Special parameters

• Categories: Math Notes

• An old paper (Über hypergeometrische Funktionen, deren letztes Element speziell ist) by W. Heymann (from a talk by Wadim Zudilin) discusses the value of an ${}_2F_1$ at special $t$’s. He does $t=-1/3,1/4,1/5$ and related values and mentions Gauss for $t=1,-1,1/2$ and Kummer for $t=-1/8,1/9,8/9$.

• For example we have this identity

  $$ F_h = F \left( -\frac{h}{2}, -\frac{h}{2} + \frac{1}{2}, h + \frac{3}{2}, -\frac{1}{3} \right) = \frac{2}{3} \cdot \left( \frac{8}{9} \right)^h \cdot \frac{\Gamma \left( \frac{1}{3} \right) \cdot \Gamma \left( h + \frac{3}{2} \right)}{\sqrt{\pi} \cdot \Gamma \left( h + \frac{4}{3} \right)} $$

  If we take $h=1/2$ we get an HGM defined over $\mathbb Q$, we get the gamma vector $[-4,1,1,2]$ and corresponding Weierstrass model

  $$ y^2+xy=x^3+\frac t{64}x. $$

  For $t=-1/3$ we get a curve with $j$-invariant $j(\sqrt{-3})=54000$ and the Frobenius traces match. We recognize the rhs as a Jacobi motive twisted by ${\mathbb Q}(\sqrt{2})$.

• With MAGMA

> J:=JacobiMotive([1/3],[1/2,1/2+1/3]);
ORDER_IS_SUBFIELD: special root
> J:=TateTwist(J,-1);
ORDER_IS_SUBFIELD: special root
> pv:=PrimesInInterval(5,100);
> [EulerFactor(J,p): p in pv];
[
    5*x^2 + 1,
    7*x^2 - 4*x + 1,
    11*x^2 + 1,
    13*x^2 - 2*x + 1,
    17*x^2 + 1,
    19*x^2 + 8*x + 1,
    23*x^2 + 1,
    29*x^2 + 1,
    31*x^2 - 4*x + 1,
    37*x^2 + 10*x + 1,
    41*x^2 + 1,
    43*x^2 + 8*x + 1,
    47*x^2 + 1,
    53*x^2 + 1,
    59*x^2 + 1,
    61*x^2 - 14*x + 1,
    67*x^2 - 16*x + 1,
    71*x^2 + 1,
    73*x^2 + 10*x + 1,
    79*x^2 - 4*x + 1,
    83*x^2 + 1,
    89*x^2 + 1,
    97*x^2 - 14*x + 1
]

• With GP

? heymanntr(p,h)=hgm(-1/3,[-h/2,-h/2+1/2],[h+3/2,1],p)
? forprime(p=5,100,print(p,"  ",recognizep(heymanntr(p,1/2)*kronecker(2,p))))
5  0
7  4
11  0
13  2
17  0
19  -8
23  0
29  0
31  4
37  -10
41  0
43  -8
47  0
53  0
59  0
61  14
67  16
71  0
73  -10
79  4
83  0
89  0
97  14

• For $t=4/5$ Heymann has

  $$ F\left( \frac{h}{2}, \frac{h}{2} + \frac{1}{2}, 3h, \frac{4}{5} \right) = k \cdot \left( \frac{27}{25} \right)^h \cdot \frac{\Gamma \left( h + \frac{1}{3} \right) \Gamma \left( h + \frac{2}{3} \right)}{\Gamma \left( h + \frac{2}{5} \right) \Gamma \left( h + \frac{3}{5} \right)}, \qquad h + \frac{1}{3} > 0, $$ $$ F\left( \frac{h}{2}, \frac{h}{2} + \frac{1}{2}, 3h - 1, \frac{4}{5} \right) = k' \cdot \left( \frac{27}{25} \right)^h \cdot \frac{\Gamma \left( h - \frac{1}{3} \right) \Gamma \left( h + \frac{1}{3} \right)}{\Gamma \left( h - \frac{1}{5} \right) \Gamma \left( h + \frac{1}{5} \right)}, \qquad h - \frac{1}{3} > 0. $$

  For $h=1/2$ we get the Artin HGM $[[1/4,3/4],[1/2,1]]$ with gamma vector $[-4,2,2]$, which has geometric model the Belyi equation

  $$ x^2(1-x)^2-2^2\cdot2^2/4^4t. $$

  For $t=4/5$ we get the polynomial

  $$ x^4 - 2x^3 + x^2 - 1/20 $$

  which gives the cyclotomic field ${\mathbb Q}(\zeta_{20})^+$ with Galois group cyclic of order four. The rhs of the formula corresponds to the Jacobi motive

  $$ [[1/2+1/3,1/2+2/3],[1/2+2/5,1/2+3/5]] $$

  twisted by the quadratic character of ${\mathbb Q}(\sqrt 3)$, times possibly a contribution from the unspecified constant $k$ in the formula (we compute it below).

• In MAGMA

> J:=JacobiMotive([1/2+1/3,1/2+2/3],[1/2+2/5,1/2+3/5]);
> pv:=PrimesInInterval(7,100);
> [[p,EulerFactor(J,p)]: p in pv];
[
    [
        7,
        -T^2 + 1
    ],
    [
        11,
        T^2 - 2*T + 1
    ],
    [
        13,
        -T^2 + 1
    ],
    [
        17,
        -T^2 + 1
    ],
    [
        19,
        T^2 - 2*T + 1
    ],
    [
        23,
        -T^2 + 1
    ],
    [
        29,
        T^2 - 2*T + 1
    ],
    [
        31,
        T^2 - 2*T + 1
    ],
    [
        37,
        -T^2 + 1
    ],
    [
        41,
        T^2 - 2*T + 1
    ],
    [
        43,
        -T^2 + 1
    ],
    [
        47,
        -T^2 + 1
    ],
    [
        53,
        -T^2 + 1
    ],
    [
        59,
        T^2 - 2*T + 1
    ],
    [
        61,
        T^2 - 2*T + 1
    ],
    [
        67,
        -T^2 + 1
    ],
    [
        71,
        T^2 - 2*T + 1
    ],
    [
        73,
        -T^2 + 1
    ],
    [
        79,
        T^2 - 2*T + 1
    ],
    [
        83,
        -T^2 + 1
    ],
    [
        89,
        T^2 - 2*T + 1
    ],
    [
        97,
        -T^2 + 1
    ]
]

BTW, we get exactly the same Euler factors if we take the conjugate motive (with parameters $[[1/2+1/3,1/2+2/3],[1/2+1/5,1/2+4/5]]$)

• With GP

? heymantr1(p)=hgm(4/5,[-1/4,1/4],[1/2,1],p)
? forprime(p=7,100,print(p,"  ",recognizep(heymantr1(p))))

 7  0
11  -2
13  0
17  0
19  2
23  0
29  -2
31  -2
37  0
41  2
43  0
47  0
53  0
59  2
61  2
67  0
71  -2
73  0
79  2
83  0
89  -2
97  0

The traces agree up to a sign, which does not seem to be a character.

Checking directly against the number field in GP

? f=x^4 - 2*x^3 + x^2 - 1/20
? nf=nfinit(f);
  *** nfinit: Warning: nonmonic polynomial. Result of the form [nf,c].
? forprime(p=7,100,print(p,"   ",nfeulerfactor(nf,p)/(1-x)/(1-kronecker(20,p)*x)))
7   x^2 + 1
11   x^2 + 2*x + 1
13   x^2 + 1
17   x^2 + 1
19   x^2 - 2*x + 1
23   x^2 + 1
29   x^2 + 2*x + 1
31   x^2 + 2*x + 1
37   x^2 + 1
41   x^2 - 2*x + 1
43   x^2 + 1
47   x^2 + 1
53   x^2 + 1
59   x^2 - 2*x + 1
61   x^2 - 2*x + 1
67   x^2 + 1
71   x^2 + 2*x + 1
73   x^2 + 1
79   x^2 - 2*x + 1
83   x^2 + 1
89   x^2 + 2*x + 1
97   x^2 + 1

This matches the HGM trace.

We can compute the constant $k$ numerically.

? ff=hypseries([1/4,3/4],[1/2,1],500);
? a=subst(truncate(ff),x,4/5.)
1.9021130325903071442328786667587642868
? b=gamma(1/2+1/3)*gamma(1/2+2/3)/gamma(1/2+2/5)/gamma(1/2+3/5)
1.0300566479164914136743113906093968629
? c=a/b
1.8466101223051520415421743456221458944
? algdep(c*5/3,4)
x^4 - 10*x^2 + 5

This constant is in the same field ${\mathbb Q}(\zeta_{20})^+$.

Actually the other constants are also algebraic in the same field.

? algdep(a,4)
x^4 - 5*x^2 + 5
? algdep(b*3/5,2)
x^2 + x - 1

• Math
• HGM