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Normalized holomorphic periods of hypergeometric Calabi–Yau families

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Numbers
$A$
$z$ 
$F_A(z)$
2.2.2.2
1:
1.118636387164187068349619257525640916794857551529361191480618646924046289663704113733002548529336059 + i * [0, 0]
comment: This value is $16L(f,2)/\pi^2$ for $f(\tau)=\eta(2\tau)^4\eta(4\tau)^4$, where $\eta$ is the Dedekind eta function [4].
2.2.2.2
1/2:
1.037915197940088916213825324962959020133073456272991528225175213296969503576069913010528534187185328 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=5$.
2.2.2.2
-1:
0.9511579157472163468392528179949360791584348927329900602410956778832142081013419868339982944593858079 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=5$.
2.2.2.2
2:
1.139096853476574636159157205540753853047546983943121404339900199862499857791953201046698731930913055 + i * 0.1846073108892278834486608541187469714917196900043806279103573348060826882403009742287489558806348530
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=5$.
2.2.3
1:
1.104571259310865345571104853107242129836708936313415116297346700614171962064133362987943559267457201 + i * [0, 0]
2.2.3
1/2:
1.033621201664808000129188029767010821054908122129869524374284946667015631625192346167337959722309043 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
2.2.3
-1:
0.9564538273747178766774266775144366282942918641652266488400652613675933159660869846592585384559705588 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
2.2.3
2:
1.124696042070001876250742474736401696144184956779354986588401186287575719305585599546787643404640357 + i * 0.1615108286040944895844280629907366362208533260578205807567104076749089570790953274938412336816192249
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=7$.
2.2.4
1:
1.087310326239675992386009373260665985917476912335913786022488047498404144136749586106185871479512844 + i * [0, 0]
2.2.4
1/2:
1.028282506588595430359218886088290737777442462095424896010071922139494469087532377909316060222465535 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
2.2.4
-1:
0.9631192507076838650744491121753459209247192442526105487847660905731776070408751027721873636008214482 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
2.2.4
2:
1.106277168604442323404718604414831450087405728885056118458239987721985487074741380873870399346029576 + i * 0.1335561573881550119177341830419905275810270790391372712431011567580226705932111784674631176404041488
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
2.2.6
1:
1.063732084253105935076110882339434398663500452352534443900792758032114247360994516139297845282345790 + i * [0, 0]
2.2.6
1/2:
1.020861811172582377115785284095350726451694754739231809007148631480296946324753118395989167993289354 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
2.2.6
-1:
0.9725364078433962484217249970972765257554351248654435679587137440658294810014611017418932297503688282 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
2.2.6
2:
1.079763671455945428197688233196553260315007073895083582794645420661055269528826112400557996177577679 + i * 0.09612217397078532195244426655010905124855798509729965533433881043619343532966631093034220448868251748
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
3.3
1:
1.092181913533199498482429626216215770221851496068080821180234841911442676691279904582856696186230987 + i * [0, 0]
3.3
1/2:
1.029814357720524538598592631353200019217835935454875581475008400482161658229309194418451972021902005 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
3.3
-1:
0.9611770395518294007425355064922050366432343339935246088395418326195224069522442034046707755141269730 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
3.3
2:
1.111742365810487619405534797010457713559886274498542139638525306312375846813260046638276218848778823 + i * 0.1412988565209175408452183650759191344374143425253484635689769583002833061502606601020727974785490267
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
3.4
1:
1.076974686281653188207964898773171746199296859573876587952368454052357030020376576049933348127744066 + i * [0, 0]
3.4
1/2:
1.025081040405271592709923148427203423847243378859880813762275883521295355970404539627523659982932342 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
3.4
-1:
0.9671211209353557004753012023975238866796045237311458888608429881363841993145080345945406953700438006 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
3.4
2:
1.095192314736970778503607606751545830627846248001892460606842477686544444789688997624469742001783679 + i * 0.1168372834905556379123167544963672054563098219165189137000671485213653016976397407411374951170512891
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
3.6
1:
1.056196516746322510914893787616944645481537587672082459689046822739440954049643457518012308966897118 + i * [0, 0]
3.6
1/2:
1.018501259548123310059133616254602539477418262743504942161035389668368606870488609765645270178836984 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
3.6
-1:
0.9755181223767834752004830517206892912997782101166973817717644508779524221879707557566264161111411850 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
3.6
2:
1.071400114577534894107625377783698690607197869926820631742303959445650353298962529588903273033629413 + i * 0.08408417447516480052535820321702371652754173812949983341319103791363616268550546079431861279185876482
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
4.4
1:
1.064285288815143746485718095463061927777179866937529906336720862562287112252979440240775718942315920 + i * [0, 0]
4.4
1/2:
1.021100120943649281328093008994669184291991634990564211461469657809125594426938281169656137648603901 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
4.4
-1:
0.9721568314298268877582758280836334068688826964105512627978585272119926240358824602220107885706616854 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
4.4
2:
1.081048109619669434969966898736619253344875769035415735128678265911060994362012616006531032859760049 + i * 0.09660512978647053315880308058009971531161957352685606178413638280135158277487436147370274655376045087
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
4.6
1:
1.046941792014333627656878400048713510930519117575563261094519909830809323370324857146403053103853123 + i * [0, 0]
4.6
1/2:
1.015565668290715282552572744965252342046117140162064469826281784124273425279723954189637418894040585 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
4.6
-1:
0.9792695334974491821470929949403871487192311398691782873910125347253849181558008436025981535841752300 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
4.6
2:
1.060746190085482032426159222470775067474722658457283098141469602889275160756004862314976548360730377 + i * 0.06951840466457809405715547456004221748963741482178742164111557032562585015571786361033246527607741802
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
6.6
1:
1.034287016964406574076735582477908303307919309503354613519781035489236889233964576280911945375648192 + i * [0, 0]
6.6
1/2:
1.011483878096793283259053940008459564662982976995380011710875627421444171654906089426911106473379801 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
6.6
-1:
0.9845671059478174960488005397818101998643539502517179725513222808865540031226053367561031759361154883 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
6.6
2:
1.045485299736774110378725988623045996435546296549922037164421171539487748194536282992258907648475633 + i * 0.05002114007662587171926670739378191186665472694815230670738436921678673335733671175918081907929328975
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
5
1:
1.070725868430155800570739558831538047590232853160144425163528234046553558519315858672206123352350932 + i * [0, 0]
5
1/2:
1.023094497870339516835429134173997772916630552482134625415800743367968518531881787515904043962839040 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
5
-1:
0.9696655045764480894309196757598702756321489090001752319353103853215921746818347846333350517179285065 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
5
2:
1.087956594781827924895467987992159251382553618708012718026687432983541546562917205365454415345876392 + i * 0.1070350798274859175421236623106674120905277364742826027178002977500194297261406611499457414814298626
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
8
1:
1.046427257680576204257289224503456298869401352156003512791060978284194710703036285628493978470607596 + i * [0, 0]
8
1/2:
1.015342910196347427814624856057414762168125434236905688203465093705851339267168058789662874100078260 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
8
-1:
0.9796268515220511581733706963308979133316440981427922935997140261366341125886328233492761979069829189 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
8
2:
1.059552365618689112893443403525873757442136780035768532208872404401293936709080057729389095491204326 + i * 0.06908645065650923650169696264048971180622742811387829267222036947177423890441597971792768404230609200
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
10
1:
1.033797388365786440627528834111156965634657540675531306916451699331336029875967549568280466754851195 + i * [0, 0]
10
1/2:
1.011270899916824788837796309744253169419358899913948267171933942459956694129670230863931085091232587 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
10
-1:
0.9849110243128746307108181842462879042938275965601863153218142218908889693047046845465564957487233541 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
10
2:
1.044350200590881269481561085674393836219917638045779861469564562324116788425026254263630614900713157 + i * 0.04962562368340939284738485346143351240061953358460834080719007857755982280579993563513369923502352311
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=3$ and the unit eigenvalue at $p=3$.
12
1:
1.033377870707763335074221917323234402875207986110277877324031080063747022611114783086132565125041893 + i * [0, 0]
12
1/2:
1.011088764946528652969541439444321808460866339318932415524148215781490270137865080034505054487816734 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
12
-1:
0.9852043611444791463800038524066482649696501121315931907216777448967431229556840024732637589139495962 + i * [0, 0]
comment: Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
12
2:
1.043377401265892409729066276839175743721002172245644734985383097301804251125369400611333638542437118 + i * 0.04928133282505547436496170914220203140660989752038506862889466676199337641031505042923057452953698714
comment: The upper-bank value retains its positive imaginary component. Local arithmetic of this fibre includes the Frobenius polynomial at $p=5$ and the unit eigenvalue at $p=5$.
Definition
The normalized holomorphic period is $F_A(z)={}_4F_3(\alpha_1,\alpha_2,\alpha_3,\alpha_4;1,1,1;z)$, with $F_A(0)=1$, for a classified hypergeometric Calabi–Yau template $A$ [1] [5].
Parameters
$A$
—   hypergeometric template ($A$ is one of the fourteen cyclotomic numerator multisets specified in (5))
$z$
—   rational fibre ($z\in\mathbb Q\setminus\{0\}$, with the boundary prescriptions in (6))
Formulas
(1)
For $|z|<1$, $F_A(z)=\sum_{n\geq0}c_nz^n$, where $c_0=1$ and $c_{n+1}=c_n\prod_{j=1}^4(n+\alpha_j)/(n+1)^4$. Equivalently, $c_n=\prod_{j=1}^4(\alpha_j)_n/(n!)^4$, with $(a)_n$ the rising factorial [1].
(2)
With $\theta=z\,d/dz$, $[\theta^4-z\prod_{j=1}^4(\theta+\alpha_j)]F_A(z)=0$ and $F_A(0)=1$ [1].
(3)
Put $S_r(z)={}_2F_1(1/r,1-1/r;1;z)$, as in Ramanujan's signature functions. For the upper multiset $(1/r,1-1/r,1/s,1-1/s)$, $F_A=S_r\star S_s$, where $(\sum a_nz^n)\star(\sum b_nz^n)=\sum a_nb_nz^n$. In particular $S_2(z)=2K(z)/\pi$, where $K$ uses the parameter $m=z$.
(4)
For $f(\tau)=\eta(2\tau)^4\eta(4\tau)^4$, a weight-four level-eight cusp form, $F_{2.2.2.2}(1)=16L(f,2)/\pi^2$. Here $\eta$ is the Dedekind eta function and $L(f,s)=\sum_{n\geq1}a_n n^{-s}$, analytically continued from its half-plane of convergence, when $f(\tau)=\sum_{n\geq1}a_n e^{2\pi i n\tau}$ [4].
Comments
(5)
The key $A=(d_1,\ldots,d_k)$ specifies all reduced fractions $a/d_j$ with $1\leq a<d_j$, including multiplicities, as the four upper parameters. Dots separate cyclotomic indices. The fourteen keys are $2.2.2.2$, $2.2.3$, $2.2.4$, $2.2.6$, $3.3$, $3.4$, $3.6$, $4.4$, $4.6$, $6.6$, $5$, $8$, $10$ and $12$ [1].
(6)
Continue the power series from zero to the plane cut along $[1,\infty)$. At $z=1$ use the finite limit as $z\to1^-$; at rational $z>1$ use the upper-bank limit, continued through the upper half-plane without winding around a singularity. The values at $z<1$ are real.
(7)
The coordinate $z$ is the classical hypergeometric argument, with singular points $0,1,\infty$. It is neither a mirror parameter nor an unscaled factorial-series variable. In Section 2.1 of [2], the variable is $x=\mu z$, where their conifold is $x=\mu$.
(8)
The sum of the upper parameters is $2$ in every template. The excess of the ${}_4F_3$ series at $1$ is therefore $1$, so the series converges absolutely there [5]. These normalized solutions are distinct from periods in an unspecified integral cycle basis.
(9)
The smooth fibres $z=1/2,-1,2$ are exactly the rational solutions of $z,1-z\in\mathbb Z[1/2]^\times$. They introduce no potentially bad prime outside $2$ beyond the template primes [3]. The Frobenius-polynomial table and the unit-root table give local arithmetic invariants of these same global fibres, with the same template and coordinate conventions.
(10)
For the ten templates assembled from complementary pairs $1/r,1-1/r$ with $r\in\{2,3,4,6\}$, the period series is the Hadamard product in (3). This is a coefficientwise product of series; it is not multiplication of their evaluated values.
Programs
(P1)
Python
import mpmath as mp
mp.mp.dps = 120
a = [mp.mpf(1)/5, mp.mpf(2)/5, mp.mpf(3)/5, mp.mpf(4)/5]
z = mp.mpf(2)
y = mp.hyper(a, [1, 1, 1], z)
if z > 1:
    y = mp.conj(y)  # Upper-bank value for real z on the cut.
print(y)
References
[1]
Ling Long, Fang-Ting Tu, Noriko Yui and Wadim Zudilin, Supercongruences for rigid hypergeometric Calabi–Yau threefolds, Table 1 and Section 5. (arXiv)
[2]
Kilian Bönisch, Albrecht Klemm, Emanuel Scheidegger and Don Zagier, D-brane masses at special fibres of hypergeometric families of Calabi-Yau threefolds, modular forms, and periods, Section 2.1 and Table 1. (arXiv)
[3]
David P. Roberts and Fernando Rodriguez Villegas, Hypergeometric Motives, Section 10. (arXiv)
[4]
M. Rogers, J. G. Wan and I. J. Zucker, Moments of elliptic integrals and critical L-values, Section 5, equation (30). (arXiv)
Links
Similar tables
Frobenius polynomials of hypergeometric Calabi–Yau motives —   gives good-prime Frobenius polynomials of the same templates and smooth rational fibres
Unit roots of hypergeometric Calabi–Yau motives —   gives the corresponding p-adic unit eigenvalues
Values of Ramanujan's signature hypergeometric functions $F_r(x)$ —   gives the lower-rank signature functions whose Hadamard products produce ten period series
Values of the Gauss hypergeometric function ${}_2F_1(a,b;c;z)$ —   gives real Gauss hypergeometric values, including the rank-two ingredients
Complete elliptic integral of the first kind $K(m)$ —   gives the elliptic integral used in the signature-two relation
Holomorphic periods of Fermat curves —   gives marked curve periods with a specified cycle and differential, a different normalization of a different family
Data properties
Entries are of type: complex number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds all fourteen classified templates at the conifold $z=1$ and at $z=1/2,-1,2$, the complete rational solution set of $z,1-z\in\mathbb Z[1/2]^\times$. The allowed extra-prime set $\{2\}$ is a noncanonical reference choice that controls possible bad reduction and aligns these analytic values with the local arithmetic companions. All templates receive negative real, interior real, singular-limit and continued complex coverage before extending any one family; the attached argument-selection note records the justification)
How they were obtained:

Computed using mpmath at 140 and 190 decimal working digits, with 100 significant digits retained in each nonzero component. The numerical intervals record agreement between computations; they are not certified truncation bounds. Real fibres have imaginary component exactly zero by analytic continuation of the real power series. The upper-bank signs are fixed by continuing through the upper half-plane.

more

All 56 values were compared with separate computations: Taylor continuation of the Picard–Fuchs equation for the smooth fibres, and a Bernoulli-polynomial asymptotic expansion of the coefficient tail, summed using Hurwitz zeta, for the conifold limits. A telescoping series controls the tail computation; the rank-two relation $S_2=2K/\pi$ controls each continuation path. The all-half conifold value was checked with the level-eight modular L-function in (4). The attached checks and numerical evidence give the actual agreement, software versions, exact coefficient checks and comparisons with values read from the draft.



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