Unitary groups and theta functions

Abstract We consider cycles on three-dimensional Shimura varieties attached to unitary groups, defined over extensions of a complex multiplication (CM) field $E$, which appear in the context of the conjectures of Gan et al. [6]. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of [8], and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda $-module constructed from Heegner points.

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2 Jacobi’s theta functions of one variable and the triple product identity

Theta functions, elliptic functions and π ◽

10.1515/9783110541915-002 ◽

2020 ◽

pp. 15-28

Keyword(s):

Theta Functions ◽

Triple Product ◽

Product Identity

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Second-order integrable Lagrangians and WDVV equations

Letters in Mathematical Physics ◽

10.1007/s11005-021-01403-3 ◽

2021 ◽

Vol 111 (2) ◽

Author(s):

E. V. Ferapontov ◽

M. V. Pavlov ◽

Lingling Xue

Keyword(s):

Lagrangian Density ◽

Theta Functions ◽

Second Order ◽

Elementary Functions ◽

Lagrange Equations ◽

Wdvv Equations ◽

Integrability Conditions ◽

Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

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Higher depth mock theta functions and q-hypergeometric series

Forum Mathematicum ◽

10.1515/forum-2021-0013 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Joshua Males ◽

Andreas Mono ◽

Larry Rolen

Keyword(s):

Theta Function ◽

Modular Forms ◽

Hypergeometric Series ◽

Theta Functions ◽

Mock Theta Function ◽

Mock Theta Functions ◽

Maass Forms ◽

Mock Modular Forms ◽

Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

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The intersection graph of a finite simple group has diameter at most 5

Archiv der Mathematik ◽

10.1007/s00013-021-01583-3 ◽

2021 ◽

Author(s):

Saul D. Freedman

Keyword(s):

Simple Group ◽

Upper Bound ◽

Finite Simple Group ◽

Unitary Groups ◽

Intersection Graph ◽

Monster Group ◽

Prime Dimension

AbstractLet G be a non-abelian finite simple group. In addition, let $$\Delta _G$$ Δ G be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of $$\Delta _G$$ Δ G has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

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