We prove that every modular function on a multilattice L with values in a topological Abelian group generates a uniformity on L which makes the multilattice operations uniformly continuous with respect to the exponent...
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We prove that every modular function on a multilattice L with values in a topological Abelian group generates a uniformity on L which makes the multilattice operations uniformly continuous with respect to the exponential uniformity on the power set of L.
We present an asymptotically fast algorithm for the numerical evaluation of modular functions such as the elliptic modular function j. Our algorithm makes use of the natural connection between the arithmetic-geometric...
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We present an asymptotically fast algorithm for the numerical evaluation of modular functions such as the elliptic modular function j. Our algorithm makes use of the natural connection between the arithmetic-geometric mean (ACM) of complex numbers and modular functions. Through a detailed complexity analysis, we prove that for a given tau, evaluating N significative bits of j(tau) can be done in time O(M(N)log N), where M(N) is the time complexity for the multiplication of two N-bit integers. However, this is only true for a fixed tau and the time complexity of this first algorithm greatly increases as Im(tau) does. We then describe a second algorithm that achieves the same time complexity independently of the value of tau in the classical fundamental domain F. We also show how our method can be used to evaluate other modular forms, such as the Dedekind eta function, with the same time complexity.
We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose q- expansions satisfy the following: fk( A;t) := q - k ( 1 + a( 1) q + a( 2) q 2 +...
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We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose q- expansions satisfy the following: fk( A;t) := q - k ( 1 + a( 1) q + a( 2) q 2 + center dot center dot center dot) + O( q), where a( n) are numbers satisfying a certain analytic condition. We show that the zeros of such fk( t) in the fundamental domain of SL2( Z) lie on | t | = 1 and are transcendental. We recover as a special case earlier work of Witten on extremal " partition" functions Zk( t). These functions were originally conceived as possible generalizations of constructions in three- dimensional quantum gravity.
The article describes the development of a formula for certain rational functions on a Hilbert modular surface and CM 0-cycles on the surface associated to a non-biquadratic quartic CM field which contains the real qu...
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The article describes the development of a formula for certain rational functions on a Hilbert modular surface and CM 0-cycles on the surface associated to a non-biquadratic quartic CM field which contains the real quadratic field of the Hilbert modular surface. One important feature is that the factorization is determined by the arithmetic of the reflex field.
Since the modular curve X(5) = Gamma(5)\h* has genus zero, we have a field isomorphism K(X(5)) approximate to C(X(2)(z)) where X(2)(z) is a product of Klein forms. We apply it to construct explicit class fields over a...
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Since the modular curve X(5) = Gamma(5)\h* has genus zero, we have a field isomorphism K(X(5)) approximate to C(X(2)(z)) where X(2)(z) is a product of Klein forms. We apply it to construct explicit class fields over an imaginary quadratic field K from the modular function j(Delta,25)(z) := X(2)(5z). And, for every integer N >= 7 we further generate ray class fields K((N)) over K with modulus N just from the two generators X(2)(z) and X(3)(z) of the function field K(X(1)(N)), which are also the product of Klein forms without using torsion points of elliptic curves.
This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu's algorit...
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This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu's algorithms, we present an algorithm to find Ramanujan-type identities for a(mn + t). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for p(11n + 6) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions p(5n + 2) and p(5n + 3) and Andrews-Paule's broken 2-diamond partition functions 2(25n+ 14) and 2(25n+ 24). It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews' singular overpartition functions Q3,1(9n + 3) and Q3,1(9n + 6) due to Shen, the 2-dissection formulas of Ramanujan, and the 8-dissection formulas due to Hirschhorn.
For a positive integer N divisible by 4, 5, 6,7 or 9, let O-1,O-N(Q) be the ring of weakly holomorphic modular functions for the congruence subgroup Gamma(1)(N) with rational Fourier coefficients. We present explicit ...
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For a positive integer N divisible by 4, 5, 6,7 or 9, let O-1,O-N(Q) be the ring of weakly holomorphic modular functions for the congruence subgroup Gamma(1)(N) with rational Fourier coefficients. We present explicit generators of the ring O-1,O-N(Q) over Q by making use of modular units which have infinite product expansions.
The Dedekind η-function is defined byand is well-known to satisfy a functional equation relating η(aτ + b/cτ + d) to η (τ), where a, b, c, d are rational integers with ad – bc = 1—see for instance Iseki [2], a...
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The Dedekind η-function is defined byand is well-known to satisfy a functional equation relating η(aτ + b/cτ + d) to η (τ), where a, b, c, d are rational integers with ad – bc = 1—see for instance Iseki [2], and the further references cited there.
One major theme of this paper concerns the expansion of modular forms and functions in terms of fractional (Puiseux) series. This theme is connected with another major theme, holonomic functions and sequences. With pa...
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One major theme of this paper concerns the expansion of modular forms and functions in terms of fractional (Puiseux) series. This theme is connected with another major theme, holonomic functions and sequences. With particular attention to algorithmic aspects, we study various connections between these two worlds. Applications concern partition congruences, Rieke-Klein relations, irrationality proofs a la Beukers, or approximations to pi studied by Ramanujan and the Borweins. As a major ingredient. to a "first guess, then prove" strategy, a new algorithm for proving differential equations for modular forms is introduced.
In 1967 the first author and Karl Stromberg published a theorem concerning generalized limits of Riemann sums on locally compact groups. The setting is a locally compact group G and an increasing sequence H-n of close...
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In 1967 the first author and Karl Stromberg published a theorem concerning generalized limits of Riemann sums on locally compact groups. The setting is a locally compact group G and an increasing sequence H-n of closed subgroups whose union is dense in G. The theorem was shown to hold provided that the restriction of the modular function on G to H-n agrees with the modular function of H-n for all large n. This hypothesis holds in many cases and, in fact, Ross and Stromberg were unable to determine whether the hypothesis was really needed for the theorem or even whether this hypothesis always holds. An example is provided which shows that this hypothesis does not always hold. It is then shown that the theorem fails without the hypothesis.
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