A276183 - OEIS

A276183

Genus of the quotient of the modular curve X_0(n) by the Fricke involution.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 2, 3, 0, 3, 1, 2, 1, 1, 1, 3, 2, 2, 2, 4, 0, 2, 2, 2, 1, 3, 2, 5, 1, 2, 1, 4, 1, 4, 3, 3, 2, 4, 1, 4, 2, 4, 4, 4, 1, 3, 3, 2, 3, 3, 1, 7

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OFFSET

1,42

COMMENTS

a(n) is the genus of quotient space H/Gamma_0*(n), where H is the upper half plane and Gamma_0*(n) = Gamma_0(n) + W Gamma_0(n) is the extension of Gamma_0(n) via the involution z <-> W(z) = -n/z (see Cohn, 1988).

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 1..54321

Harvey Cohn, Fricke's Two-Valued Modular Equations, Math. Comp. 51 (1988), 787-807.

Harvey Cohn, A Numerical Survey of the Reduction of Modular Curve Genus by Fricke's Involutions, Number Theory (New York Seminar 1989-1990), p. 100.

Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.

Andrew P. Ogg, Automorphismes de courbes modulaires, Séminaire Delange-Pisot-Poitou. Théorie des nombres, vol. 16, no. 1 (1974-1975), talk no. 7, p. 1.

FORMULA

a(n) = (1 + A001617(n))/2 - r * A000003(n)/12 for all n > 4, where r=4 for n=3 (mod 8), r=6 for n=7 (mod 8) and r=3 otherwise.

a(n) <> 4884 for all n.

EXAMPLE

G.f. = x^22 + x^28 + x^30 + x^33 + x^34 + x^37 + x^38 + x^40 + 2*x^42 + x^43 + x^44 + ...

MATHEMATICA

f[n_] := If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors@ n}] - Count[(#^2 - # + 1)/n & /@ Range@ n, _?IntegerQ]/3 - Count[(#^2 + 1)/n & /@ Range@ n, _?IntegerQ]/4];

g[n_] := Ceiling[k0 = k /. FindRoot[EllipticK[1 - k^2]/EllipticK[k^2] == Sqrt@ n, {k, 1/2, 10^-10, 1}, WorkingPrecision -> 600, MaxIterations -> 100]; Exponent[MinimalPolynomial[RootApproximant[k0^2, 24], x], x]/2];

r[n_] := If[MemberQ[{3, 7}, #], 3 + (# - 1)/2, 3] &@ Mod[n, 8]; a[n_] := If[n <= 4, 0, (1 + f@ n)/2 - r[n] g[n]/12]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 102}] (* Michael De Vlieger, Oct 28 2016, after Michael Somos at A001617 and Jean-François Alcover at A000003 *)

ClassList[n_?Negative] :=

Select[Flatten[#, 1] &@Table[

{i, j, (j^2 - n)/(4 i)}, {i, Sqrt[-n/3]}, {j, 1 - i, i}],

Mod[#3, 1] == 0 && #3 >= # &&

GCD[##] == 1 && ! (# == #3 && #2 < 0) & @@ # &]

A001617[n_] := If[n < 1, 0,

1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d,

Divisors@n}] -

Count[(#^2 - # + 1)/n & /@ Range[n], _?IntegerQ]/3 -

Count[(#^2 + 1)/n & /@ Range[n], _?IntegerQ]/4];

a[n_] := If[0 <= n <= 4, 0, (A001617[n] + 1)/2 - If[Mod[n, 8] == 3, 4, If[Mod[n, 8] == 7, 6, 3]] Length[ClassList[-4 n]]/12] (* David Jao, Sep 07 2020 *)

PROG

(PARI)

A000003(n) = qfbclassno(-4*n);

A000089(n) = {

if (n%4 == 0 || n%4 == 3, return(0));

if (n%2 == 0, n \= 2);

my(f = factor(n), fsz = matsize(f)[1]);

prod(k = 1, fsz, if (f[k, 1] % 4 == 3, 0, 2));

};

A000086(n) = {

if (n%9 == 0 || n%3 == 2, return(0));

if (n%3 == 0, n \= 3);