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Anharmonic solutions to the Riccati equation and elliptic modular functions

  • Ahmed Sebbar EMAIL logo and Oumar Wone
Published/Copyright: August 5, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 2

Abstract

We study the irreducible algebraic equation

xn+a1⁢xn-1+⋯+an=0,with n≥4,

on the differential field (𝔽=ℂ⁢(t),δ=dd⁢t). We assume that a root of the equation is a solution to the Riccati differential equation u′+B0+B1⁢u+B2⁢u2=0, where B0, B1, B2 are in 𝔽.

We show how to construct a large class of polynomials as in the above algebraic equation, i.e., we prove that there exists a polynomial Fn⁢(x,y)∈ℂ⁢(x)⁢[y] such that for almost T∈𝔽∖ℂ, the algebraic equation Fn⁢(x,T)=0 is of the same type as the above stated algebraic equation. In other words, all its roots are solutions to the same Riccati equation. On the other hand, we give an example of a degree 3 irreducible polynomial equation satisfied by certain weight 2 modular forms for the subgroup Γ⁢(2), whose roots satisfy a common Riccati equation on the differential field (ℂ⁢(E2,E4,E6),dd⁢τ), with Ei⁢(τ) being the Eisenstein series of weight i. These solutions are related to a Darboux–Halphen system. Finally, we deal with the following problem: For which “potential” q∈ℂ⁢(℘,℘′) does the Riccati equation d⁢Yd⁢z+Y2=q admit algebraic solutions over the differential field ℂ⁢(℘,℘′), with ℘ being the classical Weierstrass function? We study this problem via Darboux polynomials and invariant theory and show that the minimal polynomial Φ⁢(x) of an algebraic solution u must have a vanishing fourth transvectant τ4⁢(Φ)⁢(x).

Keywords: Riccati equation; Darboux–Halphen system; differential Galois theory; Darboux polynomial; modular forms; Cayley’s process; Wedekind theorem; transvectants
MSC 2010: 34A26; 34A34; 34C14; 12H05; 11F03; 33E05; 12H20; 34M15

Communicated by Jan Bruinier

Acknowledgements

The authors would like to thank the anonymous referee for her/his comments and careful reading.

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Received: 2017-2-6
Revised: 2017-5-9
Published Online: 2017-8-5
Published in Print: 2018-3-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The variance of divisor sums in arithmetic progressions
  3. Characterizing Lie groups by controlling their zero-dimensional subgroups
  4. CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
  5. Towards a Goldberg–Shahidi pairing for classical groups
  6. On the non-existence of cyclic splitting fields for division algebras
  7. Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
  8. Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
  9. Anharmonic solutions to the Riccati equation and elliptic modular functions
  10. A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
  11. Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
  12. On symplectic semifield spreads of PG(5,q2), q odd
  13. From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
  14. Golodness and polyhedral products for two-dimensional simplicial complexes
https://doi.org/10.1515/forum-2017-0025
Keywords for this article
Riccati equation; Darboux–Halphen system; differential Galois theory; Darboux polynomial; modular forms; Cayley’s process; Wedekind theorem; transvectants

Articles in the same Issue

  1. Frontmatter
  2. The variance of divisor sums in arithmetic progressions
  3. Characterizing Lie groups by controlling their zero-dimensional subgroups
  4. CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
  5. Towards a Goldberg–Shahidi pairing for classical groups
  6. On the non-existence of cyclic splitting fields for division algebras
  7. Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
  8. Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
  9. Anharmonic solutions to the Riccati equation and elliptic modular functions
  10. A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
  11. Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
  12. On symplectic semifield spreads of PG(5,q2), q odd
  13. From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
  14. Golodness and polyhedral products for two-dimensional simplicial complexes
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