back to table · edit · history · where entries came from · files · download
10910 bytes, as of the version from 2026-11-02 11:01 (current). Recorded here, not run.
{"Title": "Modified Mahler measures of the Hessian family", "Definition": "The modified Mahler measure CITE{S} of the Hessian family is the signed period $\\widetilde N(t)=I(t)-2J(t)$ for $-1<t<27$, with the root integrals $I(t),J(t)$ defined in CITE{formula-integrals}.", "Parameters": {"t": {"type": "R", "display": "$t$", "title": "Hessian parameter", "constraints": "$-1<t<27$"}}, "References": {"S": {"bib": "D. Samart, Mahler measure of a nonreciprocal family of elliptic curves, corrected version 3 (2023).", "arxiv": "2301.05390v3"}, "ST": {"bib": "D. Samart and Z. Tao, Determinants of Mahler measures and special values of L-functions, version 3 (2026).", "arxiv": "2409.14714v3"}, "TG": {"bib": "Z. Tao and X. Guo, CM points, class numbers, and the Mahler measures of x^3+y^3+1-kxy, version 2 (2024).", "arxiv": "2310.12510v2"}}, "Tags": ["number theory", "special values", "elliptic curves", "integral"], "Data properties": {"type": "R", "digits": "100", "rigour": "heuristic (agreement-checked)", "complete": "no", "complete-note": "it holds all ten arguments in $(-1,27)$ from the three rational CM parameters and both embeddings of the fifteen quadratic parameters in CITE{ST}, Table 1, rows 13 through 27; the degree-two catalogue is a noncanonical computational boundary chosen to cover its arithmetic orbits before moving to quartic fields, and no continuous reference mesh is used", "rigour details": "The defining quadratic-root integral is evaluated by adaptive tanh-sinh quadrature, split at the torus crossing $c=\\arccos((k-1)/2)$. Intervals are bisected when the numerical convergence estimate is insufficient. All ten values were computed at 135 and 185 working decimal digits, with at least 124.24 significant digits of relative agreement; 100 significant digits are retained. These are agreement-checked approximations, not certified error bounds.\n\nAn independent implementation evaluates CITE{formula-hypergeometric} at 125 working digits, using its convergent series or integrating its derivative expressed with two ${}_2F_1$ functions. Its retained output agrees with every defining-integral value to at least 119.40 significant digits. The quadratic integrator first reproduces the nonzero control $N(0)=\\mu_3$, HREF{Mahler_measures_of_(short_random_walks)#3}[the three-step Mahler measure], using $\\mu_3=3\\sqrt3 L(2,\\chi_{-3})/(4\\pi)$ and an independent Hurwitz-zeta evaluation of the Dirichlet value HREF{Values_of_Dirichlet_L-functions_at_positive_integers#3,2,2}[$L(2,\\chi_{-3})$]. The $t=24$ row is also checked against $-81L(C_{24},2)/(4\\pi^2)$, computed independently by PARI at 145 and 195 working digits, using CITE{TG}, Theorem 1.5 and the weight-two functional equation.\n\nAll 33 CM candidates are constructed exactly as real algebraic numbers. Their $j$-invariants satisfy the corresponding Hilbert class polynomials in exact arithmetic; these polynomials were independently recomputed with Sage and compared with HREF{Hilbert_class_polynomials}[the stored polynomials]. The model's $j$-invariant formula is checked symbolically, and exact comparison with $-1$ and $27$ selects the ten stored arguments. Attached sources contain the generator, independent checks, numerical evidence and the argument-selection justification."}, "Display properties": {"number-header": "$\\widetilde N(t)$"}, "Numbers": {"24": {"number": "-1.800714521389232519501185405743840291737296186591797905532716386162363170058159818083129625765602197", "comment": "The rational CM parameter has order discriminant $-27$; its modified period is negative."}, "18-6*sqrt(3)": {"number": "-1.096951909366771601952010997733538752297149349050759564931438930169293673386852406424059633150610001", "param-latex": "$18-6\\sqrt{3}$"}, "-27/2+27/2*sqrt(5)": {"number": "-1.513687186239743341872745049688579673087463674158772386230870101576999585487544908803560048817077913", "param-latex": "$\\frac{-27}{2}+\\frac{27}{2}\\sqrt{5}$"}, "-4320+1944*sqrt(5)": {"number": "-1.932641908064528725581449762187575135462765735637631814047882617171322950109503337595673070721830244", "param-latex": "$-4320+1944\\sqrt{5}$"}, "-864+216*sqrt(17)": {"number": "-1.914640409126734184230432088345860797426992831413936376282089809050506578263090011064672112896029418", "param-latex": "$-864+216\\sqrt{17}$"}, "-163296+35640*sqrt(21)": {"number": "-1.938187616612528602698936496141481210976320713758078558538411926519719123332956344695326978350758086", "param-latex": "$-163296+35640\\sqrt{21}$"}, "4+4*sqrt(33)": {"number": "-1.936704632266976978860771242092303076541712282719866196414683070569061135700968614279565508112022658", "param-latex": "$4+4\\sqrt{33}$"}, "-16740+2916*sqrt(33)": {"number": "-1.276728224888631527984651430908906310784113775297782331334350359993656517307210947957583367819621887", "param-latex": "$-16740+2916\\sqrt{33}$"}, "-55296+8640*sqrt(41)": {"number": "-1.937829950655483536226235978084597768154442686116267964164355182160257613499460155880071043747073648", "param-latex": "$-55296+8640\\sqrt{41}$"}, "-13500000+1431000*sqrt(89)": {"number": "-1.938392354885384622642192274639343697785362106957646937023009099408714775556172476069611853799770514", "param-latex": "$-13500000+1431000\\sqrt{89}$"}}, "Comments": {"comment-conventions": "Natural logarithms are used. The product of the two roots is $e^{i\\theta}$, so the larger root has modulus at least one; a tie gives $F_t(\\theta)=0$. This agrees with Samart's larger-root branch CITE{S}, Lemma 3, and remains defined at the removable singularity of the explicit root formula when $k=1$.", "comment-ordinary": "The ordinary logarithmic Mahler measure HREF{T649}[$N(t)$] of $P_t=x^3+y^3+1-kxy$ equals that of $Q_t=y^2+(x^2-kx)y+x$. With $I(t)=\\pi^{-1}\\int_0^cF_t(\\theta)\\,d\\theta$ and $J(t)=\\pi^{-1}\\int_c^\\pi F_t(\\theta)\\,d\\theta$, the ordinary and modified periods are the different combinations in CITE{formula-periods}. The sign of the modified period is retained.", "comment-notation": "The parameter is $t=k^3$, with the real cube root fixing $k$. Samart writes $\\tilde n(k)$ CITE{S}; Samart and Tao use $\\tilde{\\mathrm n}(t)$ and the piecewise symbol $\\nu(t)$ CITE{ST}. This table holds the modified period on its open real domain.", "comment-cm": "The arithmetic arguments are the interior members of the rational and real quadratic CM catalogue in CITE{ST}: $t=24$ and one conjugate from each of Table 1, rows 14, 17, 19, 22, 23, 24, 25, 26 and 27. Complex multiplication refers to the curve $C_t$ in CITE{formula-cm}, and the degree bound applies to $t$, not its cube root. Exact radical identities and CM discriminants are recorded in the attached generator and argument-selection note."}, "Formulas": {"formula-integrals": "$\\displaystyle I(t)=\\frac1\\pi\\int_0^{c(t)}F_t(\\theta)\\,d\\theta,\\qquad J(t)=\\frac1\\pi\\int_{c(t)}^\\pi F_t(\\theta)\\,d\\theta,$ where $k=\\sqrt[3]{t}\\in\\mathbb R$, $c(t)=\\arccos((k-1)/2)\\in[0,\\pi]$ and $F_t(\\theta)=\\log\\max\\{|y_1|,|y_2|\\}$. Here $y_1,y_2$ are the roots of $Q_t(e^{i\\theta},y)=0$, with $Q_t(x,y)=y^2+(x^2-kx)y+x$ CITE{S}. The variables $x,y$ are integration variables.", "formula-periods": "$N(t)=I(t)+J(t),\\qquad\\widetilde N(t)=I(t)-2J(t)=N(t)-3J(t).$ Here HREF{T649}[$N(t)$] is the ordinary Mahler measure, and $I(t),J(t)$ are the two root integrals in CITE{formula-integrals} CITE{ST}, Section 2.", "formula-hypergeometric": "For $0<t<27$ and $k=\\sqrt[3]{t}$, $\\displaystyle\\widetilde N(t)=-\\frac{1}{4\\pi^2}\\left[\\Gamma(1/3)^3 k\\,{}_3F_2\\!\\left(\\begin{matrix}1/3,1/3,1/3\\\\2/3,4/3\\end{matrix};\\frac{t}{27}\\right)+\\frac{\\Gamma(2/3)^3 k^2}{2}\\,{}_3F_2\\!\\left(\\begin{matrix}2/3,2/3,2/3\\\\4/3,5/3\\end{matrix};\\frac{t}{27}\\right)\\right].$ This is the convergent-series form following Theorem 1 of CITE{S}, with its first coefficient simplified by the gamma duplication and reflection formulas. The constants HREF{Values_of_the_Gamma_function_at_rational_numbers#1/3}[$\\Gamma(1/3)$] and HREF{Values_of_the_Gamma_function_at_rational_numbers#2/3}[$\\Gamma(2/3)$] are tabulated separately.", "formula-cm": "$C_t:\\ Y^2=X^3-27t^2X^2+216t^3(t-27)X-432t^4(t-27)^2,\\qquad j(C_t)=\\frac{t(t+216)^3}{(t-27)^3}.$ This model is nonsingular for $t\\ne0,27$ CITE{ST}. Its selected CM arguments satisfy $H_D(j(C_t))=0$ for the corresponding imaginary quadratic order discriminant $D$, using HREF{Hilbert_class_polynomials}[the Hilbert class polynomials]."}, "Similar tables": [{"table": "HREF{T649}[Mahler measures of the Hessian family]", "relation": "holds the ordinary period $I(t)+J(t)$ for the same polynomial and parameter normalization, whereas this table holds $I(t)-2J(t)$"}, {"table": "HREF{Hilbert_class_polynomials}", "relation": "holds $H_D$, which characterizes the CM $j$-invariants used to select the arithmetic arguments"}, {"table": "HREF{Values_of_the_Gamma_function_at_rational_numbers}", "relation": "holds the gamma constants in the convergent hypergeometric evaluation of the modified period"}], "Keywords": ["Hesse", "Hessian", "Mahler measure", "modified Mahler measure", "x^3+y^3+1-kxy", "y^2+(x^2-kx)y+x", "Samart", "CM", "Samart-Tao"], "Programs": {"program-python": {"language": "Python", "code": "import mpmath as mp\n\ndef modified_hessian(argument, working=135):\n # A callable constructs a radical afresh at the working precision.\n with mp.workdps(working):\n t = mp.mpf(argument() if callable(argument) else argument)\n if not -1 < t < 27:\n raise ValueError('Require -1 < t < 27.')\n k = mp.sign(t)*mp.root(abs(t), 3)\n c = mp.acos((k-1)/2)\n def f(theta):\n x = mp.exp(mp.j*theta)\n B = x*x-k*x\n D = mp.sqrt(B*B-4*x)\n return mp.log(max(abs((-B+D)/2), abs((-B-D)/2)))\n tolerance = mp.power(10, -working+20)\n def integrate(a, b, depth=0):\n value, error = mp.quad(f, [a, b], error=True, maxdegree=8)\n if mp.isfinite(value) and error <= tolerance*(b-a)/mp.pi:\n return value\n if depth == 22:\n raise ArithmeticError('Quadrature did not converge.')\n m = (a+b)/2\n return integrate(a, m, depth+1)+integrate(m, b, depth+1)\n I = integrate(mp.mpf(0), c/2)+integrate(c/2, c)\n J = integrate(c, (c+mp.pi)/2)+integrate((c+mp.pi)/2, mp.pi)\n return (I-2*J)/mp.pi\n\nlow = modified_hessian('24', 135)\nhigh = modified_hessian('24', 185)\nwith mp.workdps(200):\n assert abs(low-high) < mp.mpf('1e-115')*abs(high)\n print(mp.nstr(high, 100))", "comment": "The standalone mpmath routine evaluates the defining signed period at a supplied real argument. The example compares 135 and 185 working digits before printing 100 significant digits. These are numerical agreement checks, not certified error bounds; the attached generator enumerates the ten selected CM arguments."}}}