Hecke Resolvent

This note studies the exact characteristic polynomial and resolvent of the proved 7x7 Hecke kernel on the Heegner set.

7x7 characteristic polynomial

lam**7 - 1792*lam**6 + 1101734*lam**5 - 287890132*lam**4 + 32967160008*lam**3 - 1634357991536*lam**2 + 29925111424384*lam - 168144119646208

7x7 resolvent

f(x) = (-29925111424384*x**6 + 3268715983072*x**5 - 98901480024*x**4 + 1151560528*x**3 - 5508670*x**2 + 10752*x - 7)/(168144119646208*x**7 - 29925111424384*x**6 + 1634357991536*x**5 - 32967160008*x**4 + 287890132*x**3 - 1101734*x**2 + 1792*x - 1)

Special values: - 1_over_137: 6.4581433199465661236227817313532542299300583284587 - 1_over_163: 4.3950509661179098698178665307605517329796578370242 - 1_over_26: 4.5609723202357422630037779389819331123292300130978 - 1_over_13: 2.4781670572630033478555818548161669123308940689734 - alpha: 6.4542481735169000613073588855673257765879972580797

Numerical solutions of f(x) = c: - f(x)=137: [0.0012077563443659658, 0.0020186394157720154, 0.0036205313226137245, 0.010116581824767951, 0.012930211596244651, 0.05106173907063287, 0.0957184557349151] - f(x)=137.036: [0.0012077589855210098, 0.002018643629081197, 0.0036205386606521363, 0.010116603146578527, 0.012930235861014397, 0.05106183972731209, 0.09571863657169911] - f(x)=163: [0.0012093439827160055, 0.0020211819630056482, 0.0036249699885287424, 0.010129439996961559, 0.012944981498285508, 0.051122759443785915, 0.09582845251930398]

13x13 operator viewpoint

Using the saved 13 x 7 Hecke-eigenvalue matrix A, the full operator is A A^T on the cusp-form embedding space. Its nonzero eigenvalues match those of G = A^T A, so

g(x) = Tr((I - x A A^T)^(-1)) = f(x) + 6.

The 13x13 characteristic polynomial in the spectral variable λ is

lam**13 - 1792*lam**12 + 1101734*lam**11 - 287890132*lam**10 + 32967160008*lam**9 - 1634357991536*lam**8 + 29925111424384*lam**7 - 168144119646208*lam**6

because there are 6 zero eigenvalues. The formal polynomial has degree 13, but the number of free spectral parameters is still only 7, not 2g = 26.

Taylor expansion

traces Tr(G^n) for n=0..12: [7, 1792, 1007796, 695333500, 519744990248, 404539626927832, 322018984332670752, 259663892341453733616, 210973448498913560470816, 172182257141545179549603232, 140898330410509307271212178816, 115481829854899015648999345071936, 94740150570455236672453230173740928]
recurrence coefficients from the degree-7 characteristic polynomial: [-1792, 1101734, -287890132, 32967160008, -1634357991536, 29925111424384, -168144119646208]
recurrence verified for n=7..25: True

QED comparison

QED benchmark coefficients: ['1/2', '-0.32800000000000000000000000000000000000000000000000', '1.1810000000000000000000000000000000000000000000000']
Hecke coefficients Tr(G^n) start [7, 1792, 1007796, 695333500, 519744990248, 404539626927832, 322018984332670752]

No natural x makes the raw Hecke resolvent expansion look like the QED perturbation series: the constant term is rigidly 7 (or 13 on the full space), so it cannot match the QED tree-level 0.5 by tuning x.