Eisenstein series uniform convergence (#10377)

We show that the sum defining an Eisenstein Series converges locally uniformly. This is the basis for later proving that they are holomorphic (see #11013 ) and bounded at infinity (see #12456), which combine to show they are modular forms (see #12604).




Co-authored-by: Chris Birkbeck <[email protected]>
Co-authored-by: David Loeffler <[email protected]>

Mathlib.lean

@@ -3075,6 +3075,7 @@ import Mathlib.NumberTheory.Modular
 import Mathlib.NumberTheory.ModularForms.Basic
 import Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
 import Mathlib.NumberTheory.ModularForms.EisensteinSeries.Basic
+import Mathlib.NumberTheory.ModularForms.EisensteinSeries.UniformConvergence
 import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
 import Mathlib.NumberTheory.ModularForms.JacobiTheta.Manifold
 import Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable

Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean

@@ -0,0 +1,197 @@
+/-
+Copyright (c) 2024 Chris Birkbeck. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Chris Birkbeck, David Loeffler
+-/
+
+import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
+import Mathlib.Analysis.NormedSpace.FunctionSeries
+import Mathlib.Analysis.PSeries
+import Mathlib.Order.Interval.Finset.Box
+import Mathlib.NumberTheory.ModularForms.EisensteinSeries.Basic
+
+/-!
+# Uniform convergence of Eisenstein series
+
+We show that the sum of `eisSummand` converges locally uniformly on `ℍ` to the Eisenstein series
+of weight `k` and level `Γ(N)` with congruence condition `a : Fin 2 → ZMod N`.
+
+## Outline of argument
+
+The key lemma `r_mul_max_le` shows that, for `z ∈ ℍ` and `c, d ∈ ℤ` (not both zero),
+`|c z + d|` is bounded below by `r z * max (|c|, |d|)`, where `r z` is an explicit function of `z`
+(independent of `c, d`) satisfying `0 < r z < 1` for all `z`.
+
+We then show in `summable_one_div_rpow_max` that the sum of `max (|c|, |d|) ^ (-k)` over
+`(c, d) ∈ ℤ × ℤ` is convergent for `2 < k`. This is proved by decomposing `ℤ × ℤ` using the
+`Finset.box` lemmas.
+-/
+
+noncomputable section
+
+open Complex UpperHalfPlane Set Finset
+
+open scoped BigOperators
+
+variable (z : ℍ)
+
+namespace EisensteinSeries
+
+lemma norm_eq_max_natAbs (x : Fin 2 → ℤ) : ‖x‖ = max (x 0).natAbs (x 1).natAbs := by
+  rw [← coe_nnnorm, ← NNReal.coe_natCast, NNReal.coe_inj, Nat.cast_max]
+  refine eq_of_forall_ge_iff fun c ↦ ?_
+  simp only [pi_nnnorm_le_iff, Fin.forall_fin_two, max_le_iff, NNReal.natCast_natAbs]
+
+section bounding_functions
+
+/-- Auxiliary function used for bounding Eisenstein series, defined as
+  `z.im ^ 2 / (z.re ^ 2 + z.im ^ 2)`. -/
+def r1 : ℝ := z.im ^ 2 / (z.re ^ 2 + z.im ^ 2)
+
+lemma r1_eq : r1 z = 1 / ((z.re / z.im) ^ 2 + 1) := by
+  rw [div_pow, div_add_one (by positivity), one_div_div, r1]
+
+lemma r1_pos : 0 < r1 z := by
+  dsimp only [r1]
+  positivity
+
+/-- For `c, d ∈ ℝ` with `1 ≤ d ^ 2`, we have `r1 z ≤ |c * z + d| ^ 2`.  -/
+lemma r1_aux_bound (c : ℝ) {d : ℝ} (hd : 1 ≤ d ^ 2) :
+    r1 z ≤ (c * z.re + d) ^ 2 + (c * z.im) ^ 2 := by
+  have H1 : (c * z.re + d) ^ 2 + (c * z.im) ^ 2 =
+    c ^ 2 * (z.re ^ 2 + z.im ^ 2) + d * 2 * c * z.re + d ^ 2 := by ring
+  have H2 : (c ^ 2 * (z.re ^ 2 + z.im ^ 2) + d * 2 * c * z.re + d ^ 2) * (z.re ^ 2 + z.im ^ 2)
+    - z.im ^ 2 = (c * (z.re ^ 2 + z.im ^ 2) + d * z.re) ^ 2 + (d ^ 2 - 1) * z.im ^ 2 := by ring
+  rw [r1, H1, div_le_iff (by positivity), ← sub_nonneg, H2]
+  exact add_nonneg (sq_nonneg _) (mul_nonneg (sub_nonneg.mpr hd) (sq_nonneg _))
+
+/-- This function is used to give an upper bound on the summands in Eisenstein series; it is
+defined by `z ↦ min z.im √(z.im ^ 2 / (z.re ^ 2 + z.im ^ 2))`. -/
+def r : ℝ := min z.im √(r1 z)
+
+lemma r_pos : 0 < r z := by
+  simp only [r, lt_min_iff, im_pos, Real.sqrt_pos, r1_pos, and_self]
+
+lemma r_lower_bound_on_verticalStrip {A B : ℝ} (h : 0 < B) (hz : z ∈ verticalStrip A B) :
+    r ⟨⟨A, B⟩, h⟩ ≤ r z := by
+  apply min_le_min hz.2
+  rw [Real.sqrt_le_sqrt_iff (by apply (r1_pos z).le)]
+  simp only [r1_eq, div_pow, one_div]
+  rw [inv_le_inv (by positivity) (by positivity), add_le_add_iff_right]
+  apply div_le_div (sq_nonneg _) _ (by positivity) (pow_le_pow_left h.le hz.2 2)
+  simpa only [even_two.pow_abs] using pow_le_pow_left (abs_nonneg _) hz.1 2
+
+lemma auxbound1 {c : ℝ} (d : ℝ) (hc : 1 ≤ c ^ 2) : r z ≤ Complex.abs (c * z + d) := by
+  rcases z with ⟨z, hz⟩
+  have H1 : z.im ≤ √((c * z.re + d) ^ 2 + (c * z).im ^ 2) := by
+    rw [Real.le_sqrt' hz, im_ofReal_mul, mul_pow]
+    exact (le_mul_of_one_le_left (sq_nonneg _) hc).trans <| le_add_of_nonneg_left (sq_nonneg _)
+  simpa only [r, abs_apply, normSq_apply, add_re, re_ofReal_mul, coe_re, ← pow_two, add_im, mul_im,
+    coe_im, ofReal_im, zero_mul, add_zero, min_le_iff] using Or.inl H1
+
+lemma auxbound2 (c : ℝ) {d : ℝ} (hd : 1 ≤ d ^ 2) : r z ≤ Complex.abs (c * z + d) := by
+  have H1 : √(r1 z) ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2) :=
+    (Real.sqrt_le_sqrt_iff (by positivity)).mpr (r1_aux_bound _ _ hd)
+  simpa only [r, abs_apply, normSq_apply, add_re, re_ofReal_mul, coe_re, ofReal_re, ← pow_two,
+    add_im, im_ofReal_mul, coe_im, ofReal_im, add_zero, min_le_iff] using Or.inr H1
+
+lemma div_max_sq_ge_one (x : Fin 2 → ℤ) (hx : x ≠ 0) :
+    1 ≤ (x 0 / ‖x‖) ^ 2 ∨ 1 ≤ (x 1 / ‖x‖) ^ 2 := by
+  refine (max_choice (x 0).natAbs (x 1).natAbs).imp (fun H0 ↦ ?_) (fun H1 ↦ ?_)
+  · have : x 0 ≠ 0 := by
+      rwa [← norm_ne_zero_iff, norm_eq_max_natAbs, H0, Nat.cast_ne_zero, Int.natAbs_ne_zero] at hx
+    simp only [norm_eq_max_natAbs, H0, Int.cast_natAbs, Int.cast_abs, div_pow, _root_.sq_abs, ne_eq,
+      OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, Int.cast_eq_zero, this, div_self,
+      le_refl]
+  · have : x 1 ≠ 0 := by
+      rwa [← norm_ne_zero_iff, norm_eq_max_natAbs, H1, Nat.cast_ne_zero, Int.natAbs_ne_zero] at hx
+    simp only [norm_eq_max_natAbs, H1, Int.cast_natAbs, Int.cast_abs, div_pow, _root_.sq_abs, ne_eq,
+      OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, Int.cast_eq_zero, this, div_self,
+      le_refl]
+
+lemma r_mul_max_le {x : Fin 2 → ℤ} (hx : x ≠ 0) : r z * ‖x‖ ≤ Complex.abs (x 0 * z + x 1) := by
+  have hn0 : ‖x‖ ≠ 0 := by rwa [norm_ne_zero_iff]
+  have h11 : x 0 * (z : ℂ) + x 1 = (x 0 / ‖x‖ * z + x 1 / ‖x‖) * ‖x‖ := by
+    rw [div_mul_eq_mul_div, ← add_div, div_mul_cancel₀ _ (mod_cast hn0)]
+  rw [norm_eq_max_natAbs, h11, map_mul, Complex.abs_ofReal, abs_norm, norm_eq_max_natAbs]
+  gcongr
+  · rcases div_max_sq_ge_one x hx with H1 | H2
+    · simpa only [norm_eq_max_natAbs, ofReal_div, ofReal_intCast] using auxbound1 z (x 1 / ‖x‖) H1
+    · simpa only [norm_eq_max_natAbs, ofReal_div, ofReal_intCast] using auxbound2 z (x 0 / ‖x‖) H2
+
+/-- Upper bound for the summand `|c * z + d| ^ (-k)`, as a product of a function of `z` and a
+function of `c, d`. -/
+lemma summand_bound {k : ℝ} (hk : 0 ≤ k) (x : Fin 2 → ℤ) :
+    Complex.abs (x 0 * z + x 1) ^ (-k) ≤ (r z) ^ (-k) * ‖x‖ ^ (-k) := by
+  by_cases hx : x = 0
+  · simp only [hx, Pi.zero_apply, Int.cast_zero, zero_mul, add_zero, ← norm_eq_abs, norm_zero]
+    by_cases h : -k = 0
+    · rw [h, Real.rpow_zero, Real.rpow_zero, one_mul]
+    · rw [Real.zero_rpow h, mul_zero]
+  · rw [← Real.mul_rpow (r_pos _).le (norm_nonneg _)]
+    exact Real.rpow_le_rpow_of_nonpos (mul_pos (r_pos _) (norm_pos_iff.mpr hx)) (r_mul_max_le z hx)
+      (neg_nonpos.mpr hk)
+
+variable {z} in
+lemma summand_bound_of_mem_verticalStrip {k : ℝ} (hk : 0 ≤ k) (x : Fin 2 → ℤ)
+    {A B : ℝ} (hB : 0 < B) (hz : z ∈ verticalStrip A B) :
+    Complex.abs (x 0 * z + x 1) ^ (-k) ≤ r ⟨⟨A, B⟩, hB⟩ ^ (-k) * ‖x‖ ^ (-k) := by
+  refine (summand_bound z hk x).trans (mul_le_mul_of_nonneg_right ?_ (by positivity))
+  exact Real.rpow_le_rpow_of_nonpos (r_pos _) (r_lower_bound_on_verticalStrip z hB hz)
+    (neg_nonpos.mpr hk)
+
+end bounding_functions
+
+section summability
+
+/-- The function `ℤ ^ 2 → ℝ` given by `x ↦ ‖x‖ ^ (-k)` is summable if `2 < k`. We prove this by
+splitting into boxes using `Finset.box`. -/
+lemma summable_one_div_norm_rpow {k : ℝ} (hk : 2 < k) :
+    Summable fun (x : Fin 2 → ℤ) ↦ ‖x‖ ^ (-k) := by
+  rw [← (finTwoArrowEquiv _).symm.summable_iff, summable_partition _ Int.existsUnique_mem_box]
+  · simp only [finTwoArrowEquiv_symm_apply, Function.comp_def]
+    refine ⟨fun n ↦ (hasSum_fintype (β := box (α := ℤ × ℤ) n) _).summable, ?_⟩
+    suffices Summable fun n : ℕ ↦ ∑' (_ : box (α := ℤ × ℤ) n), (n : ℝ) ^ (-k) by
+      refine this.congr fun n ↦ tsum_congr fun p ↦ ?_
+      simp only [← Int.mem_box.mp p.2, Nat.cast_max, norm_eq_max_natAbs, Matrix.cons_val_zero,
+        Matrix.cons_val_one, Matrix.head_cons]
+    simp only [tsum_fintype, univ_eq_attach, sum_const, card_attach, nsmul_eq_mul]
+    apply ((Real.summable_nat_rpow.mpr (by linarith : 1 - k < -1)).mul_left
+      8).of_norm_bounded_eventually_nat
+    filter_upwards [Filter.eventually_gt_atTop 0] with n hn
+    rw [Int.card_box hn.ne', Real.norm_of_nonneg (by positivity), sub_eq_add_neg,
+      Real.rpow_add (Nat.cast_pos.mpr hn), Real.rpow_one, Nat.cast_mul, Nat.cast_ofNat, mul_assoc]
+  · exact fun n ↦ Real.rpow_nonneg (norm_nonneg _) _
+
+/-- The sum defining the Eisenstein series (of weight `k` and level `Γ(N)` with congruence
+condition `a : Fin 2 → ZMod N`) converges locally uniformly on `ℍ`. -/
+theorem eisensteinSeries_tendstoLocallyUniformly {k : ℤ} (hk : 3 ≤ k) {N : ℕ} (a : Fin 2 → ZMod N) :
+    TendstoLocallyUniformly (fun (s : Finset (gammaSet N a)) ↦ (∑ x in s, eisSummand k x ·))
+      (eisensteinSeries a k ·) Filter.atTop := by
+  have hk' : (2 : ℝ) < k := by norm_cast
+  have p_sum : Summable fun x : gammaSet N a ↦ ‖x.val‖ ^ (-k) :=
+    mod_cast (summable_one_div_norm_rpow hk').subtype (gammaSet N a)
+  simp only [tendstoLocallyUniformly_iff_forall_isCompact, eisensteinSeries]
+  intro K hK
+  obtain ⟨A, B, hB, HABK⟩ := subset_verticalStrip_of_isCompact hK
+  refine (tendstoUniformlyOn_tsum (hu := p_sum.mul_left <| r ⟨⟨A, B⟩, hB⟩ ^ (-k : ℝ))
+    (fun p z hz ↦ ?_)).mono HABK
+  simpa only [eisSummand, one_div, ← zpow_neg, norm_eq_abs, abs_zpow, ← Real.rpow_intCast,
+    Int.cast_neg] using summand_bound_of_mem_verticalStrip (by positivity) p hB hz
+
+/-- Extend a function on `ℍ` arbitrarily to a function on all of `ℂ`. -/
+local notation "↑ₕ" f => f ∘ (PartialHomeomorph.symm (openEmbedding_coe.toPartialHomeomorph _))
+
+/-- Variant of `eisensteinSeries_tendstoLocallyUniformly` formulated with maps `ℂ → ℂ`, which is
+nice to have for holomorphicity later. -/
+lemma eisensteinSeries_tendstoLocallyUniformlyOn {k : ℤ} {N : ℕ} (hk : 3 ≤ k)
+    (a : Fin 2 → ZMod N) : TendstoLocallyUniformlyOn (fun (s : Finset (gammaSet N a )) ↦
+      ↑ₕ(fun (z : ℍ) ↦ ∑ x in s, eisSummand k x z )) (↑ₕ(eisensteinSeries_SIF a k).toFun)
+          Filter.atTop (UpperHalfPlane.coe '' ⊤) := by
+  apply TendstoLocallyUniformlyOn.comp (s := ⊤) _ _ _ (PartialHomeomorph.continuousOn_symm _)
+  · simp only [SlashInvariantForm.toFun_eq_coe, Set.top_eq_univ, tendstoLocallyUniformlyOn_univ]
+    apply eisensteinSeries_tendstoLocallyUniformly hk
+  · simp only [OpenEmbedding.toPartialHomeomorph_target, Set.top_eq_univ, mapsTo_range_iff,
+    Set.mem_univ, forall_const]
+
+end summability

Mathlib/Topology/UniformSpace/UniformConvergence.lean

@@ -757,6 +757,11 @@ theorem tendstoLocallyUniformlyOn_iff_forall_isCompact [LocallyCompactSpace α]
   (tendstoLocallyUniformlyOn_TFAE F f p hs).out 0 1
 #align tendsto_locally_uniformly_on_iff_forall_is_compact tendstoLocallyUniformlyOn_iff_forall_isCompact
 
+lemma tendstoLocallyUniformly_iff_forall_isCompact [LocallyCompactSpace α]  :
+    TendstoLocallyUniformly F f p ↔ ∀ K : Set α, IsCompact K → TendstoUniformlyOn F f p K := by
+  simp only [← tendstoLocallyUniformlyOn_univ,
+    tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_univ, Set.subset_univ, forall_true_left]
+
 theorem tendstoLocallyUniformlyOn_iff_filter :
     TendstoLocallyUniformlyOn F f p s ↔ ∀ x ∈ s, TendstoUniformlyOnFilter F f p (𝓝[s] x) := by
   simp only [TendstoUniformlyOnFilter, eventually_prod_iff]