ θ if π is unramified at ∞,

where π is an automorphic cuspidal rep'n of GL_{2}(𝔸_{ℚ})
with unitary central character and local Hecke eigenvalues

α_{π}^{(j)}(p) for p < ∞ and μ_{π}^{(j)}(∞) for p = ∞, j = 1, 2.

The best known result with

θ = \frac{7}{64}

is due to Kim + Sarnak.
□ \underline{Remark 4}:  In general, there may be a finite number
of "bad primes" p for which

V^{0}(ℤ/pℤ) = V (ℤ/pℤ).

\underline{Lemma} (general).  Let q(x_{1}, x_{2}, x_{3}) be an integral quadratic
form with square-free determinant Δ, and t an integer.
If p is a bad prime for the equation

q(x_{1}, x_{2}, x_{3}) = t 19

then p = 2, 3, 5 or p|(Δ, t). 20

                5. Proof of Thm 2
Consider the level distribution sum

S = ∑_{n≡0( mod d)} a_{n}(T) = ∑_{{x∈V (ℤ) \atop x1x2x3≡0( mod d)}} F_{T} (x)

■ Step 1:  application of theory of quadratic forms
Let G/ℚ be the spin double cover of the orthogonal
group SO(q)/ℚ.  The form q being anisotropic implies
that G consists of the elements of norm 1 in a quaternion
division algebra D_{q} defined over ℚ.  Let Λ denote
the unit group of integral quaternions of norm 1.  The
image of Λ under the projection to SO(q) acts on V

preserving V (ℤ).  It acts with finitely many orbits:

V (ℤ) = ⨆_{j=1}^{h} y^{(j)}Λ.

Also, apply the strong approximation theorem for such
spin groups...→ ...

S = ∑_{{ζ∈V (ℤ/dℤ) \atop ζ1ζ2ζ3≡0( mod d)}} ∑_{j=1}^{h} ∑_{β∈δ_{j,ζ}^{−1}Λ^{(j)}δ_{j,ζ}\Λ_{ζ}} F_{T} (y^{(j)}δ_{j,ζ}β).

where Λ^{(j)} is the stabilizer of y^{(j)} in Λ, δ_{j,ζ} ∈ Λ
satisfies

y^{(j)}δ_{j,ζ} ≡ ζ(modd). 22

Note that δ_{j,ζ}^{−1}Λ^{(j)}δ_{j,ζ} is a subgroup of

Λ_{ζ} = {γ ∈ Λ : ζγ ≡ ζ(modd)}, ζ ∈ V (ℤ/dℤ).

Also, Λ_{ζ} is a congruence subgroup of Λ.
■ Step 2:  Application of Jacquet + Langlands
According to the Jacquet + Langlands correspondence
from our division algebra D/ℚ to GL_{2}/ℚ, we conclude
that

λ_{1}(Λ_{ζ}\ℍ) ≥ \frac{1}{4} − θ^{2}.

■ Step 3:  Application of Selberg

Let Γ be a congruence subgroup of finite index of
a fixed co-compact lattice Λ.  Let

0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ …

be the spectrum and

ϕ_{0}, ϕ_{1}, ϕ_{2}, …

a corresponding orthonormal basis of eigenfunctions
of the Laplacian Δ on

X_{Γ} = Γ\ℍ.

Write

λ_{j} = 1/4 + t_{j}^{2} with t_{j} ≥ 0 or t_{j} ∈ iℝ^{+}. 24

□ Selberg (1956):  For k(z, w) a point pair on ℍ×
ℍ which is compact support in d(z, w), we have

∑_{γ∈Γ} k(γz, w) = ∑_{j} h_{R}(t_{j})ϕ_{j}(z)\overline{ϕ_{j}(w)},

at least in the L^{2}-sense.  Here h_{R}(t) is the spherical
transform of k;

h_{R}(t) := ∫_{ℍ} k(z, i)y^{1/2+it}\frac{dxdy}{y^{2}}.

Selberg's \underline{footnote}:  "This formula can be used for
estimation of the number of points γz in large regions
with rotational symmetry about the point w."

In particular,

∑_{γ∈Γ} I_{R}(γz, w) = ∑_{j} h_{R}(t_{j})ϕ_{j}(z)\overline{ϕ_{j}(w)}

with

I_{R}(z, w) = \{ \begin{matrix}1 & if d(z, w) ≤ R, \\ 0 & otherwise. \end{matrix}

and

h_{R}(t) = ∫_{d(z,i)≤R} y^{1/2+it}\frac{dxdy}{y^{2}}.

□ Define the "smooth-out" of I_{R}(z, w) by

k_{R,ϕ}(z) = ∫_{ℍ} I_{R}(z, w)ϕ(w)dA(w).

Integrate Selberg against ϕ(w),

∑_{γ∈Γ} k_{R,ϕ}(γz) = ∑_{j} h_{R}(t_{j})ϕ_{j}(z)⟨ϕ, ϕ_{j}⟩.

The right-hand side now converges rapidly; also

ϕ_{0}(z) = \frac{1}{\sqrt{Area(Γ\ℍ)}},

h_{R} (\frac{i}{2}) = Area(B_{R}),

and

⟨ϕ, ϕ_{0}⟩ = \frac{1}{\sqrt{Area(Γ\ℍ)}}.

Thus the contribution from j = 0 yields the main term.
By the theory of spherical functions,

|h_{R}(t_{j})| ≤ Area(B_{R})^{1/2+θ}

since

t_{j} ∈ ℝ

or

|Im(t_{j})| ≤ θ.

Hence,

| ∑_{j≠0} h_{R}(t_{j})ϕ(z)⟨ϕ, ϕ_{j}⟩| ≤ Area(B_{R})^{1/2+θ} ∑_{j≠0} |ϕ_{j}(z)||⟨ϕ, ϕ_{j}⟩|.

□ Hence the j ≠ 0 sum is at most

O{Area(B_{R})^{1/2+θ}}, 28

which establishes the following
\underline{Lemma}.  For any z ∈ ℍ and R ≥ 1,

∑_{γ∈Γ} k_{R,ϕ}(γz) = \frac{Area(B_{R})}{Area(Γ\ℍ)} + O_{ϕ}\{Area(B_{R})^{1/2+θ}\}.

Note that this is independent of Γ.
■ Inserting back, we get Thm 2.

■ Thm 1 follows from Thm 2 via 3-dim sieve.
□ Recall our philosophy:
prime ← sieves (combinatorial) ← equi-distribution
← harmonic analysis ← representation theory for GL(2)
                   ← algebra.