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35
On rationally parametrized modular equations
"... Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformati ..."
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Cited by 15 (0 self)
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Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformation of elliptic curve periods, which are parametrized by a Hauptmodul (function field generator). Since the periods satisfy a Picard–Fuchs equation, which is of hypergeometric, Heun, or more general type, the new equations can be viewed as algebraic transformation formulas for special functions. The ones for N = 4,3, 2 yield parametrized modular transformations of Ramanujan’s elliptic integrals of signatures 2, 3,4. The case of signature 6 will require an extension of the present theory, to one of modular equations for general elliptic surfaces.
Transformations of some Gauss Hypergeometric Functions
, 2004
"... This paper presents explicit algebraic transformations of some Gauss hypergeometric functions. Specifically, the transformations considered apply to hypergeometric solutions of hypergeometric differential equations with the local exponent differences 1/k, 1/ℓ,1/m such that k, ℓ, m are positive integ ..."
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Cited by 12 (3 self)
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This paper presents explicit algebraic transformations of some Gauss hypergeometric functions. Specifically, the transformations considered apply to hypergeometric solutions of hypergeometric differential equations with the local exponent differences 1/k, 1/ℓ,1/m such that k, ℓ, m are positive integers and 1/k + 1/ℓ + 1/m < 1. All algebraic transformations of these Gauss hypergeometric functions are considered.
Solving second order linear differential equations with Klein’s theorem
 In ISSAC’05
, 2005
"... Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem ..."
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Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein’s theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semiinvariants. 1.
Explicit formula for the generating series of diagonal 3D rook paths
 SEMIN. LOTHAR. COMBIN
, 2011
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The modular curve as the space of stability conditions of a CY3 algebra. Preprint, available at arXiv:1111.4184
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Darboux evaluations of algebraic Gauss hypergeometric functions. Available at http://arxiv.org/math.CA/0504264
, 2005
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Renormalization, isogenies and rational symmetries of differential equations
, 911
"... Abstract. We give an example of infinite order rational transformation that leaves a linear differential equation covariant. This example can be seen as a nontrivial but still simple illustration of an exact representation of the renormalization group. ..."
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Abstract. We give an example of infinite order rational transformation that leaves a linear differential equation covariant. This example can be seen as a nontrivial but still simple illustration of an exact representation of the renormalization group.
Transformations of algebraic Gauss hypergeometric functions
, 2003
"... A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pullback of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are hypergeometric equations with tetrahedral, octahedral or icosahedra ..."
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Cited by 5 (2 self)
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A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pullback of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are hypergeometric equations with tetrahedral, octahedral or icosahedral monodromy groups. We give an algorithm for computing Klein’s pullback coverings in these cases, based on certain explicit expressions (Darboux evaluations) of algebraic hypergeometric functions. The explicit expressions can be computed from a data base (covering the Schwarz table) and using contiguous relations. Klein’s pullback transformations also induce algebraic transformations between hypergeometric solutions and a standard hypergeometric function with the same finite monodromy group.
Belyi functions for hyperbolic hypergeometrictoHeun transformations
, 1212
"... One place where Belyi functions occur is pullback transformations of hypergeometric differential equations to Fuchsian equations with few singularities. This paper presents a complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations (i.e., canonic ..."
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One place where Belyi functions occur is pullback transformations of hypergeometric differential equations to Fuchsian equations with few singularities. This paper presents a complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations (i.e., canonical Fuchsian equations with 4 singularities). The considered hypergeometric equations have the local exponent differences 1/k, 1/ℓ, 1/m that satisfy k, ℓ, m ∈ N and the hyperbolic condition 1/k + 1/ℓ + 1/m < 1. In total, we find 872 such Belyi functions up to Möbius transformations, in 366 Galois orbits. Their maximal degree is 60, which is well beyond reach of standard computational methods. To obtain these Belyi functions, we developed two efficient algorithms that exploit the implied hypergeometrictoHeun transformations.