Special hypergeometric motives and their $L$-functions: Asai recognition
Lassina Demb\'el\'e, Alexei Panchishkin, John Voight, Wadim Zudilin
TL;DR
This paper identifies special hypergeometric motives as arising from Asai L-functions of Hilbert modular forms, connecting classical Ramanujan-inspired motives with modern automorphic forms.
Contribution
It establishes a novel link between hypergeometric motives and Asai L-functions, expanding the understanding of their origins and properties.
Findings
Hypergeometric motives are related to Asai L-functions.
Connection between classical motives and automorphic forms.
New insights into the structure of special motives.
Abstract
We recognize certain special hypergeometric motives, related to and inspired by the discoveries of Ramanujan more than a century ago, as arising from Asai -functions of Hilbert modular forms.
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Special hypergeometric motives and their -functions: Asai recognition
Lassina Dembélé
Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA
,
Alexei Panchishkin
Institut Fourier, Université Grenoble-Alpes, B.P. 74, 38402 St.-Martin d’Hères, France
[email protected] https://www-fourier.ujf-grenoble.fr/~panchish/ ,
John Voight
Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA
[email protected] http://www.math.dartmouth.edu/~jvoight/ and
Wadim Zudilin
Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, Netherlands
[email protected] http://www.math.ru.nl/~wzudilin/
Abstract.
We recognize certain special hypergeometric motives, related to and inspired by the discoveries of Ramanujan more than a century ago, as arising from Asai -functions of Hilbert modular forms.
1. Introduction
Motivation
The generalized hypergeometric functions are a familiar player in arithmetic and algebraic geometry. They come quite naturally as periods of certain algebraic varieties, and consequently they encode important information about the invariants of these varieties. Many authors have studied this rich interplay, including Igusa [27], Dwork [13], and Katz [29]. More recently, authors have considered hypergeometric motives (HGMs) defined over , including Cohen [8], Beukers–Cohen–Mellit [3], and Roberts–Rodriguez-Villegas–Watkins [36]. A hypergeometric motive over arises from a parametric family of varieties with certain periods (conjecturally) satisfying a hypergeometric differential equation; the construction of this family was made explicit by Beukers–Cohen–Mellit [3] based on work of Katz [29]. Following the analogy between periods and point counts (Manin’s “unity of mathematics” [7]), counting points on the reduction of these varieties over finite fields is accomplished via finite field hypergeometric functions, a notion originating in work of Greene [17] and Katz [29]. These finite sums are analogous to truncated hypergeometric series in which Pochhammer symbols are replaced with Gauss sums, and they provide an efficient mechanism for computing the -functions of hypergeometric motives. (Verifying the precise connection to the hypergeometric differential equation is usually a difficult task, performed only in some particular cases.)
In this paper, we illustrate some features of hypergeometric motives attached to particular arithmetically significant hypergeometric identities for and . To motivate this study, we consider the hypergeometric function
[TABLE]
where we define the Pochhammer symbol (rising factorial) by
[TABLE]
Ramanujan [34, eq. (36)] more than a century ago proved the delightful identity
[TABLE]
involving a linear combination of the hypergeometric series (1.1) and its -derivative (a different, but contiguous hypergeometric function). Notice the practicality of this series for computing the quantity on the right-hand side of (1.3), hence for computing and itself.
The explanation for the identity (1.3) was already indicated by Ramanujan: the hypergeometric function can be parametrized by modular functions (see (2.4) below), and the value arises from evaluation at a complex multiplication (CM) point! Put into the framework above, we observe that the HGM of rank with parameters and corresponds to the Fermat–Dwork pencil of quartic K3 surfaces of generic Picard rank defined by the equation
[TABLE]
whose transcendental -function is related to the symmetric square -function attached to a classical modular form (see Elkies–Schütt [14]). At the specialization , the K3 surface is singular, having Picard rank ; it arises as the Kummer surface of , where is the elliptic -curve LMFDB label 144.1-b1 defined over attached to the CM order of discriminant , and . The corresponding classical modular form with LMFDB label 144.2.c.a has CM, and we have the identity
[TABLE]
where denotes the transcendental lattice of (as a Galois representation). The rare event of CM explains the origin of the formula (1.1): for more detail, see Example 3.12 below.
Main result
With this motivation, we seek in this paper to explain similar hypergeometric Ramanujan-type formulas for in higher rank. Drawing a parallel between these examples, our main result is to experimentally identify that the -function of certain specializations of hypergeometric motives (coming from these formulas) have a rare property: they arise from Asai -functions of Hilbert modular forms of weight over real quadratic fields.
For example, consider the higher rank analogue
[TABLE]
given by Guillera [19]; the question mark above a relation indicates that it has been experimentally observed, but not proven. Here, we suggest that (1.6) is ‘explained’ by the existence of a Hilbert modular form over of weight and level in the sense that we experimentally observe that
[TABLE]
where notation is explained in section 3. (By contrast, specializing the hypergeometric -series at other values generically yields a primitive -function of degree .) Our main result, stated more generally, can be found in Conjecture 5.1.
In spite of a visual similarity between Ramanujan’s formula (1.3) for and Guillera’s formula (1.6) for , the structure of the underlying hypergeometric motives is somewhat different. Motives attached to hypergeometric functions are reasonably well understood (see e.g. Zudilin [39, Observation 4]), and we review them briefly in Section 2. By contrast, the motives associated with similar formulas had not been linked explicitly to modular forms. In Conjecture 5.1, we propose that they are related to Hilbert modular forms, and we experimentally establish several other formulas analogous to (1.7).
More generally, for a hypergeometric family, we expect interesting behavior (such as a formula involving periods) when the motivic Galois group at a specialization is smaller than the motivic Galois group at the generic point. We hope that experiments in our setting leading to this kind of explanation will lead to further interesting formulas and, perhaps, a proof.
Organization
The paper is organized as follows. After a bit of setup in section 2, we quickly review hypergeometric motives in section 3. In section 4 we discuss Asai lifts of Hilbert modular forms, then in section 5 we exhibit the conjectural hypergeometric relations. We conclude in section 6 with some final remarks.
Acknowledgements
This project commenced during Zudilin’s visit in the Fourier Institute in Grenoble in June 2017, followed by the joint visit of Dembélé, Panchishkin, and Zudilin to the Max Planck Institute for Mathematics in Bonn in July 2017, followed by collaboration between Voight and Zudilin during the trimester on Periods in Number Theory, Algebraic Geometry and Physics at the Hausdorff Research Institute for Mathematics in Bonn in March–April 2018. We thank the staff of these institutes for providing such excellent conditions for research.
Dembélé and Voight were supported by a Simons Collaboration grant (550029, to Voight).
It is our pleasure to thank Frits Beukers, Henri Cohen, Vasily Golyshev, Jesús Guillera, Günther Harder, Yuri Manin, Anton Mellit, David Roberts, Alexander Varchenko, Fernando Rodriguez-Villegas, Mark Watkins and Don Zagier for valuable feedback, stimulating discussions, and crucial observations. The authors would also like to thank the anonymous referee for their stimulating report, insightful comments, and helpful suggestions.
2. Hypergeometric functions
In this section, we begin with some basic setup. For and , define the generalized hypergeometric function
[TABLE]
These functions possess numerous features that make them unique in the class of special functions. It is convenient to abbreviate (2.1) as
[TABLE]
where and are called the parameters of the hypergeometric function: they are multisets (that is, sets with possibly repeating elements), with the additional element introduced to reflect the appearance of in the denominator in (2.1). The hypergeometric function (2.1) satisfies a linear homogeneous differential equation of order :
[TABLE]
Among many arithmetic instances of the hypergeometric functions, there are those that can be parameterized by modular functions. One particular example, referenced in the introduction, is
[TABLE]
for with , where
[TABLE]
and denotes the Dedekind eta function with . Taking the CM point , we obtain and the evaluation [26, Example 3]
[TABLE]
As indicated by Ramanujan [34], CM evaluations of hypergeometric functions like (2.6) are accompanied by formulas for , like (1.3) given in the introduction.
Remark 2.7*.*
Less is known about the conjectured congruence counterpart of (2.6),
[TABLE]
for primes , where
[TABLE]
The congruence (2.8) is in line with a general prediction of Roberts–Rodriguez-Villegas [35], though stated there for only.
Ramanujan’s and Ramanujan-type formulas for corresponding to rational values of are tabulated in [6, Tables 3–6]. Known identities for are due to Guillera [18, 19, 20, 21, 23], also in collaboration with Almkvist [1] and Zudilin [25]. We list the corresponding hypergeometric data and for them in Table 2, we have in all these cases.
[TABLE]
Remark 2.11*.*
Some other entries in Table 2 nicely pair up with Ramanujan’s and Ramanujan-type formulas for [19]. Apart from case #9 from Table 2 discussed above, we highlight another instance [19, eq. (2-4)]:
[TABLE]
underlying entry #13, which shares similarities with the Ramanujan-type formula
[TABLE]
Remark 2.14*.*
The specialization points in Table 2 exhibit significant structure: writing and , so that , we already see -triples of good quality! But more structure is apparent: see Remark 5.9.
3. Hypergeometric motives
In this section, we quickly introduce the theory of hypergeometric motives over .
Definition
Analogous to the generalized hypergeometric function (2.1), a hypergeometric motive is specified by hypergeometric data, consisting of two multisets and with satisfying and . Herein, we consider only those hypergeometric motives that are defined over , which means that the polynomials
[TABLE]
have coefficients in —that is, they are products of cyclotomic polynomials.
Let be a prime power that is coprime to the least common denominator of , and let be a finite field with elements. Let be a generator of the character group on , and let be a nontrivial (additive) character. For , define the Gauss sum
[TABLE]
then is periodic in with period .
When for all , we define the finite field hypergeometric sum for by
[TABLE]
by direct analogy with the generalized hypergeometric function. More generally, Beukers–Cohen–Mellit [3, Theorem 1.3] have extended this definition to include all prime powers that are coprime to the least common denominator of .
There exist such that
[TABLE]
and we define
[TABLE]
Computing the local -factors at good primes is completely automated in the Magma [4] package of hypergeometric motives.
Motive and -function
The finite field hypergeometric sums arose in counting points on algebraic varieties over finite fields, and they combine to give motivic -functions following Beukers–Cohen–Mellit [3], as follows. For a parameter , let be the pencil of varieties in weighted projective space defined by the equations
[TABLE]
and subject to . The pencil is affine and singular [3, Section 5]; in fact, it is smooth outside of , where it acquires an ordinary double point.
Theorem 3.7**.**
Suppose that and . Then there exists a suitable completion of such that
[TABLE]
*and where is explicitly given. *
The completion provided in Theorem 3.7 may still be singular, and a nonsingular completion is not currently known in general; we expect that has only quotient singularities, and hence behaves like a smooth manifold with respect to rational cohomology, by the nature of the toric (partial) desingularization. In any event, this theorem shows that the sums (3.3) have an explicit connection to arithmetic geometry and complex analysis.
We accordingly define hypergeometric -functions, as follows. Let be the set of primes dividing the numerator or denominator in together with the primes dividing the numerator or denominator of or . A prime is called good (for ). For a good prime , we define the formal series
[TABLE]
Corollary 3.9**.**
For and , we have
[TABLE]
Proof.
The zeta function of over is a rational function by work of Dwork; the exponential series for is also rational, so the result follows from Theorem 3.7. ∎
Remark 3.10*.*
In fact, we expect that is a polynomial of degree ; it should follow from the construction in Theorem 3.7 or from work of Katz [28], but we could not find a published proof. We establish this property in the cases we consider, as a byproduct of our analysis.
Globalizing, we define the incomplete -series
[TABLE]
a Dirichlet series that converges in a right half-plane, but otherwise remains rather mysterious. Our goal in what follows will be to match such -functions (coming from geometry, rapidly computable) with -functions of modular forms in certain cases, so that the former can be completed to inherit the good properties of the latter.
Examples
We conclude this section with two examples.
Example 3.12**.**
We return to our motivating example, with the parameters and , we find and . Then eliminating in (3.6) gives
[TABLE]
and Theorem 3.7 yields
[TABLE]
We make a change of parameters and consider the pencil of quartic K3 hypersurfaces with generically smooth fibers defined by
[TABLE]
as in (1.4), with generic Picard rank . The family 3.13 is known as the Fermat–Dwork family and is well studied (going back to Dwork [13, §6j, p. 73]; see e.g. Doran–Kelly–Salerno–Sperber–Voight–Whitcher [12, §1.5] for further references). In the context of mirror symmetry, one realizes as the mirror of [11, §5.2] in the following way, due to Batyrev: there is an action of on , and is birational to . We see again that the finite field hypergeometric sum contributes nontrivially to the point counts [12, Main Theorem 1.4.1(a)].
In either model, the holomorphic periods of or are given by the hypergeometric series
[TABLE]
As mentioned in the introduction, at the specialization , the K3 surface is singular, with Picard number —it is this rare event that explains the formula (1.3). Computing the local -factors, we find
[TABLE]
for , where is the quadratic character attached to and defined in (2.9). Indeed, this factorization agrees with the fact that the global -series can be completed to
[TABLE]
where is the classical modular form with LMFDB label 144.2.c.a: more generally, see Elkies–Schütt [14], Doran–Kelly–Salerno–Sperber–Voight–Whitcher [11, Theorem 5.1.3], or Zudilin [39, Observation 4]. Consequently, the completed hypergeometric -series inherits analytic continuation and functional equation.
Example 3.15**.**
We consider the hypergeometric data attached to Ramanujan-type formula (2.12), corresponding to in Table 2 and with parameters and . This example is, in many aspects, runs parallel to Example 3.12 and the related mirror symmetry construction of the famous quintic threefold [5]. We have
[TABLE]
and Theorem 3.7 implies
[TABLE]
Alternatively, we consider the pencil of sextic fourfolds
[TABLE]
in . Under the change of parameter , we find that is birational to where . The are generically Calabi–Yau fourfolds. A computation (analogous to Candelas–de la Ossa–Greene–Parks [5]) shows that the Picard–Fuchs differential operator is given by
[TABLE]
The unique (up to scalar) holomorphic solution near zero is the hypergeometric function
[TABLE]
Using the Magma implementation, we compute the first few (good) -factors:
[TABLE]
and observe that for ,
[TABLE]
Moreover, when then and the quartic polynomial factors as
[TABLE]
whereas for it is generically irreducible. This suggests again a rare event which we seek to explain using modular forms.
4. The Asai transfer of a Hilbert modular form
Having defined -functions arising from hypergeometric motives in the previous sections, over the next two sections we follow the predictions of the Langlands philosophy and seek to identify these -functions as coming from modular forms in the cases of interest. More precisely, we confirm experimentally a match with the Asai transfer of certain Hilbert modular forms over quadratic fields. We begin in this section by setting up the needed notation and background. As general references for Hilbert modular forms, consult Freitag [15] or van der Geer [16]; for a computational take, see Dembélé–Voight [9].
Let be a real quadratic field of discriminant with ring of integers and Galois group . By a prime of we mean a nonzero prime ideal . Let be the two embeddings of into . For we write , and for we write for the coordinate-wise application of . An element is totally positive if ; we write for the group of totally positive elements. The group
[TABLE]
acts on the product of upper half-planes by embedding-wise linear fractional transformations .
Let , write , and let and . Let be a nonzero ideal. Let denote the (finite-dimensional) -vector space of Hilbert cusp forms of weight , level , and central character . Hilbert cusp forms are the analogue of classical cusp forms, but over the real quadratic field . When the narrow class number of is equal to (i.e., every nonzero ideal of is principal, generated by a totally positive element) and is the trivial character, a Hilbert cusp form is a holomorphic function , vanishing at infinity, such that
[TABLE]
for all such that .
The space is equipped with an action of pairwise commuting Hecke operators indexed by nonzero primes . A Hilbert cusp form is a newform if is an eigenform for all Hecke operators and does not arise from with a proper divisor.
Let be a newform. For , we have with a totally real algebraic integer (the Hecke eigenvalue), and we factor
[TABLE]
where is the absolute norm. Then .
For prime with , following Asai [2] we define, abbreviating ,
[TABLE]
We call the factors the good -factors of . The partial Asai -function of is the Dirichlet series defined by the Euler product
[TABLE]
where .
The key input we need is the following theorem. For a newform , let be the newform of weight and level with , with central character . Finally, for central character (of the idele class group of ) let denote its restriction (to the ideles of ).
Theorem 4.5** (Krishnamurty [31], Ramakrishnan [33]).**
Let be a Hilbert newform, and suppose that is not a twist of . Then the partial -function can be completed to a -automorphic -function
[TABLE]
of degree , conductor , with central character .
More precisely, there exists a cuspidal automorphic representation of such that for all . In particular, is entire and satisfies a functional equation with .
The automorphic representation in Theorem 4.5 goes by the name Asai transfer, Asai lift, or tensor induction of the automorphic representation attached to , and we write .
Proof.
We may identify , with -group
[TABLE]
where . We define the -dimensional representation
[TABLE]
For a place of , let be the restriction of to .
Let be the cuspidal automorphic representation of attached to . Then is an admissible representation of corresponding to an -parameter
[TABLE]
where is the Weil–Deligne group of . We define to be the irreducible admissible representation of attached to by the local Langlands correspondence, and we combine these to
[TABLE]
By a theorem of Ramakrishnan [33, Theorem D] or Krishnamurty [31, Theorem 6.7], is an automorphic representation of whose -function is defined by
[TABLE]
whose good -factors agree with (4.3) [31, §4]. Under the hypothesis that is not a twist of , we conclude that is cuspidal [33, Theorem D(b)]. Consequently, we may take in the theorem. ∎
Remark 4.11*.*
Some authors also define the representation , which is the quadratic twist of by the quadratic character attached to .
In addition to the direct construction (4.4) and the automorphic realization in Theorem 4.5, one can also realize the Asai -function via Galois representations. By Taylor [37, Theorem 1.2], attached to is a Galois representation
[TABLE]
such that for each prime , we have
[TABLE]
Then there is a natural extension of to , a special case of multiplicative induction (or tensor induction) [32, §7] defined as follows: for a lift of to which by abuse is also denoted , we define [30, p. 1363] (taking a left action)
[TABLE]
Up to isomorphism, this representation does not depend on the choice of lift . A direct computation [30, Lemma 3.3.1] then verifies that as defined in (4.4).
The bad -factors and conductor of are uniquely determined by the good -factors, but they are not always straightforward to compute.
5. Matching the hypergeometric and Asai -functions
We now turn to the main conjecture of this paper.
Main conjecture
We propose the following conjecture.
Conjecture 5.1**.**
Let be a set of parameters from Table 2 and let . Then there exist quadratic Dirichlet characters and a Hilbert cusp form over a real quadratic field of weight such that for all good primes we have
[TABLE]
In particular, we have the identity
[TABLE]
We can be more precise in Conjecture 5.1 for some of the rows, as follows. Let be a row in Table 2 with . Then we conjecture that the central character of is a quadratic character of the class group of induced from a Dirichlet character; and the conductors of , the discriminant of , and the level of are indicated in Table 5.
[TABLE]
Evidence
We verified Conjecture 5.1 for the rows indicated in Table 5 using Magma [4]; the algorithms for hypergeometric motives were implemented by Watkins, algorithms for -functions implemented by Tim Dokchitser, and algorithms for Hilbert modular forms by Dembélé, Donnelly, Kirschmer, and Voight. The code is available online [10].
Moreover, using the -factor data in Table 5, we have confirmed the functional equation for up to 20 decimal digits for all but #13. When the discriminant and the level are coprime, we observe that the conductor of is .
[TABLE]
Remark 5.4*.*
In a recent arithmetic study of his formulas for , Guillera [24] comes up with an explicit recipe to cook up the two quadratic characters for each such formula. He calls them and and records them in [24, Table 3]. Quite surprisingly, they coincide with our and in Table 5.
Example 5.5**.**
Consider row . In the space of Hilbert cusp forms over of weight and level with trivial central character, we find a unique newform with first few Hecke eigenvalues , , , , and , giving for example
[TABLE]
we then match
[TABLE]
We matched -factors for all good primes such that a prime of lying over has .
Example 5.6**.**
For row , the space of Hilbert cusp forms over of weight and level has dimension with a newspace of dimension . We find a form with Hecke eigenvalues , , …; accordingly, we find
[TABLE]
and so on. We again matched Hecke eigenvalues up to prime norm .
Remark 5.8*.*
To match row in Table 2 with a candidate Hilbert modular form, we would need to extend the implementation of hypergeometric motives to apply for specialization at points ; we expect this extension to be straightforward, given the current implementation of finite field hypergeometric sums.
By contrast, to match the final rows and –, we run into difficulty with computing spaces of Hilbert modular forms: we looked for forms in low level, but the dimensions grow too quickly with the level. We also currently lack the ability to efficiently compute with arbitrary nontrivial central character. We plan to return to these examples with a new approach to computing systems of Hecke eigenvalues for Hilbert modular forms in future work.
Remark 5.9*.*
Returning to Remark 2.14, we observe structure in the specialization points from Table 2: beyond patterns in the factorization of and , we also note that for these points the completed -function typically has unusually small conductor , as in Table 5. (Perhaps a twist of #15 has smaller conductor?) Some general observations that may explain this conductor drop:
- •
Factor where consists of the product of primes that divide the least common denominator of or the numerator or denominator of . Then should be the squarefree part of the numerator of ; this numerator is divisible by a nontrivial square in ten of the fifteen cases.
- •
The power of dividing the numerator or denominator of is itself a multiple of for most primes dividing a denominator in .
- •
For a prime , define if is coprime to and otherwise let . If is a multiple of , then tends to be especially small.
These last two phenomena were first observed by Rodriguez–Villegas; we thank the referee for these observations.
While not making any assertions about completeness, these observations give some indication of why our Table 2 is so short: the specialization points like those listed are quite rare, and they seem to depend on a pleasing but remarkable arithmetic confluence. It would be certainly valuable to be able to predict more generally and precisely the conductor of hypergeometric -functions.
Method
We now discuss the recipe by which we found a match. For simplicity, we exclude the case and suppose that the central character is trivial. In a nutshell, our method uses good split ordinary primes to recover the Hecke eigenvalues up to sign.
We start with the hypergeometric motive and compute for many good primes . We first guess and by factoring : for primes that are split in , we usually have irreducible whereas and for inert primes we find . We observe in many cases that is (up to squares) the numerator of . Combining this information gives us a good guess for and .
We now try to guess the Hecke eigenvalues of a candidate Hilbert newform of weight . Let be a good split prime, and suppose that is ordinary for , i.e., the normalized valuations are as small as possible, or equivalently, factoring
[TABLE]
we may choose so that are -adic units. We expect that such primes will be abundant, though that seems difficult to prove. Then has Hodge–Tate weights (i.e., reciprocal roots with valuations) (adding pairwise) so the Tate twist has Hodge–Tate weights and coefficients with valuations , matching that of the hypergeometric motive.
So we factor over the -adic numbers, identifying ordinary when the roots have corresponding valuations . Then we have the equations
[TABLE]
and two similar equations for . Therefore
[TABLE]
so
[TABLE]
and this determines the Hecke eigenvalue
[TABLE]
up to sign.
We then go hunting in Magma by slowly increasing the level and looking for newforms whose Hecke eigenvalues match the value in (5.14) up to sign. With a candidate in hand, we then compute all good -factors using (4.3) to identify a precise match. The bottleneck in this approach is the computation of systems of Hecke eigenvalues for Hilbert modular forms.
6. Conclusion
The story brings many more puzzles into investigation, as formulas discussed in this note do not exhaust the full set of mysteries. Some of them are associated with the special evaluations of , like the intermediate one in the trio
[TABLE]
Here the first equation is from Ramanujan’s list [34, eq. (42)], the second one is recently established by Guillera [22, eq. (1.6)], while the third one corresponds to Entry #15 in Table 2 and is given in [19, eq. (2-5)]. There is also one formula for , due to B. Gourevich (2002),
[TABLE]
(observe that ). And the pattern extends even further with the support of the experimental findings
[TABLE]
due to Yue Zhao [38] (September 2017). On the top of these examples there are ‘divergent’ hypergeometric formulas for and coming from ‘reversing’ Zhao’s experimental formulas for and in [38], and corresponding to the hypergeometric data
[TABLE]
respectively. We hope to address the arithmetic-geometric origins of the underlying motives in the near future.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Almkvist, G. , Guillera, J. , Ramanujan-like series for 1 / π 2 1 superscript 𝜋 2 1/\pi^{2} and string theory, Experiment. Math. 21 (2012), no. 3, 223–234.
- 2[2] Asai, T. , On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method, Math. Ann. 226 (1977), 81–94.
- 3[3] Beukers, F. , Cohen, H. , Mellit, A. , Finite hypergeometric functions, Pure Appl. Math. Q. 11 :4 (2015), 559–589.
- 4[4] Bosma, W. , Cannon, J. , Playoust, C. , The Magma algebra system. I. The user language, J. Symbolic Comput. 24 :3–4 (1997), 235–265.
- 5[5] Candelas, P. , de la Ossa, X. , Green, P.S. , Parkes, L. , A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21–74.
- 6[6] Chan, H.H. , Cooper, S. , Rational analogues of Ramanujan’s series for 1 / π 1 𝜋 1/\pi , Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 2, 361–383.
- 7[7] Clemens, C. H. , A scrapbook of complex curve theory , 2nd ed., Grad. Studies in Math., vol. 55, Amer. Math. Soc., Providence, RI (2003).
- 8[8] Cohen, H. , Computing L 𝐿 L -functions: a survey, J. Théor. Nombres Bordeaux 27 (2015), no. 3, 699–726.