{ "Title": "Zagier traces of singular moduli", "Definition": "Zagier's trace of singular moduli is $t_m(d)=\\sum_{[Q]\\in\\mathcal Q_d/\\Gamma}J_m(\\alpha_Q)/w_Q$ CITE{Zagier}, equation (16). Here $\\mathcal Q_d$ contains all positive definite integral forms of discriminant $-d$, including imprimitive forms, and $\\Gamma=\\mathrm{PSL}_2(\\mathbb Z)$.", "Parameters": { "m": { "type": "Z", "display": "$m$", "title": "Hecke order", "constraints": "$m\\geq1$" }, "d": { "type": "Z", "display": "$d$", "title": "discriminant magnitude", "constraints": "$d>0$, $d\\equiv0,3\\pmod4$" } }, "Tags": [ "number theory", "modular function", "modular form", "elliptic curves" ], "Data properties": { "type": "Z", "rigour": "exact", "complete": "no", "complete-note": "it holds every $1\\leq m\\leq4$ and every $3\\leq d\\leq200$ with $d\\equiv0,3\\pmod4$, including all nonfundamental discriminants and all imprimitive contributions. These noncanonical finite bounds cover the ordinary trace, the first two prime Hecke orders and the first composite order across small CM discriminants of varied class number and conductor. The choice balances breadth across this discrete arithmetic family with rapidly growing integers; the longest entry has 78 characters", "rigour details": "Computed from certified Hilbert class polynomials using SageMath's FLINT/Arb implementation CITE{Sage}. Faber polynomials, quotient-algebra traces and square-divisor weights use exact integer or rational arithmetic, and every final result is required to be an integer.\n\nAll 400 entries agree with an independent exact calculation from the weight-$3/2$ coefficient recurrences and Hecke operators in CITE{Zagier}, §§2 and 6. They also lie in direct CM orbit sums computed by summing Hecke translates of $j-744$ in 1024-bit complex ball arithmetic. Each component is finite with radius less than $10^{-110}$; each real interval isolates the recorded integer. Known-answer controls and deliberately incorrect answers precede these comparisons. The integers are stored exactly, not inferred by rounding those balls.\n\nThe checks compare 100 class polynomials with HREF{Hilbert_class_polynomials}[the companion table], all thirteen rational CM cases with HREF{Rational_singular_moduli}[the rational singular moduli], and the three higher Faber polynomials with HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[the identity-class polynomials]. They check the Laurent principal parts against HREF{Q-expansion_of_the_j-invariant}[the stored j expansion], all sixteen ordinary traces printed in CITE{Zagier}, §1, the explicit polynomials in §6, and the displayed generating identity through discriminant 200. The attached checker repeats these checks on values read from the draft." }, "Numbers": { "1": { "3": "-248", "4": "492", "7": "-4119", "8": "7256", "11": "-33512", "12": "53008", "15": "-192513", "16": "287244", "19": "-885480", "20": "1262512", "23": "-3493982", "24": "4833456", "27": "-12288992", "28": "16576512", "31": "-39493539", "32": "52255768", "35": "-117966288", "36": "153541020", "39": "-331534572", "40": "425691312", "43": "-884736744", "44": "1122626864", "47": "-2257837845", "48": "2835861520", "51": "-5541103056", "52": "6896878512", "55": "-13136687601", "56": "16220381536", "59": "-30197680312", "60": "37017882624", "63": "-67515206970", "64": "82226601996", "67": "-147197952744", "68": "178211037024", "71": "-313645814923", "72": "377674773768", "75": "-654403831496", "76": "784073551152", "79": "-1339190286960", "80": "1597178431536", "83": "-2691907586232", "84": "3196800943968", "87": "-5321761716339", "88": "6294842638512", "91": "-10359073015248", "92": 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"16600400961966941114914838933284983052419529248625262886119980840612036752", "183": "67261234780942745685602045060984193128759012508682491234787591514425572711", "184": "106956017450638627974583770845250773401728174438018397984793754350673789024", "187": "426839022278304967378679651648336145269240587591197239209803479432536129584", "188": "675373809731224565690709663492004405758300537340900328858382666168402739670", "191": "2655983923753357777149068967696080626544605641465794244310330445814376619874", "192": "4182281838345107106479800308537355245077631569553845329810484284083588381808", "195": "16214951737145317372845465170353550477186337906082375410713065931040833667168", "196": "25414163239542327782004582598407782039831237800657475432470511091938079891564", "199": "97182485494388278334047141966022437195768447404078099245667128708970662590644", "200": "151628818650476778864481406700333223368298657340988337399166388168491887804584" } }, "Comments": { "comment-roots": "For $Q=[a,b,c]$, the CM root is $\\alpha_Q=(-b+i\\sqrt d)/(2a)$, with $\\sqrt d>0$. The discriminant of the form is $-d$. Forms are equivalent under $\\mathrm{PSL}_2(\\mathbb Z)$.", "comment-weights": "The stabilizer order $w_Q$ is $3$ for a form equivalent to a positive integer multiple of $[1,1,1]$, $2$ for a multiple of $[1,0,1]$, and $1$ otherwise. These are stabilizers in $\\mathrm{PSL}_2$, so the central sign of $\\mathrm{SL}_2$ contributes no additional factor. The exceptional weights also apply to imprimitive forms CITE{Zagier}, §1.", "comment-normalization": "The function $J_m$ is the unique level-one modular function holomorphic on the upper half-plane with expansion $J_m(\\tau)=q^{-m}+O(q)$, where $q=e^{2\\pi i\\tau}$ CITE{Zagier}, §6. With the classical HREF{Q-expansion_of_the_j-invariant}[$j$-invariant], put $J=j-744$. Then $J_1=J$ and $J_m=P_{m,1A}(J)$, using the identity-class HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[Hecke-type Faber polynomials]; $P_{1,1A}(X)=X$. No division by $m$ is made.", "comment-coefficients": "The ordinary traces are Fourier coefficients of a weakly holomorphic modular form of weight $3/2$ for $\\Gamma_0(4)$ CITE{BF}. More generally, $t_m(d)$ is the negative of the coefficient denoted $B_m(1,d)$ in CITE{Zagier}, Theorem 5(ii), using that paper's Hecke normalization.", "comment-small": "The HREF{Rational_singular_moduli}[rational singular moduli] give the smallest cases: $j(\\alpha_{[1,1,1]})=0$ yields $t_1(3)=-248$, and $j(i)=1728$ yields $t_1(4)=492$. At $d=12$, the form $[2,2,2]$ contributes $-248$ in addition to the primitive form $[1,0,3]$, giving $t_1(12)=53008$." }, "Formulas": { "formula-class-polynomials": "Let $H_{-e}(x)$ be the primitive-order HREF{Hilbert_class_polynomials}[Hilbert class polynomial], and let $\\operatorname{Tr}_{\\mathbb Q[x]/(H_{-e})}$ denote the trace of multiplication in this quotient algebra. Then $t_m(d)=\\sum_{f\\geq1,\\ f^2\\mid d,\\ e=d/f^2\\equiv0,3\\ (\\mathrm{mod}\\ 4)}\\frac{1}{w(e)}\\operatorname{Tr}_{\\mathbb Q[x]/(H_{-e})}\\!\\bigl(P_{m,1A}(x-744)\\bigr)$, where $w(3)=3$, $w(4)=2$ and $w(e)=1$ otherwise. Here $P_{m,1A}(X)$ is the identity-class HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[Faber polynomial].", "formula-hecke": "$J_m(\\tau)=\\sum_{ab=m,\\ a,b>0}\\sum_{r=0}^{b-1}J\\!\\left(\\frac{a\\tau+r}{b}\\right)$, where $J=j-744$ CITE{Zagier}, proof of Theorem 5(ii).", "formula-generating": "$$\\begin{aligned}&-q^{-1}+2+\\sum_{d>0}t_1(d)q^d\\\\&=-q^{-1}\\left(\\sum_{r\\in\\mathbb Z}(-1)^r q^{r^2}\\right)\\left(1+240\\sum_{n\\geq1}\\sigma_3(n)q^{4n}\\right)\\prod_{n\\geq1}(1-q^{4n})^{-6},\\end{aligned}$$ where $\\sigma_3(n)=\\sum_{a\\mid n,\\ a>0}a^3$ and $t_1(d)=0$ when $d\\equiv1,2\\pmod4$ CITE{Zagier}, equation (3) and Theorem 1.", "formula-j-trace": "$\\sum_{[Q]\\in\\mathcal Q_d/\\mathrm{PSL}_2(\\mathbb Z)}j(\\alpha_Q)/w_Q=t_1(d)+744H(d)$, where $H(d)=\\sum_{[Q]\\in\\mathcal Q_d/\\mathrm{PSL}_2(\\mathbb Z)}1/w_Q$ is the Hurwitz–Kronecker class number CITE{Zagier}, §1." }, "References": { "Zagier": { "bib": "Don Zagier, Traces of singular moduli, in Motives, Polylogarithms and Hodge Theory, Part I, International Press Lecture Series 3, 2002, 211–244, §§1, 2 and 6.", "url": "https://people.mpim-bonn.mpg.de/zagier/files/tex/TracesSingModuli/fulltext.pdf" }, "BF": { "bib": "Jan Hendrik Bruinier and Jens Funke, Traces of CM values of modular functions, Introduction, equations (1.1)–(1.2).", "url": "https://www.maths.dur.ac.uk/users/jens.funke/cmtraces.pdf" } }, "Links": { "Sage": { "title": "SageMath: certified Hilbert class polynomial computation", "url": "https://doc.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/cm.html#sage.schemes.elliptic_curves.cm.hilbert_class_polynomial" } }, "Similar tables": [ { "table": "HREF{Hilbert_class_polynomials}[Hilbert class polynomials]", "relation": "Their roots are the primitive CM orbits used in the square-divisor decomposition of these weighted traces." }, { "table": "HREF{Rational_singular_moduli}[Rational singular moduli]", "relation": "The degree-one CM orbits provide exact special cases, with the same classical j normalization." }, { "table": "HREF{Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[Hecke-type Faber polynomials]", "relation": "The identity-class polynomial of degree m evaluates to the modular function J_m whose CM trace is stored here." }, { "table": "HREF{Q-expansion_of_the_j-invariant}[$q$-expansion of the $j$-invariant]", "relation": "These coefficients determine J=j-744 and its identity-class Faber polynomials." }, { "table": "HREF{Values_of_monstrous_McKay_Thompson_series_at_rectangular_CM_points}[McKay–Thompson values at rectangular CM points]", "relation": "These are individual principal CM values for nonidentity Monster classes, whereas this table sums full classical CM orbits." }, { "table": "HREF{Zeros_of_Hecke-type_Faber_polynomials_of_monstrous_McKay_Thompson_series}[Zeros of Hecke-type Faber polynomials]", "relation": "These are roots of the same polynomial family, rather than weighted sums of its CM evaluations." } ], "Keywords": [ "Zagier trace", "Hecke traces", "CM traces", "traces of singular moduli", "half-integral-weight Fourier coefficients" ], "Display properties": { "number-header": "$t_m(d)$", "group parameters": [ [ "m" ], [ "d" ] ] }, "Programs": { "program-sage": { "language": "Sage", "code": "# Exact traces for Hecke orders 1 through 4.\nR. = PolynomialRing(QQ)\nF = [x-744,\n x^2-1488*x+159768,\n x^3-2232*x^2+1069956*x-36866976,\n x^4-2976*x^3+2533680*x^2-561444608*x+8507424792]\ndef zagier_trace(m, d):\n assert 1 <= m <= 4 and d > 0 and d % 4 in (0, 3)\n result = QQ(0)\n for f in divisors(d):\n if d % f^2:\n continue\n e = d // f^2\n if e % 4 not in (0, 3):\n continue\n H = R(hilbert_class_polynomial(-e, algorithm='arb'))\n w = 3 if e == 3 else 2 if e == 4 else 1\n result += sum(((x^k*F[m-1]) % H)[k] for k in range(H.degree())) / QQ(w)\n return ZZ(result)\nzagier_trace(1, 203) # another discriminant beyond the stored range\n" } } }