{
  "Title": "Bloch elliptic dilogarithm at CM torsion points",
  "Definition": "Bloch's elliptic dilogarithm at the torsion point $u=(r\\tau_\\Delta+s)/N$ of the marked CM torus is $D_q(u)=\\sum_{n\\in\\mathbb Z}D(e^{2\\pi i u}q^n)$, where $q=e^{2\\pi i\\tau_\\Delta}$ and $D$ is the Bloch-Wigner dilogarithm CITE{Brunault}.",
  "Parameters": {
    "Delta": {
      "type": "Z",
      "display": "$\\Delta$",
      "title": "CM order discriminant",
      "constraints": "$\\Delta<0$ is an imaginary quadratic order discriminant"
    },
    "N": {
      "type": "Z",
      "display": "$N$",
      "title": "exact order of the torsion point",
      "constraints": "$N\\geq3$"
    },
    "r": {
      "type": "Z",
      "display": "$r$",
      "title": "$\\tau_\\Delta$ coordinate numerator",
      "constraints": "$0\\leq r<N$; together with $s$, it satisfies CITE{comment-representatives}"
    },
    "s": {
      "type": "Z",
      "display": "$s$",
      "title": "real coordinate numerator",
      "constraints": "$0\\leq s<N$; together with $r$, it satisfies CITE{comment-representatives}"
    }
  },
  "Comments": {
    "comment-marking": "Put $b=0$ when $4\\mid\\Delta$ and $b=1$ otherwise, $\\tau_\\Delta=(-b+i\\sqrt{|\\Delta|})/2$, and $E_\\Delta=\\mathbb C/(\\mathbb Z+\\tau_\\Delta\\mathbb Z)$. The square root is positive, and $u=(r\\tau_\\Delta+s)/N$. The marking is part of the convention: it is not replaced by a modularly equivalent representative. In particular, $q=e^{2\\pi i\\tau_\\Delta}$ is negative for odd $\\Delta$.",
    "comment-representatives": "The condition $\\gcd(N,r,s)=1$ makes $N$ the exact order of the point. Among $(r,s)$ and $((-r)\\bmod N,(-s)\\bmod N)$, the lexicographically smaller pair is listed. The condition $2s-br\\not\\equiv0\\pmod N$ removes the real-$x$ locus, where every summand vanishes. The origin and all points of order two also vanish. Omitted opposite points have the negative of the listed value; signs are retained.",
    "comment-branches": "All logarithms are natural. The principal dilogarithm and principal logarithm in $D$ give the standard single-valued Bloch-Wigner function. Its continuous value is zero on the real axis. Classical polylogarithm values appear in HREF{Values_of_the_polylogarithm_at_rational_arguments}.",
    "comment-cm": "The current discriminants are exactly the thirteen CM orders with HREF{Rational_singular_moduli}[rational singular modulus], including nonmaximal orders CITE{SageCM}. Thus $\\Delta$ is an order discriminant and need not be a fundamental field discriminant. At these orders, the principal lattice represents the unique ideal class.",
    "comment-relations": "On the unit circle, $D(e^{i\\theta})=\\operatorname{Cl}_2(\\theta)$, the HREF{Values_of_the_Clausen_functions_at_rational_multiples_of_pi}[classical Clausen function]. The elliptic average has additional nonzero terms. Guillera and Rogers connect the points $u=1/3,1/4,1/6$ to HREF{Mahler_measures_of_the_Hessian_family}[Hessian Mahler measures] and HREF{Mahler_measures_of_x_x-1_y_y-1_k}[reciprocal Laurent Mahler measures] by theta parametrizations CITE{GR}, Theorem 4.",
    "comment-sign": "For the Eisenstein-Kronecker phase $e^{2\\pi i(n\\xi-m\\eta)}$ at $u=\\xi\\tau+\\eta$, this definition follows the negative sign in CITE{Brunault}, Proposition 6. The positive sign printed with that phase in the introductory regulator formula of CITE{Samart} is inconsistent with the defining $D$-series. This distinction does not change the shared $D$-series definition."
  },
  "Formulas": {
    "formula-bloch-wigner": "$D(z)=\\operatorname{Im}\\bigl(\\operatorname{Li}_2(z)+\\log|z|\\log(1-z)\\bigr)$, continuously extended by $D(0)=D(1)=0$ CITE{GR}.",
    "formula-paired": "$D_q(u)=D(x)+\\sum_{n=1}^{\\infty}\\bigl(D(xq^n)-D(q^n/x)\\bigr)$ for $x=e^{2\\pi i u}$ with $|q|<|x|\\leq1$. This follows from $D(1/z)=-D(z)$ CITE{GR}.",
    "formula-symmetry": "$D_q(-u)=-D_q(u)$. For real $q$, complex conjugation of $x$ negates the value; in the stated coordinates, $D_q((r\\tau_\\Delta+(br-s))/N)=-D_q((r\\tau_\\Delta+s)/N)$. Coordinates are interpreted modulo $N$."
  },
  "Programs": {
    "program-sage": {
      "language": "Sage",
      "code": "from generate import Table\ng = Table()\nprint(g.value({'Delta': -4, 'N': 4, 'r': 0, 's': 1}, 100))",
      "comment": "Download the attached generate.py into the working directory. This example computes one signed value with a rigorous error bound. The default enumeration in Table covers the selection stated in Data properties."
    }
  },
  "Similar tables": [
    {
      "table": "HREF{Rational_singular_moduli}",
      "relation": "gives the rational singular moduli of the same thirteen CM orders"
    },
    {
      "table": "HREF{Values_of_the_Clausen_functions_at_rational_multiples_of_pi}",
      "relation": "gives the classical unit-circle values $D(e^{i\\theta})$ used in the elliptic average"
    },
    {
      "table": "HREF{Values_of_the_polylogarithm_at_rational_arguments}",
      "relation": "gives classical polylogarithm values, including the weight-two function used to define $D$"
    },
    {
      "table": "HREF{Mahler_measures_of_x_x-1_y_y-1_k}",
      "relation": "contains the reciprocal Laurent family connected to elliptic dilogarithms at points of order four by CITE{GR}, Theorem 4"
    },
    {
      "table": "HREF{Mahler_measures_of_the_Hessian_family}",
      "relation": "contains ordinary Hessian Mahler measures connected to elliptic dilogarithms at points of order three and six by CITE{GR}, Theorem 4"
    }
  ],
  "Links": {
    "Brunault": {
      "title": "François Brunault: On Zagier's conjecture for base changes of elliptic curves, section 2 and Proposition 6",
      "url": "https://perso.ens-lyon.fr/francois.brunault/recherche/LEF.pdf"
    },
    "SageCM": {
      "title": "SageMath: Complex multiplication for elliptic curves, cm_orders",
      "url": "https://doc.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/cm.html"
    }
  },
  "References": {
    "GR": {
      "bib": "Jesús Guillera and Mathew Rogers, Mahler measure and the WZ algorithm (2013), section 3, Theorem 4.",
      "arxiv": "1006.1654v2"
    },
    "Samart": {
      "bib": "Detchat Samart, The elliptic trilogarithm and Mahler measures of K3 surfaces (2014), section 1.",
      "arxiv": "1309.7730v2"
    }
  },
  "Keywords": [
    "Bloch",
    "Bloch-Wigner",
    "elliptic polylogarithm",
    "elliptic regulator",
    "complex multiplication",
    "CM torsion"
  ],
  "Tags": [
    "special values",
    "special functions",
    "elliptic curves",
    "period"
  ],
  "Data properties": {
    "type": "R",
    "digits": "100",
    "rigour": "proven",
    "complete": "no",
    "complete-note": "it holds the 362 signed values on the principal marked lattices of all thirteen class-number-one orders, $\\Delta=-3,-4,-7,-8,-11,-12,-16,-19,-27,-28,-43,-67,-163$, at exact orders $3\\leq N\\leq6$, with all torsion directions modulo inversion except the symmetry-forced zeros specified in CITE{comment-representatives}. This arithmetic reference covers rectangular and rhombic CM tori, the orders $3,4,6$ occurring in Mahler-measure identities, the next prime order $5$, and the first mixed-prime order $6$. The cutoff six and the representative convention are noncanonical choices independent of numeral notation; no separate application is claimed for every combination. Higher class numbers and higher torsion orders remain outside this finite coverage",
    "rigour details": "Each value is enclosed by Arb ball evaluation of the paired Bloch-Wigner series, including uncertainty in the CM argument. For $|z|=t<1$, the bound $|D(z)|\\leq t(1+|\\log t|)/(1-t)$ gives an explicit geometric tail bound. The generator adds that bound to the computed ball; the enclosures justify the 100 significant digits recorded. It uses 128 extra working bits, and the least relative accuracy measured over the selection was 421 bits. Between 7 and 57 terms on each side were needed.\n\nEvery entry was recomputed with 30 extra decimal working digits and compared with an independent Fourier expansion whose unit-circle term is a finite Hurwitz-zeta sum. Relative differences were below $5.2\\times10^{-128}$. Point inversion and real-$q$ conjugation identities were checked for all entries, and the 132 omitted inversion representatives were checked as zeros. A separate ball integral for the reciprocal Laurent Mahler measure at the square-lattice point $u=1/4$ agrees with CITE{GR}, equation (26). Its nonzero integration control was $\\int_0^1t^2\\,dt=1/3$. A directly truncated Eisenstein-Kronecker sum with an explicit tail bound determines the negative lattice sign; this last check establishes the sign, not the recorded precision. The attached checks.py and numerical-evidence.json give the methods and measurements."
  },
  "Display properties": {
    "number-header": "$D_q(u)$"
  },
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