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A hypergeometric version of the modularity of rigid Calabi-Yau manifolds. (English) Zbl 1456.11073
This paper considers the fourteen one-parameter families of Calabi-Yau threefolds whose periods are expressed in terms of hypergeometric functions. For these fourteen families, periods are solutions of hypergeometric equations with parameter \((r, 1-r, t, 1-t)\), where \begin{multline*} (r,t)=\Big(\frac{1}{2},\frac{1}{2}\Big),\Big(\frac{1}{2},\frac{1}{3}\Big),\Big(\frac{1}{2},\frac{1}{4}\Big), \Big(\frac{1}{2},\frac{1}{6}\Big),\Big(\frac{1}{3}\Big),\Big(\frac{1}{3},\frac{1}{4}\Big),\Big(\frac{1}{3},\frac{1}{6}\Big),\\ \Big(\frac{1}{4},\frac{1}{4}\Big),\Big(\frac{1}{4},\frac{1}{6}\Big),\Big(\frac{1}{6},\frac{1}{6}\Big),\Big(\frac{1}{5},\frac{2}{5}\Big), \Big(\frac{1}{8},\frac{3}{8}\Big),\Big(\frac{1}{10},\frac{3}{10}\Big),\Big(\frac{1}{12},\frac{5}{12}\Big). \end{multline*} At a conifold point, any of these Calabi-Yau threefolds becomes rigid, and the \(p\)-th coefficient \(a(p)\) of the corresponding modular form of weight \(4\) can be recovered from the truncated partial sums of the corresponding hypergeometric series modulo a higher power of \(p\), where \(p\) is any good prime \(>5\).
This paper discusses relationships between the critical values of the \(L\)-series of the modular form and the values of a related basis of solutions to the hypergeometric differential equation. It is numerically observed that the critical \(L\)-values are \(\mathbb{Q}\)-proportional to the hypergeometric values \(F_1(1), F_2(1), F_3(1)\), where \(F_j(z)\) are solutions of the hypergeometric equation for the hypergeometric function \(F_0(z)=_4F_3(z)\) with parameters \((r, 1-r, t, 1-t)\). This confirms the prediction of Golyshev concerning gamma structures [V. Golyshev and A. Mellit, J. Geom. Phys. 78, 12–18 (2014; Zbl 1284.33001)].

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11T24 Other character sums and Gauss sums
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J33 Mirror symmetry (algebro-geometric aspects)
33C20 Generalized hypergeometric series, \({}_pF_q\)
Citations:
Zbl 1284.33001
Software:
LMFDB
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References:
[1] Ahlgren, Scott and Ono, Ken, A {G}aussian hypergeometric series evaluation and {A}p\'ery number congruences, Journal f\"ur die Reine und Angewandte Mathematik. [Crelle’s Journal], 518, 187-212, (2000) · Zbl 0940.33002
[2] Berndt, Bruce C., Ramanujan’s notebooks, {P}art {V}, xiv+624, (1998), Springer-Verlag, New York · Zbl 0886.11001
[3] Beukers, F., Another congruence for the {A}p\'ery numbers, Journal of Number Theory, 25, 2, 201-210, (1987) · Zbl 0614.10011
[4] Beukers, Frits and Cohen, Henri and Mellit, Anton, Finite hypergeometric functions, Pure and Applied Mathematics Quarterly, 11, 4, 559-589, (2015) · Zbl 1397.11162
[5] Cooper, Shaun, Inversion formulas for elliptic functions, Proceedings of the London Mathematical Society. Third Series, 99, 2, 461-483, (2009) · Zbl 1248.11031
[6] Damerell, R. M., {\(L\)}-functions of elliptic curves with complex multiplication. {I}, Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica, 17, 287-301, (1970) · Zbl 0209.24603
[7] Evans, R. J., Review of \cite{AO00}
[8] Frechette, Sharon and Ono, Ken and Papanikolas, Matthew, Gaussian hypergeometric functions and traces of {H}ecke operators, International Mathematics Research Notices, 2004, 60, 3233-3262, (2004) · Zbl 1088.11029
[9] Glaisher, J. W. L., On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, The Quarterly Journal of Pure and Applied Mathematics, 38, 1-62, (1906) · JFM 37.0214.03
[10] Golyshev, Vasily and Mellit, Anton, Gamma structures and {G}auss’s contiguity, Journal of Geometry and Physics, 78, 12-18, (2014) · Zbl 1284.33001
[11] Greene, John, Hypergeometric functions over finite fields, Transactions of the American Mathematical Society, 301, 1, 77-101, (1987) · Zbl 0629.12017
[12] Guillera, Jesus, Bilateral sums related to {R}amanujan-like series · Zbl 1278.33003
[13] Guillera, Jesus and Rogers, Mathew, Ramanujan series upside-down, Journal of the Australian Mathematical Society, 97, 1, 78-106, (2014) · Zbl 1305.33014
[14] Kilbourn, Timothy, An extension of the {A}p\'ery number supercongruence, Acta Arithmetica, 123, 4, 335-348, (2006) · Zbl 1170.11008
[15] {The LMFDB Collaboration}, The \(L\)-functions and modular forms database
[16] Long, Ling, Hypergeometric evaluation identities and supercongruences, Pacific Journal of Mathematics, 249, 2, 405-418, (2011) · Zbl 1215.33002
[17] Long, L. and Tu, F.-T. and Yui, N. and Zudilin, W., Supercongruences for rigid hypergeometric {C}alabi–{Y}au threefolds
[18] McCarthy, Dermot, Extending {G}aussian hypergeometric series to the {\(p\)}-adic setting, International Journal of Number Theory, 8, 7, 1581-1612, (2012) · Zbl 1253.33024
[19] Osburn, R. and Straub, A. and Zudilin, W., A modular supercongruence for {\(_6F_5\)}: an {A}p\'ery-like story
[20] Roberts, D. and Rodriguez Villegas, F., Hypergeometric supercongruences · Zbl 1443.11022
[21] Rodriguez Villegas, Fernando, Hypergeometric families of {C}alabi–{Y}au manifolds, Calabi–{Y}au Varieties and Mirror Symmetry ({T}oronto, {ON, Fields Inst. Commun., 38, 223-231, (2003), Amer. Math. Soc., Providence, RI} · Zbl 1062.11038
[22] Rodriguez Villegas, Fernando, Hypergeometric motives · Zbl 1062.11038
[23] Rogers, M. and Wan, J. G. and Zucker, I. J., Moments of elliptic integrals and critical {\(L\)}-values, Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan, 37, 1, 113-130, (2015) · Zbl 1383.11048
[24] Scheidegger, Emanuel, Analytic continuation of hypergeometric functions in the resonant case · Zbl 1226.81246
[25] Shimura, Goro, The special values of the zeta functions associated with cusp forms, Communications on Pure and Applied Mathematics, 29, 6, 783-804, (1976) · Zbl 0348.10015
[26] Shimura, Goro, On the periods of modular forms, Mathematische Annalen, 229, 3, 211-221, (1977) · Zbl 0363.10019
[27] Silverman, Joseph H., The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, xx+513, (2009), Springer, Dordrecht · Zbl 1194.11005
[28] Slater, Lucy Joan, Generalized hypergeometric functions, xiii+273, (1966), Cambridge University Press, Cambridge
[29] Stienstra, Jan, Mahler measure variations, {E}isenstein series and instanton expansions, Mirror Symmetry. {V}, AMS/IP Stud. Adv. Math., 38, 139-150, (2006), Amer. Math. Soc., Providence, RI · Zbl 1118.11047
[30] Stienstra, Jan and Beukers, Frits, On the {P}icard–{F}uchs equation and the formal {B}rauer group of certain elliptic {\(K3\)}-surfaces, Mathematische Annalen, 271, 2, 269-304, (1985) · Zbl 0539.14006
[31] Swisher, Holly, On the supercongruence conjectures of van {H}amme, Research in the Mathematical Sciences, 2, Art. 18, 21 pages, (2015) · Zbl 1337.33005
[32] van Hamme, L., Some conjectures concerning partial sums of generalized hypergeometric series, {\(p\)}-Adic Functional Analysis ({N}ijmegen, 1996), Lecture Notes in Pure and Appl. Math., 192, 223-236, (1997), Dekker, New York · Zbl 0895.11051
[33] Verrill, H. A., Congruences related to modular forms, International Journal of Number Theory, 6, 6, 1367-1390, (2010) · Zbl 1209.11047
[34] Zagier, Don B., Arithmetic and topology of differential equations, Proceedings of the Seventh European Congress of Mathematics (Berlin, July 18-22, 2016), 717-776, (2018), European Mathematical Society, Berlin · Zbl 1471.11167
[35] Zeilberger, Doron, Gauss’s {\({}_2F_1(1)\)} cannot be generalized to {\({}_2F_1(x)\)}, Journal of Computational and Applied Mathematics, 39, 3, 379-382, (1992) · Zbl 0764.33001
[36] Zudilin, Wadim, Ramanujan-type formulae for {\(1/\pi \)}: a second wind?, Modular Forms and String Duality, Fields Inst. Commun., 54, 179-188, (2008), Amer. Math. Soc., Providence, RI · Zbl 1159.11053
[37] Zudilin, Wadim, Ramanujan-type supercongruences, Journal of Number Theory, 129, 8, 1848-1857, (2009) · Zbl 1231.11147
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