A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds
We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The p-th coefficients a(p) of the corresponding modular form can often be read off, at least conjecturally, from the truncated partial sums of the underlying hypergeom...
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| Цитувати: | A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds / W. Zudilin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. |
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| author_facet | Zudilin, W. |
| citation_txt | A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds / W. Zudilin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The p-th coefficients a(p) of the corresponding modular form can often be read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of p and from Weil's general bounds |a(p)| ≤ 2p⁽ᵐ⁻¹⁾/², where m is the weight of the form. Furthermore, the critical L-values of the modular form are predicted to be Q-proportional to the values of a related basis of solutions to the hypergeometric differential equation.
|
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 086, 16 pages
A Hypergeometric Version
of the Modularity of Rigid Calabi–Yau Manifolds
Wadim ZUDILIN †‡§
† Department of Mathematics, IMAPP, Radboud University,
PO Box 9010, 6500 GL Nijmegen, The Netherlands
E-mail: w.zudilin@math.ru.nl
URL: http://www.math.ru.nl/~wzudilin/
‡ School of Mathematical and Physical Sciences, The University of Newcastle,
Callaghan, NSW 2308, Australia
E-mail: wadim.zudilin@newcastle.edu.au
§ Laboratory of Mirror Symmetry and Automorphic Forms,
National Research University Higher School of Economics,
6 Usacheva Str., 119048 Moscow, Russia
E-mail: wzudilin@gmail.com
Received May 03, 2018, in final form August 13, 2018; Published online August 17, 2018
https://doi.org/10.3842/SIGMA.2018.086
Abstract. We examine instances of modularity of (rigid) Calabi–Yau manifolds whose
periods are expressed in terms of hypergeometric functions. The p-th coefficients a(p) of the
corresponding modular form can be often read off, at least conjecturally, from the truncated
partial sums of the underlying hypergeometric series modulo a power of p and from Weil’s
general bounds |a(p)| ≤ 2p(m−1)/2, where m is the weight of the form. Furthermore, the
critical L-values of the modular form are predicted to be Q-proportional to the values of
a related basis of solutions to the hypergeometric differential equation.
Key words: hypergeometric equation; bilateral hypergeometric series; modular form; Calabi–
Yau manifold
2010 Mathematics Subject Classification: 11F33; 11T24; 14G10; 14J32; 14J33; 33C20
To Noriko Yui, with wishes to count more points
on algebraic varieties rather than years!
1 A prototype
In [32] L. van Hamme stated some supercongruence analogues of Ramanujan’s formulas. The
very last observation on van Hamme’s list, Conjecture (M.2) (stated here in an equivalent form),
does not seem to be linked to a known formula though:
p−1∑
k=0
(12)4k
k!4
≡ a(p)
(
mod p3
)
, (1)
where a(n) denote the Fourier coefficients of the unique cusp (eigen) form of weight 4 on Γ0(8),
f(τ) =
∞∑
n=1
a(n)qn = η(2τ)4η(4τ)4 = q
∞∏
m=1
(
1− q2m
)4(
1− q4m
)4
. (2)
This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko
Yui. The full collection is available at http://www.emis.de/journals/SIGMA/modular-forms.html
mailto:w.zudilin@math.ru.nl
http://www.math.ru.nl/~wzudilin/
mailto:wadim.zudilin@newcastle.edu.au
mailto:wzudilin@gmail.com
https://doi.org/10.3842/SIGMA.2018.086
http://www.emis.de/journals/SIGMA/modular-forms.html
2 W. Zudilin
Here and below we use the standard hypergeometric notation including (r)k = Γ(r+ k)/Γ(r) =
k−1∏
j=0
(r + j) for Pochhammer’s symbol; also the congruence c1 ≡ c2 (mod p`) for two rational
numbers is understood as c1−c2 ∈ p`Zp. The conjecture (1) was later established by T. Kilbourn
in [14] built on an earlier work of S. Ahlgren and K. Ono in [1] on the modularity of the Calabi–
Yau threefold
4∑
j=1
(
xj + x−1j
)
= 0.
Interestingly enough, the work of Ahlgren and Ono was motivated by proving a different
family of supercongruences for the Apéry numbers
A(n) =
∞∑
k=0
(
n
k
)2(n+ k
k
)2
= 4F3
(
−n, −n, n+ 1, n+ 1
1, 1, 1
∣∣∣∣ 1
)
=
n∑
k=0
(
n
k
)2(n+ k
k
)2
for n = 0, 1, 2, . . .
conjectured by F. Beukers in [3] and established modulo p there:
A
(
p− 1
2
)
≡ a(p)
(
mod p2
)
. (3)
It is not hard to observe that
A
(
p− 1
2
)
=
(p−1)/2∑
k=0
(1−p2 )2k(
1+p
2 )2k
k!4
≡
(p−1)/2∑
k=0
(12)4k
k!4
≡
p−1∑
k=0
(12)4k
k!4
(
mod p2
)
,
so that (3) follows from (1). On the other hand, the Apéry sequence and the modular paramet-
rization of its generating series
∞∑
n=0
A(n)zn gives one a natural way to construct the right-hand
side of (3) (namely, the eigenform (2) whose Fourier coefficients show up) modulo p. This
construction is performed in [3] and nicely explained in a certain generality in [33]. More
recently, V. Golyshev and D. Zagier [34, Section 7] show that the p-adic interpolation of the
coefficients a(p) of the newform f(τ) = η(2τ)4η(4τ)4 is part of a much more general picture
that, in particular, predicts that
A(−1/2) = 4F3
(
1
2 ,
1
2 ,
1
2 ,
1
2
1, 1, 1
∣∣∣∣ 1
)
=
∞∑
k=0
(12)4k
k!4
is rationally proportional to L(f, 2)/π2, where L(f, s) denotes the L-function of the modular
form. Furthermore, they prove [34] that
4F3
(
1
2 ,
1
2 ,
1
2 ,
1
2
1, 1, 1
∣∣∣∣ 1
)
=
16L(f, 2)
π2
, (4)
the identity which was independently established in [23] via a systematic expressing of critical
L-values attached to cuspidal η-products through hypergeometric functions. Note that the
identity (4) is the missing non-p-adic counterpart (M.1) of Conjecture (M.2) from [32]; the
latest edition of van Hamme’s list can be found in [31] together with the details about proofs.
One of the principal results in [1] is a summation formula for Greene’s hypergeometric func-
tion, which serves as a finite-field analogue of the classical hypergeometric series given in (4).
Curiously enough, R. Evans in his review [7] of [1] mentions that no summation formula is known
Modularity of Calabi–Yau Manifolds 3
for this 4F3-value in (4); the evaluation (4) established in [23, 34] thus fills in this gap in the
hypergeometric literature.
A principal goal of this note is to put the pair (1), (4) in a broader context of relationship
between classical generalized hypergeometric functions and the L-values of modular forms. This
is performed here more in the spirit of Golyshev’s gamma structures [10] rather than hyper-
geometric motives [20, 22] of F. Rodriguez Villegas and others. At the same time, we do not
pretend to be too broad in our exposition, mainly highlighting certain specific arithmetic and
analytical perspectives which we find aesthetically appealing.
2 Modularity of Calabi–Yau threefolds
The Calabi–Yau threefold in Section 1 comes as a part of the complete intersection of four
degree 2 surfaces in P8; the periods of the latter family of threefolds satisfy the hypergeometric
equation whose unique analytical solution is
4F3
(
1
2 ,
1
2 ,
1
2 ,
1
2
1, 1, 1
∣∣∣∣ z) =
∞∑
k=0
(12)4k
k!4
zk.
Namely, the fiber z = 1 corresponds to the rigid Calabi–Yau threefold
4∑
j=1
(
xj + x−1j
)
= 0.
There are fourteen ‘hypergeometric’ families of Calabi–Yau threefolds whose periods are
solutions of hypergeometric equations with parameters (r, 1− r, t, 1− t), where
(r, t) =
(
1
2 ,
1
2
)
,
(
1
2 ,
1
3
)
,
(
1
2 ,
1
4
)
,
(
1
2 ,
1
6
)
,
(
1
3 ,
1
3
)
,
(
1
3 ,
1
4
)
,
(
1
3 ,
1
6
)
,(
1
4 ,
1
4
)
,
(
1
4 ,
1
6
)
,
(
1
6 ,
1
6
)
,
(
1
5 ,
2
5
)
,
(
1
8 ,
3
8
)
,
(
1
10 ,
3
10
)
,
(
1
12 ,
5
12
)
,
and the modularity from Section 1 is expected to be extendable to all families as follows.
Observation 1. Let a pair (r, t) be from the list. For a prime p not dividing the denominators
of r and t, define a(p) to be the smallest (in absolute value) integer residue modulo p3 of the
partial sum
p−1∑
k=0
(r)k(1− r)k(t)k(1− t)k
k!4
of the hypergeometric series
4F3
(
r, 1− r, t, 1− t
1, 1, 1
∣∣∣∣ 1
)
=
∞∑
k=0
(r)k(1− r)k(t)k(1− t)k
k!4
.
Then |a(p)| ≤ 2p3/2 and a(p) are the Fourier coefficients of a suitable eigenform f(τ) = q +
a(2)q2 + · · · of weight 4 for some congruence subgroup of PSL2(Z).
Furthermore, introduce a special (normalized Frobenius) basis of solutions of the differential
equation for
F0(z) = 4F3
(
r, 1− r, t, 1− t
1, 1, 1
∣∣∣∣ z)
as the first coefficients in the Taylor ε-expansion of the (bilateral) hypergeometric function
1
Γ(r)Γ(1− r)Γ(t)Γ(1− t)
×
∞∑
n=−∞
Γ(r + ε+ n)Γ(1− r + ε+ n)Γ(t+ ε+ n)Γ(1− t+ ε+ n)
Γ(1 + ε+ n)4
zn+ε
4 W. Zudilin
=
1
Γ(r)Γ(1− r)Γ(t)Γ(1− t)
×
∞∑
n=0
Γ(r + ε+ n)Γ(1− r + ε+ n)Γ(t+ ε+ n)Γ(1− t+ ε+ n)
Γ(1 + ε+ n)4
zn+ε +O
(
ε4
)
= F0(z) + F1(z)ε+ F2(z)ε
2 + F3(z)ε
3 +O
(
ε4
)
as ε→ 0. (5)
Then numerical calculations suggest conjectural inclusions
L(f, 1)
F1(1)
∈ Q,
L(f, 2)
F2(1)
∈ Q and
L(f, 3)
F3(1)
∈ Q. (6)
Remark 1. Observation 1 contains an explicit algorithm for reconstructing the Hecke eigenval-
ues a(p), so it is straighforward to compute them numerically for good primes p from the partial
sums. This supercongruence part has been already exploited by F. Rodriguez Villegas in [21]
who noticed that the truncated hypergeometric sums are congruent to a(p) modulo p3 and used
this fact to identify the corresponding eigenforms f(τ) and their levels. The knowledge of Hecke
eigenvalues a(p) allows one to reconstruct all Fourier coefficients of f(τ) =
∞∑
n=1
a(n)qn from the
Euler product of the L-function L(f, s) =
∞∑
n=1
a(n)n−s. Missing finitely many a(p) in the Euler
product has no effect on the inclusions (6).
Table 1. Eigenforms for rigid Calabi–Yau manifolds.
(r, t) f(τ) level LMFDB label [15](
1
2 ,
1
2
)
η42η
4
4 8 = 23 8.4.1.a(
1
2 ,
1
3
)
η146 /
(
η32η
3
18
)
− 3η32η
2
6η
3
18 36 = 22 · 32 36.4.1.a(
1
2 ,
1
4
)
η164 /
(
η42η
4
8
)
16 = 24 16.4.1.a(
1
2 ,
1
6
)
72 = 23 · 32 72.4.1.b(
1
3 ,
1
3
)
η31η
4
3η9 − 27η3η
4
9η
3
27 27 = 33 27.4.1.a(
1
3 ,
1
4
)
η83 9 = 32 9.4.1.a(
1
3 ,
1
6
)
108 = 22 · 33 108.4.1.a(
1
4 ,
1
4
)
η104 /η
2
8 − 8η108 /η
2
4 32 = 25 32.4.1.a(
1
4 ,
1
6
)
η3212/
(
η126 η
12
24
)
+ 16η46η
4
24 144 = 24 · 32(
1
6 ,
1
6
)
216 = 23 · 33(
1
5 ,
2
5
)
η105 /(η1η25) + 5η21η
4
5η
2
25 25 = 52 25.4.1.b(
1
8 ,
3
8
)
128 = 27(
1
10 ,
3
10
)
200 = 23 · 52(
1
12 ,
5
12
)
864 = 25 · 33
Remark 2. The prediction about the relationship between the critical L-values and the hyper-
geometric values F1(1), F2(1), F3(1) is due to V. Golyshev, and it is a part of general phe-
nomenon. The fact that the coefficients Fj(z) are solutions of the hypergeometric equation
for F0(z) is established in [10, Section 3]; we survey some information about this from a ‘hyper-
geometric’ perspective in Section 3. None of the relations in (6) seem to be proved.
Accidentally, when r = t = 1
2 , we have an extra rational relation F0(1) = F2(1)/
(
2π2
)
, and
it is this equality that originates the anticipated equality (4) (rigorously established!). It is the
only case when F0(1) is linearly dependent over Q with Fj(1)/πj for j = 1, 2, 3.
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/8/4/1/a/
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/36/4/1/a/
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/16/4/1/a/
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/72/4/1/b/
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/27/4/1/a/
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/9/4/1/a/
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/108/4/1/a/
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/32/4/1/a/
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/25/4/1/b/
Modularity of Calabi–Yau Manifolds 5
Remark 3. The case (r, t) =
(
1
3 ,
1
4
)
in Observation 1 corresponds to a particularly simple CM
modular form of level 9, namely, to f(τ) = η(3τ)8. Its critical L-values possess closed-form
evaluation
L
(
η(3τ)8, 2
)
=
Γ(1/3)9
96π4
and L
(
η(3τ)8, 3
)
=
Γ(1/3)9
144
√
3π3
(the strategy for this computation is set up in Damerell’s work [6]). All 14 cases correspond to
rigid Calabi–Yau threefolds defined over Q and hence they do correspond to modular forms of
weight 4 for some congruence subgroups of PSL2(Z). Table 1 records the instances of modular
forms, for which we know their eta-product expressions; the notation ηm stands for η(mτ).
Remark 4. The eigenform f(τ) in Observation 1, namely, the eigenvalues a(p), are related to
the counting of points modulo p on the (rigid) Calabi–Yau threefold corresponding to z = 1
in the family. This counting naturally leads to representations of a(p) by means of finite-
field hypergeometric functions – due to J. Greene [11], D. McCarthy [18] and, in a greater
generality, F. Beukers, H. Cohen, A. Mellit [4] – the representations that are used in the proof
of Observation 1 in the case r = t = 1
2 . All 14 cases in the observation, namely the modulo p3
supercongruences, are now proved simultaneously and rigorously in the joint paper [17] with
L. Long, F.-T. Tu and N. Yui.
3 Bilateral hypergeometric functions and hypertrigonometry
In this section we will examine the bilateral hypergeometric sum
mHm
(
a1, . . . , am
b1, . . . , bm
∣∣∣∣ z; ε) =
m∏
j=1
Γ(bj)
m∏
j=1
Γ(aj)
∞∑
n=−∞
m∏
j=1
Γ(aj + ε+ n)
m∏
j=1
Γ(bj + ε+ n)
zn+ε (7)
from both classical [28, Chapter 6] and recent [10] perspectives. For fixed ε ∈ C (different from
the poles of the gamma functions Γ(aj + ε + n)) and a generic set of complex parameters aj
and bj , j = 1, . . . ,m, satisfying
Re(b1 + · · ·+ bm) > Re(a1 + · · ·+ am)
the defining series converges on the unit circle |z| = 1. Our principal interest will be in the case
b1 = · · · = bm = 1. On using(
z
d
dz
+ a
)
zn+ε = (a+ ε+ n)zn+ε
and the basic property of the gamma function we arrive at the following.
Lemma 1. The function (7) satisfies the (linear differential) hypergeometric equationz m∏
j=1
(
z
d
dz
+ aj
)
−
m∏
j=1
(
z
d
dz
+ bj − 1
)mHm(z; ε) = 0 (8)
on the circle |z| = 1.
6 W. Zudilin
The function (7) can be analytically continued from the unit circle to the C-plane with
cuts along the real intervals (−∞, 0] and [1,+∞) by relating it to the bilateral hypergeometric
function [28, equation (6.1.2.3)],
mHm
(
a1, . . . , am
b1, . . . , bm
∣∣∣∣ z) =
∞∑
n=−∞
(a1)n · · · (am)n
(b1)n · · · (bm)n
czn
= m+1Fm
(
1, a1, . . . , am
b1, . . . , bm
∣∣∣∣ z)
+
(b1 − 1) · · · (bm − 1)
(a1 − 1) · · · (am − 1)
m+1Fm
(
1, 2− b1, . . . , 2− bm
2− a1, . . . , 2− am
∣∣∣∣ 1
z
)
,
where the (extended to negative) Pochhammer symbol is
(a)n =
Γ(a+ n)
Γ(a)
=
1 if n = 0,
a(a+ 1) · · · (a+ n− 1) if n > 0,
1
(a− 1)(a− 2) · · · (a− (−n))
if n < 0.
Lemma 2 (see also [10]). As function of z, the function (7) is continued analytically to C \
(−∞, 0] ∪ [1,+∞) by means of the hypergeometric functions as follows:
mHm
(
a1, . . . , am
b1, . . . , bm
∣∣∣∣ z; ε) =
zε
m∏
j=1
Γ(aj + ε) Γ(bj)
m∏
j=1
Γ(aj) Γ(bj + ε)
{
m+1Fm
(
1, a1 + ε, . . . , am + ε
b1 + ε, . . . , bm + ε
∣∣∣∣ z)
+
m∏
j=1
bj + ε− 1
aj + ε− 1
m+1Fm
(
1, 2− b1 − ε, . . . , 2− bm − ε
2− a1 − ε, . . . , 2− am − ε
∣∣∣∣ z−1)},
and the analytic continuation satisfies the hypergeometric equation (8).
In particular, the lemma implies that
1
m∏
j=1
Γ(aj)
∞∑
n=−∞
m∏
j=1
Γ(aj + ε+ n)
Γ(1 + ε+ n)m
zn+ε
=
zε
m∏
j=1
Γ(aj + ε)
Γ(1 + ε)m
m∏
j=1
Γ(aj)
m+1Fm
(
1, a1 + ε, . . . , am + ε
1 + ε, . . . , 1 + ε
∣∣∣∣ z)+O
(
εm
)
,
the reduction we used in computation (5) of Section 2.
Finally, notice that the sum in (7) is invariant under the shifts of ε by integers, and the
principal result of [10] can be stated as follows.
Lemma 3. As function of ε, the function (7) is periodic with period 1. Furthermore, its nor-
malization
m∏
j=1
sinπ(aj + ε)× mHm
(
a1, . . . , am
b1, . . . , bm
∣∣∣∣ z; ε) (9)
Modularity of Calabi–Yau Manifolds 7
is a C-linear combination of eπikε, where |k| ≤ m and k ≡ m (mod 2). This means that the
Fourier expansion of the latter function is a finite Fourier polynomial, whose coefficients depend
only on z.
Proof. Using the reflection property of the gamma function we find
Γ(a+ ε+ n) =
π
sinπ(a+ ε+ n)
1
Γ(1− a− ε− n)
=
π
sinπ(a+ ε)
(−1)n
Γ(1− a− ε− n)
,
so that
mHm
(
a1, . . . , am
b1, . . . , bm
∣∣∣∣ z; ε) =
zεπm
m∏
j=1
Γ(bj)
m∏
j=1
Γ(aj) sinπ(aj + ε)
×
∞∑
n=−∞
(−1)mnzn
m∏
j=1
Γ(1− aj − ε− n)Γ(bj + ε+ n)
.
It remains to notice that the functions
1
m∏
j=1
Γ(1− aj − ε− n)Γ(bj + ε+ n)
are entire and estimate their growth as ε→∞ (see [10, Theorem 1.5]). �
Remark 5. Though Lemma 3 (and the estimates from [10]) guarantee that at most m+1 terms
show up in the Fourier expansion of (9), in reality one does not get the term e−πimε (or eπimε)
when Re z > 0 (or Re z < 0, respectively). In the case when z is real from the interval 0 < z < 1,
we still need to specify along which bank of the real line we proceed; for convenience, from now
on we agree to use the upper bank.
Even more, if z = 1 and the corresponding bilateral hypergeometric series converge at this
special point then the both terms e−πimε and eπimε in the Fourier expansion of (9) do not show
up. This allows one to rigorously establish that F1(1) and F3(1)/π2 are rationally proportional –
something that could follow from (6) complemented with the Manin–Shimura relation of the
critical L-values [25, 26].
4 A hypergeometric modularity of elliptic curves
Probably, the most classical version of the observation above refers to the modularity of elliptic
curves (that is, Calabi–Yau onefolds). Our principal illustration will deal with the family
Ez : y2 = x(1− x)(x− z), z ∈ C \ {0, 1,∞},
which is a twist of the classical Legendre family of elliptic curves
Êz : y2 = x(x− 1)(x− z), z ∈ C \ {0, 1,∞}.
In fact, performing the change x 7→ 1− x we see that the curves E1−z and Êz are isomorphic.
Let p be an odd prime and z ∈ Q be p-integral not equal to 0 or 1. By Hasse’s theorem [27,
Theorem V.1.1] the number of points on the curve Êz/Fp satisfies∣∣#(Êz/Fp)− (p+ 1)
∣∣ ≤ 2
√
p.
8 W. Zudilin
On the other hand, it follows from the proof of Theorem V.4.1(b) in [27] that
#
(
Êz/Fp
)
− 1 ≡ (−1)(p−1)/2
(p−1)/2∑
k=0
(
(p− 1)/2
k
)2
zk (mod p)
≡ (−1)(p−1)/2
(p−1)/2∑
k=0
(12)2k
k!2
zk ≡ (−1)(p−1)/2
p−1∑
k=0
(12)2k
k!2
zk (mod p).
By combining the two results above we conclude that the integer â(p) = â(p; z) = #(Êz/Fp)−
(p+ 1) satisfies Weil’s bound |â(p)| ≤ 2
√
p and the congruence
â(p) ≡
(
−4
p
) p−1∑
k=0
(12)2k
k!2
zk (mod p),
where
(−4
·
)
denotes the quadratic character modulo 4. By the modularity theorem the num-
bers â(p) build up to the L-function of the elliptic curve Êz,
L
(
Êz, s
)
=
∏
p
(
1− â(p)p−s + εpp
1−2s)−1 =
∞∑
n=1
â(n)
ns
, εp ∈ {0, 1}.
Furthermore, the central (critical) value of L
(
Êz, s
)
is rationally proportional to a period of the
curve Êz, namely, to the period
Re
∫ ∞
1
dx√
x(x− 1)(x− z)
= Re
∫ 1
0
dt√
t(1− t)(1− zt)
= π Re 2F1
(
1
2 ,
1
2
1
∣∣∣∣ z),
where we made the change of variable x = 1/t in the former integral. The real part can be
omitted when z < 1.
In order to state the above for the family of elliptic curves Ez ' Ê1−z we notice first that the
above calculation of the Hasse invariant from [27] implies the congruence(
−4
p
) p−1∑
k=0
(12)2k
k!2
zk ≡
p−1∑
k=0
(12)2k
k!2
(1− z)k (mod p). (10)
Second, writing for a real r in the range 0 < r < 1,
F (z; ε) =
1
Γ(r) Γ(1− r)
∞∑
n=−∞
Γ(r + ε+ n) Γ(1− r + ε+ n)
Γ(1 + ε+ n)2
zn+ε
=
1
Γ(r)Γ(1− r)
∞∑
n=0
Γ(r + ε+ n)Γ(1− r + ε+ n)
Γ(1 + ε+ n)2
zn+ε +O
(
ε2
)
= F0(z) + F1(z)ε+O
(
ε2
)
as ε→ 0, (11)
where
F0(z) = 2F1
(
r, 1− r
1
∣∣∣∣ z),
and applying the monodromy of the hypergeometric function we obtain
F1(z) = −Γ(r)Γ(1− r)F0(1− z) = − π
sinπr
F0(1− z).
Modularity of Calabi–Yau Manifolds 9
This relation is valid in the cut C-plane C \ (−∞, 0]∪ [1,∞) but also along the respective banks
of the cuts; in particular, for the real parts, the identity
ReF1(z) = − π
sinπr
ReF0(1− z) (12)
is true for any complex z 6= 0, 1. Using (12) with r = 1
2 we can summarize our findings as
follows.
Observation 2. Let p > 2 be a prime not dividing the denominator of a given z ∈ Q \ {0, 1}.
Define the integer a(p) = a(p; z) as the absolutely smallest residue modulo p of the partial sum
p−1∑
k=0
(12)2k
k!2
zk
(so that −p/2 < a(p) < p/2) of the hypergeometric function
F0(z) = 2F1
(
1
2 ,
1
2
1
∣∣∣∣ z) =
∞∑
k=0
(12)2k
k!2
zk.
Then the number satisfies Weil’s estimate |a(p)| < 2
√
p.
Furthermore, form the associated L-function
L(z, s) =
∏
p
(
1− a(p)p−s + p1−2s
)−1
=
∞∑
n=1
a(n)
ns
,
where the product is over primes p > 2 that do not divide the denominator of z. Then
L(z, 1)
ReF1(z)
= − L(z, 1)
π ReF0(1− z)
∈ Q, (13)
where F1(z) originates from the ε-expansion (11).
Note that a(p) constructed in Observation 2 may in fact differ, by a multiple of p, from the
p-th Fourier coefficient of the modular form associated with Ez for the range p ≤ 13. However
the change (or omission) of finite set of factors in the product defining L(Ez, s) contributes by
a nonzero rational factor in L(z, 1), so that relation (13) is seen to be equivalent to
L(Ez, 1)
ReF1(z)
= − L(Ez, 1)
π ReF0(1− z)
∈ Q.
We also stress on the fact that L(Ez, 1), therefore L(z, 1) in (13), vanishes when the (analytic)
rank of the elliptic curve Ez is positive. In such situations, numerics suggests no relation between
the hypergeometric functions F0(z), F1(z) in question and the first nonzero derivative of L(Ez, s)
(or of L(z, s)) at s = 1.
A similar analysis applies to three other classical hypergeometric series
2F1
(
r, 1− r
1
∣∣∣∣ z) =
∞∑
k=0
(r)k(1− r)k
k!2
zk, where r ∈
{
1
3 ,
1
4 ,
1
6
}
. (14)
They are known to represent the periods of suitable families of elliptic curves, for example, of
the pencils of elliptic curves
X2Y + Y 2Z + Z2X = z1/3XY Z,
X4 + Y 2 + Z4 = z1/4XY Z and X3 + Y 2 + Z6 = z1/6XY Z,
10 W. Zudilin
respectively, in weighted projective planes [29]. The corresponding Weierstrass forms are
y2 = x3 − 3(9− 8z)x+ 2
(
27− 36z + 8z2
)
,
y2 = x3 − 27(1 + 3z)x+ 54(1− 9z) and y2 = x3 − 27x+ 54(1− 2z).
Observation 3. Take r ∈
{
1
3 ,
1
4 ,
1
6
}
and z ∈ Q \ {0, 1}. Let p be a prime not dividing the
denominators of r and z. Define the integer a(p) = a(p; r, z) as the absolutely smallest residue
modulo p of the partial sum
p−1∑
k=0
(r)k(1− r)k
k!2
zk
of the hypergeometric function (14). Then the number satisfies Weil’s estimate |a(p)| < 2
√
p.
Form the associated L-function
L(z, s) =
∏
p
(
1− a(p)p−s + p1−2s
)−1
=
∞∑
n=1
a(n)
ns
,
where the product is over primes p that do not divide the denominators of r and z. Then
L(z, 1)
ReF1(z)
= − L(z, 1)
Γ(r)Γ(1− r) ReF0(1− z)
∈ Q,
where F1(z) originates from the corresponding ε-expansion (11).
We can also point out the symmetry property a(p; r, z) = χ(p)a(p; r, 1 − z) valid for r ∈{
1
2 ,
1
3 ,
1
4 ,
1
6
}
(see (10) for r = 1
2) and all admissible primes p with the corresponding choice of
the quadratic character
χ( · ) =
(
−4
·
)
,
(
−3
·
)
,
(
−2
·
)
or
(
−4
·
)
for r =
1
2
,
1
3
,
1
4
,
1
6
, respectively.
Remark 6. With each modular form f(τ) of integral weight at least 2 one can canonically
associate two periods ω− and ω+. When the weight higher than 2 shows up, and these are
examples from Section 2 above and Section 5 below, the critical L-values L(f,m)/πm represent
the both periods ω− and ω+ of the modular form, so that twisting the Hecke eigenvalues a(p)
by an odd character is equivalent to changing the parity of m or swapping the periods. This is
an immediate consequence of the Manin–Shimura description of the critical L-values [25, 26].
In situations covered in this section the modular forms f(τ) have weight 2; thus, the symmetry
a(p; r, z) = χ(p)a(p; r, 1 − z) under the involution z 7→ 1 − z displays the interchange of the
periods ω− and ω+ on the corresponding elliptic curve in the family.
The potentials of the hypergeometric description of the modularity are at least two-fold. First,
they provide us with a new class of summation theorems for arithmetic instances of classical
Euler–Gauss hypergeometric function (cf. [35]). Second, they allow one to deal with elliptic
curves defined over algebraic extensions of Q as the hypergeometric machinery works for not
necessarily rational z, at least formally.
5 Other modularity instances
One interesting message coming from Observation 1 is that z = 1 always corresponds to a rigid
Calabi–Yau threefold in each hypergeometric family. Note that z = 1 happens to be a singular
Modularity of Calabi–Yau Manifolds 11
point of the related hypergeometric differential equation, so an expectation is that Observation 1
can be suitably extended to some non-hypergeometric families and the Calabi–Yau manifolds cor-
responding to some singularities of the underlying Picard–Fuchs differential equations. But rigid
Calabi–Yau manifolds can correspond to non-singular points z as the observations in Section 4
demonstrate. We can also record vaguely the following observation about potential instances of
the modularity of Calabi–Yau twofolds (that is, K3 surfaces with Picard rank 20), where some
non-singular points show up.
Observation 4. Let r ∈
{
1
2 ,
1
3 ,
1
4 ,
1
6
}
and let rational z be 1 or ‘arithmetically special’ (that
is, corresponding to CM cases of the underlying modular parametrization – we address this
point more specifically in Remark 7). For a prime p not dividing the denominators of r and z,
define a(p) to be the absolute smallest integer residue modulo p2 of the partial sum
p−1∑
k=0
(12)k(r)k(1− r)k
k!3
zk
of the hypergeometric series
3F2
(
1
2 , r, 1− r
1, 1
∣∣∣∣ z) =
∞∑
k=0
(12)k(r)k(1− r)k
k!3
zk.
Then |a(p)| ≤ 2p and a(p) are the Fourier coefficients of a suitable eigenform f(τ) = q+a(2)q2+
· · · of weight 3 for some congruence subgroup of PSL2(Z). Furthermore, in several cases we have
L(f, 2)
π2
∈ Q
[√
d
]
Re 3F2
(
1
2 , r, 1− r
1, 1
∣∣∣∣ z)
and then also a similar inclusion for L(f, 1)/π. Here d ∈ Z depend on the data r, z and on the
choice of m in L(f,m)/πm.
The following illustrations all correspond to the choice r = 1
2 and are motivated by the results
established in [30]. The corresponding character χ is trivial and we have
p−1∑
k=0
(12)3k
k!3
≡ a1(p)
(
mod p2
)
=
{
2
(
a2 − b2
)
if p = a2 + b2, a odd,
0 if p ≡ 3 (mod 4),
p−1∑
k=0
(12)3k
k!3
(−1)k ≡ a2(p)
(
mod p2
)
,
p−1∑
k=0
(12)3k
k!3
4k ≡ a3(p)
(
mod p2
)
,
where a1(n) denote the Fourier coefficients of the cusp form of weight 3 on Γ1(16),
f1(τ) =
∞∑
n=1
a1(n)qn = η(4τ)6 = q
∞∏
m=1
(
1− q4m
)6
,
while
f2(τ) =
∞∑
n=1
a2(n)qn = η(τ)2η(2τ)η(4τ)η(8τ)2, f3(τ) =
∞∑
n=1
a3(n)qn = η(2τ)3η(6τ)3
12 W. Zudilin
are the cusp forms on Γ1(8) and Γ1(12), respectively. In addition, on using some hypergeometric
summations and [23, Theorem 5] we obtain
3F2
(
1
2 ,
1
2 ,
1
2
1, 1
∣∣∣∣ 1
)
=
π
Γ(3/4)4
=
16L(f1, 2)
π2
=
8L(f1, 1)
π
,
3F2
(
1
2 ,
1
2 ,
1
2
1, 1
∣∣∣∣ −1
)
=
Γ(1/8)2Γ(3/8)2
27/2π3
=
12
√
2L(f2, 2)
π2
=
12L(f2, 1)
π
,
Re 3F2
(
1
2 ,
1
2 ,
1
2
1, 1
∣∣∣∣ 4
)
=
3Γ(1/3)6
211/3π4
=
12L(f3, 2)
π2
=
4
√
3L(f3, 1)
π
.
Also notice that algebraic transformations of underlying hypergeometric functions correspond
to the ‘coincidences’ of the type
p−1∑
k=0
(12)k(
1
3)k(
2
3)k
k!3
(
2
27
)k
≡
p−1∑
k=0
(12)3k
k!3
4k ≡ a3(p)
(
mod p2
)
for p > 3 and
p−1∑
k=0
(12)k(
1
4)k(
3
4)k
k!3
≡
(
−4
p
) p−1∑
k=0
(12)3k
k!3
(−1)k ≡
(
−4
p
)
a2(p)
(
mod p2
)
for p > 2. The last example is of importance in relation with the computation in [24].
Remark 7. Behind such examples in Observation 4, there is Clausen’s classical identity
2F1
(
r, 1− r
1
∣∣∣∣ z)2
= 3F2
(
1
2 , r, 1− r
1, 1
∣∣∣∣ 4z(1− z)
)
(15)
valid in a neighbourhood of z = 0. If we write the corresponding ε-expansions (11) and
F̃ (z; ε) =
1
Γ
(
1
2
)
Γ(r)Γ(1− r)
∞∑
n=−∞
Γ
(
1
2 + ε+ n
)
Γ(r + ε+ n)Γ(1− r + ε+ n)
Γ(1 + ε+ n)3
(4z(1− z))n+ε
= F̃0(z) + F̃1(z)ε+ F̃2(z)ε
2 +O
(
ε3
)
as ε→ 0
then F̃0(z) = F0(z)
2 (as in (15)) but also F̃1(z) = F0(z)F1(z),
F̃2(z) =
1
2
( π
sinπr
)2
F0(z)
2 +
1
2
F1(z) =
1
2
F1(1− z)2 +
1
2
F1(z)
2.
The relations follow from the particular structure of the bilateral hypergeometric functions
F (z; ε) and F̃ (z; ε), which we outlined in Section 3, and the following generalized Clausen
identity:
2F̃ (z; ε) cosπε = F (z; ε)2e−πiε
(
1− sin2 πε
sin2 πr
)
+ F (z; 0)2eπiε (16)
valid for all ε ∈ R. The identity (16) follows from the fact that the hypergeometric differential
equation for F̃ (z; ε) is the symmetric square of the differential equation for F (z; ε).
Finally, we would like to point out some heuristics about why modular instances of K3
surfaces with Picard rank 20 correspond to the CM cases of the underlying hypergeometric
functions. Notice that the functional equation for L(f, s) in the case of a modular form of
weight 3 and level ` implies that, for the critical values, L(f, 2)/L(f, 1) = ±2π/
√
`. If we expect
Modularity of Calabi–Yau Manifolds 13
that a hypergeometric 3F2 function is linked to a modular K3 surface (with Picard rank 20),
then we must have F̃2(z)/
(
πF̃1(z)
)
to be of the form
√
lQ for some positive integer `. With the
help of the generalized Clausen identity we then conclude that the quantity
τ = τ(z) = − iF1(z)
2πF0(z)
=
iF0(1− z)
2 sinπr F0(z)
must be an imaginary quadratic irrationality, hence its functional inversion – the modular func-
tion z = z(τ) admits a singular modulus value at this point. The fact that z(τ) is a modular
parametrization of the corresponding hypergeometric function
F0(z) = 2F1
(
r, 1− r
1
∣∣∣∣ z)
for each r ∈
{
1
2 ,
1
3 ,
1
4 ,
1
6
}
is classical – see, for example, [2, p. 91]; one also has
1
2πi
dz
dτ
= z(1− z)F0(z),
the result already known to Ramanujan [2, Chapter 33], [5].
A different way to explain the modularity of K3 surfaces with Picard number 20 is kindly
communicated to us by N. Yui: Such K3 surfaces are all motivically modular in the sense that
the lattice of transcendental cycles is of rank 2 and corresponds to a modular form of weight 3
with character for some congruence subgroup of PSL2(Z). They are all of CM type as the
endomorphism algebra of the transcendental lattice is an imaginary quadratic field over Q. In
particular, this means that the underlying hypergeometric functions are also of CM type.
Another interesting instance corresponds to choosing z = 1 in the hypergeometric series
F0(z) = 6F5
(
1
2 ,
1
2 ,
1
2 ,
1
2 ,
1
2 ,
1
2
1, 1, 1, 1, 1
∣∣∣∣ z) =
∞∑
k=0
(12)6k
k!6
zk
related to a Calabi–Yau fivefold – a complete intersection of six degree 2 surfaces in P12; the
associated Hodge structure for each fiber z of the family can be conjecturally computed with
a help of the hypergeometric motives [22]. Consider the newform
g(τ) =
∞∑
n=1
b(n)qn = q + 20q3 − 74q5 − 24q7 + 157q9 + 124q11 + · · ·
= η(2τ)12 + 32η(2τ)4η(8τ)8
of weight 6 on Γ0(8). Its coefficients satisfy Weil’s bound |b(p)| ≤ 2p5/2 and numerics suggest
that
p−1∑
k=0
(12)6k
k!6
≡ b(p)
(
mod p5
)
(17)
is true for all primes p > 2. The explicit expression for g(τ) was kindly informed to us by J. Wan
who also noticed its historical cast in [9] (see the last column of the table on p. 56 there). As we
learned later, the conjecture (17) was reported in [8] and attributed to E. Mortenson; it is now
shown to be true modulo p3 in the joint work [19] with R. Osburn and A. Straub. Numerically,
the Taylor ε-expansion
1
Γ(12)6
∞∑
n=−∞
Γ(12 + ε+ n)6
Γ(1 + ε+ n)6
zn+ε =
5∑
k=0
Fk(z)ε
k +O
(
ε6
)
as ε→ 0
14 W. Zudilin
can be related, at z = 1, to the critical L-values as follows:
L(g, 1)
F1(1)
= −1
8
,
L(g, 2)
F2(1)
=
1
32
,
L(g, 3)
F3(1)
= − 3
448
,
L(g, 4)
F4(1)
=
1
640
and
L(g, 5)
F5(1)
= − 5
12032
.
As pointed out to us by F. Rodriguez Villegas and D. Roberts the related hypergeometric motive
is also linked to the modular form f(τ) from the introduction, defined in (2). Armed by this
hint, we have found the related instances
p−1∑
k=0
(4k + 1)
(12)6k
k!6
≡ pa(p)
(
mod p4
)
for p > 2
proved in [16, Theorem 1.2] and
∞∑
k=0
(4k + 1)
(12)6k
k!6
=
32
π2
L(f, 1)
established in [23, equation (33)].
Our final – and personal favourite – family of examples is about known Ramanujan(-type)
formulas [36] for 1/π, 1/π2 and their generalizations. Those fit a general picture highlighted in
the observations above, except that the modular form f(τ) is replaced by a quadratic character
so that a critical L-value L(f,m) is replaced by the critical value of the corresponding Dirichlet
L-series. This is transparent from supercongruence observations in [37] and, in addition, from
a noncongruence (bilateral) counterpart experimentally discovered by J. Guillera in [12] (see
also the related prequel [13]).
Acknowledgements
Feedback of many colleagues has been extremely helpful in preparation of this manuscript.
I would like to thank Frits Beukers, Henri Cohen, Jesús Guillera, Günter Harder, Ling Long,
Anton Mellit, Alexei Panchishkin, David Roberts, Emanuel Scheidegger, Duco van Straten,
Alexander Varchenko, Fernando Rodriguez Villegas, John Voight, James Wan, Noriko Yui and
Don Zagier for their comments and responses to my questions. Special thanks are expressed
to Vasily Golyshev for his clarification to me the link between the critical L-values and the
corresponding hypergeometrics, which underlies so-called gamma structures [10], and to Michael
Somos for his powerful help in making some entries in Table 1 explicit. Finally, I am indebted
to the anonymous referees for several helpful comments and corrections.
This note grew up from the author’s talk at the BIRS Workshop “Modular Forms in String
Theory” held in September 2016, and related discussions there. Later parts of this work were
performed during the author’s visits in research institutions whose hospitality and scientific
atmosphere were crucial to success of the project. I thank the staff of the following institutes
for providing such excellent conditions for research: BIRS (Banff, Canada, September 2016);
MATRIX (Creswick, Australia, January 2017); ESI (Vienna, Austria, March 2017); MPIM
(Bonn, Germany, December 2016 and July–August 2017); HIM (Bonn, Germany, March–April
2018).
The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF go-
vernment grant, ag. no. 14.641.31.0001.
Modularity of Calabi–Yau Manifolds 15
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1 A prototype
2 Modularity of Calabi–Yau threefolds
3 Bilateral hypergeometric functions and hypertrigonometry
4 A hypergeometric modularity of elliptic curves
5 Other modularity instances
References
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| id | nasplib_isofts_kiev_ua-123456789-209764 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:30:27Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Zudilin, W. 2025-11-26T11:24:24Z 2018 A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds / W. Zudilin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 11F33; 11T24; 14G10; 14J32; 14J33; 33C20 arXiv: 1805.00544 https://nasplib.isofts.kiev.ua/handle/123456789/209764 https://doi.org/10.3842/SIGMA.2018.086 We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The p-th coefficients a(p) of the corresponding modular form can often be read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of p and from Weil's general bounds |a(p)| ≤ 2p⁽ᵐ⁻¹⁾/², where m is the weight of the form. Furthermore, the critical L-values of the modular form are predicted to be Q-proportional to the values of a related basis of solutions to the hypergeometric differential equation. Feedback from many colleagues has been extremely helpful in the preparation of this manuscript. I would like to thank Frits Beukers, Henri Cohen, Jesús Guillera, Günter Harder, Ling Long, Anton Mellit, Alexei Panchishkin, David Roberts, Emanuel Scheidegger, Duco van Straten, Alexander Varchenko, Fernando Rodriguez Villegas, John Voight, James Wan, Noriko Yui, and Don Zagier for their comments and responses to my questions. Special thanks are expressed to Vasily Golyshev for his clarification of the link between the critical L-values and the corresponding hypergeometrics, which underlie so-called gamma structures [10], and to Michael Somos for his powerful help in making some entries in Table 1 explicit. Finally, I am indebted to the anonymous referees for several helpful comments and corrections. This note grew out of the author’s talk at the BIRS Workshop “Modular Forms in String Theory” held in September 2016, and related discussions there. Later parts of this work were performed during the author’s visits to research institutions whose hospitality and scientific atmosphere were crucial to the success of the project. I thank the staff of the following institutes for providing such excellent conditions for research: BIRS (Banff, Canada, September 2016); MATRIX (Creswick, Australia, January 2017); ESI (Vienna, Austria, March 2017); MPIM (Bonn, Germany, December 2016 and July–August 2017); HIM (Bonn, Germany, March–April 2018). The author is partially supported by the Laboratory of Mirror Symmetry, NRU HSE, RF government grant, ag. no. 14.641.31.0001. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds Article published earlier |
| spellingShingle | A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds Zudilin, W. |
| title | A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds |
| title_full | A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds |
| title_fullStr | A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds |
| title_full_unstemmed | A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds |
| title_short | A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds |
| title_sort | hypergeometric version of the modularity of rigid calabi-yau manifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209764 |
| work_keys_str_mv | AT zudilinw ahypergeometricversionofthemodularityofrigidcalabiyaumanifolds AT zudilinw hypergeometricversionofthemodularityofrigidcalabiyaumanifolds |