Modular form tendsto at Im Infty
modular_form_tendsto_atImInfty
Plain-language statement
A modular form tends to its value at infinity as z → i∞.
Exact Lean statement
lemma modular_form_tendsto_atImInfty {k : ℤ} (f : ModularForm (Gamma 1) k) :
Filter.Tendsto f.toFun atImInfty (nhds ((qExpansion 1 f).coeff 0))Formal artifact
Lean source
lemma modular_form_tendsto_atImInfty {k : ℤ} (f : ModularForm (Gamma 1) k) : Filter.Tendsto f.toFun atImInfty (nhds ((qExpansion 1 f).coeff 0)) := by obtain ⟨c, hc, hO⟩ := ModularFormClass.exp_decay_sub_atImInfty' f have hΓ : (1 : ℝ) ∈ (↑(CongruenceSubgroup.Gamma 1) : Subgroup (GL (Fin 2) ℝ)).strictPeriods := CongruenceSubgroup.Gamma_one_coe_eq_SL ▸ one_mem_strictPeriods_SL rw [qExpansion_coeff_zero (by norm_num : (0 : ℝ) < 1) (ModularFormClass.analyticAt_cuspFunction_zero f (by norm_num) hΓ) (periodic_comp_ofComplex f hΓ)] simpa using (tendsto_zero_of_exp_decay hc hO).add_const (valueAtInfty f.toFun)- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/EisensteinAsymptotics.lean:54-62
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Related declarations
Anti Der Pos
antiDerPos
Plain-language statement
If is a modular form where is positive for sufficiently large (i.e. constant term is positive) and the derivative is positive, then is also positive.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Anti Serre Der Pos
antiSerreDerPos
Plain-language statement
Let be a holomorphic function where is real for all . Assume that Serre derivative is positive on the imaginary axis. If is positive for sufficiently large , then is positive for all .
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Closed Ball center subset upper Half Plane
closedBall_center_subset_upperHalfPlane
Plain-language statement
Closed ball centered at z with radius z.im/2 is contained in the upper half plane.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.