Browsing by Author "Yang, Yifan"

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A basis for S (k) (I" (0)(4)) and representations of integers as sums of squares
(2012-05-01)
Fukuhara, Shinji
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
In this paper, we find a basis for the space S (k) (I" (0)(4)) of cusp forms of even weight k for the congruence subgroup I" (0)(4) in terms of Eisenstein series. As an application, we obtain formulas for r (4s) (n), the number of ways to represent a nonnegative integer n as a sum of 4s integral squares.
A SHORT PROOF OF MILNE'S FORMULAS FOR SUMS OF INTEGER SQUARES
(2005-12-01)
Long, Ling
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
We give a short proof of Milne's formulas for sums of 4n(2) and 4n(2) + 4n integer squares using the theory of modular forms. Other identities of Milne are also discussed.
ALGEBRAIC TRANSFORMATIONS OF HYPERGEOMETRIC FUNCTIONS AND AUTOMORPHIC FORMS ON SHIMURA CURVES
(2013-12-01)
Tu, Fang-Ting
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
In this paper, we will obtain new algebraic transformations of the F-2(1)-hypergeometric functions. The main novelty in our approach is the interpretation of identities among F-2(1)-hypergeometric functions as identities among automorphic forms on different Shimura curves.
Apery limits and special values of L-functions
(2008-07-01)
Yang, Yifan
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應用數學系
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Department of Applied Mathematics
We describe a general method to determine the Apery limits of a differential equation that has a modular-function origin. As a by-product of our analysis, we discover a family of identities involving the special values of L-functions associated with modular forms. The proof of these identities is independent of differential equations and Apery limits. (C) 2008 Elsevier Inc. All rights reserved.
Automorphisms of hyperelliptic modular curves X(0)(N) in positive characteristic
(2010-01-01)
Kontogeorgis, Aristides
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
We study the automorphism groups of the reduction X(0)(N) x (F) over bar (p) of a modular curve X(0)(N) over primes p inverted iota N.
Automorphisms of hyperelliptic modular curves X-0(N) in positive characteristic
(2010-01-01)
Kontogeorgis, Aristides
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
We study the automorphism groups of the reduction X-0(N) x (F) over bar (p) of a modular curve X-0(N) over primes p inverted iota N.
BASES FOR S-k(Gamma(1)(4)) AND FORMULAS FOR EVEN POWERS OF THE JACOBI THETA FUNCTION
(2013-12-01)
Fukuhara, Shinji
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
We find a basis for the space S-k(Gamma(1)(4)) of cusp forms of weight k for the congruence subgroup G1(4) in terms of Eisenstein series. As an application, we obtain formulas for r(2k)(n), the number of ways to represent a non-negative integer n as sums of 2k integer squares.
CONGRUENCES MODULO 5 AND 7 FOR 4-COLORED GENERALIZED FROBENIUS PARTITIONS
(2017-10-01)
Chan, Heng Huat
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Wang, Liuquan
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
Let c phi(k)(n) denote the number of k-colored generalized Frobenius partitions of n. Recently, new Ramanujan-type congruences associated with c phi(4)(n) were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as c phi(4)(14n + 6) 0 mod 7 and Seller's congruence c phi(4)(10n + 6) 0 mod 5.
Congruences of the Partition Function
(2011)
Yang, Yifan
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應用數學系
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Department of Applied Mathematics
Let p(n) denote the partition function. In this article, we will show that congruences of the form p(ml(k)n+B)= 0 mod m for all n >= 0 exist for all primes m and l satisfying m >= 13 and l l = 2, 3, m, where B is a suitably chosen integer depending on m and l. Here, the integer k depends on the Hecke eigenvalues of a certain invariant subspace of S(m/2-1)(G(0)(576), chi(12)) and can be explicitly computed. More generally, we will show that for each integer i > 0 there exists an integer k such that with a properly chosen B the congruence p(m(i) l(k) n+B) equivalent to 0 mod m(i) holds for all integers n not divisible by l.
Defining equations of modular curves
(2006-08-20)
Yang, Yifan
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應用數學系
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Department of Applied Mathematics
We obtain defining equations of modular curves X-0(N), X-1(N), and X(N) by explicitly constructing modular functions using generalized Dedekind eta functions. As applications, we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup. We also use our model of X-0(37) to find explicit modular parameterization of rational elliptic curves of conductor 37. (C) 2005 Elsevier Inc. All rights reserved.
DEFINING EQUATIONS OF X(0)(2(2n))
(2009-03-01)
Tu, Fang-Ting
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
In this note we will obtain defining equations of modular curves X(0)(2(2n)). The key ingredient is a recursive formula for certain generators of the function fields on X(0)(2(2n)).
Determination of hauptmoduls and construction of abelian extensions of quadratic number fields
(2007-09-01)
Chiang-Hsieh, Hung-Jen
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
We obtain Hauptmoduls of genus zero congruence subgroups of the type T-0(+)(p) := T-0(p) + wp, where p is a prime and wp is the Atkin-Lehner involution. We then use the Hauptmoduls, along with modular functions on T, (p) to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.
Differential equations satisfied by modular forms and K3 surfaces
(2007-06-01)
Yang, Yifan
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Yui, Noriko
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應用數學系
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Department of Applied Mathematics
We study differential equations satisfied by modular forms of two variables associated to Gamma(1) x Gamma(2), where Gamma(i) (i = 1, 2) are genus zero subgroups of SL2(R) commensurable with SL2(Z), e.g., Gamma(0)(N) or Gamma(0)(N)* for some N. In some examples, these differential equations are realized as the Picard-Fuchs differential equations of families of K3 surfaces with large Picard numbers, e.g., 19,18,17,16. Our method rediscovers some of the Lian-Yau examples of "modular relations" involving power series solutions to the second and the third order differential equations of Fuchsian type in [14], [15].
Equations of hyperelliptic Shimura curves
(2017-01)
Guo, Jia-Wei
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
By constructing suitable Borcherds forms on Shimura curves and using Schofer\'s formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves X-0(D) (N). As a byproduct, we also address the problem of whether a modular form on Shimura curves X-0(D) (N)/W-D,W-N with a divisor supported on CM divisors can be realized as a Borcherds form, where X-0(D) (N)/W-D,W-N denotes the quotient of X-0(D) (N) by all of the Atkin-Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.
Jacobsthal identity for Q(root-2)
(2012-11-01)
Hashimoto, Ki-ichiro
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Long, Ling
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Yang, Yifan
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應用數學系
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Department of Applied Mathematics
Let p be a prime congruent to 1 or 3 modulo 8 so that the equation p = a(2) + 2b(2) is solvable in integers. In this paper, we obtain closed-form expressions for a and b in terms of Jacobsthal sums. This is analogous to a classical identity of Jacobsthal.
MODULAR FORMS OF HALF-INTEGRAL WEIGHTS ON SL(2, Z)
(2014-09-01)
Yang, Yifan
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應用數學系
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Department of Applied Mathematics
In this paper, we prove that, for an integer r with (r, 6) = 1 and 0 < r < 24 and a nonnegative even integer s, the set {eta(24 tau)(r) f(24 tau) is an element of M-s (1)} is isomorphic to S-r+2s-1(new) (6, -(8/r), -(12/r)) circle times (12/.) as Hecke modules under the Shimura correspondence. Here M-s (1) denotes the space of modular forms of weight s on Gamma(0)(1) = SL(2, Z), S-2k(new) (6, is an element of(2), is an element of(3)) is the space of newforms of weight 2k on Gamma(0)(6) that are eigenfunctions with eigenvalues is an element of(2) and is an element of(3) for Atkin-Lehner involutions W-2 and W-3 respectively, and the notation circle times(12/.). There is also an analogous results for the cases (r, 6) = 3.
Modular units and cuspidal divisor class groups of X(1)(N)
(2009-07-15)
Yang, Yifan
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應用數學系
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Department of Applied Mathematics
In this article, we consider the group F(1)(infinity)(N) of modular units on X(1)(N) that have divisors supported on the cusps lying over infinity of X(0)(N), called the infinity-cusps. For each positive integer N, we will give an explicit basis for the group F(1)(infinity)(N). This enables us to compute the group structure of the rational torsion subgroup l(1)(infinity)(N) of the Jacobian J(1)(N) of X(1)(N) generated by the differences of the infinity-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the p-primary part of l(1)(infinity)(p(n)) for a regular prime p. (C) 2009 Elsevier Inc. All rights reserved.
Modular units and cuspidal divisor class groups of X-1(N)
(2009-07-15)
Yang, Yifan
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應用數學系
;
Department of Applied Mathematics
In this article, we consider the group F-1(infinity)(N) of modular units on X-1(N) that have divisors supported on the cusps lying over infinity of X-0(N), called the infinity-cusps. For each positive integer N, we will give an explicit basis for the group F-1(infinity)(N). This enables us to compute the group structure of the rational torsion subgroup l(1)(infinity)(N) of the Jacobian J(1)(N) of X-1(N) generated by the differences of the infinity-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the p-primary part of l(1)(infinity)(p(n)) for a regular prime p. (C) 2009 Elsevier Inc. All rights reserved.
Neutrino physics with JUNO
(2016-03)
An, Fengpeng
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An, Guangpeng
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An, Qi
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Antonelli, Vito
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Baussan, Eric
The Jiangmen Underground Neutrino Observatory (JUNO), a 20 kton multipurpose underground liquid scintillator detector, was proposed with the determination of the neutrino mass hierarchy (MH) as a primary physics goal. The excellent energy resolution and the large fiducial volume anticipated for the JUNO detector offer exciting opportunities for addressing many important topics in neutrino and astro-particle physics. In this document, we present the physics motivations and the anticipated performance of the JUNO detector for various proposed measurements. Following an introduction summarizing the current status and open issues in neutrino physics, we discuss how the detection of antineutrinos generated by a cluster of nuclear power plants allows the determination of the neutrino MH at a 3-4 sigma significance with six years of running of JUNO. The measurement of antineutrino spectrum with excellent energy resolution will also lead to the precise determination of the neutrino oscillation parameters sin(2) theta(12), Delta m(21)(2), and vertical bar Delta m(ee)(2)vertical bar to an accuracy of better than 1%, which will play a crucial role in the future unitarity test of the MNSP matrix. The JUNO detector is capable of observing not only antineutrinos from the power plants, but also neutrinos/antineutrinos from terrestrial and extra-terrestrial sources, including supernova burst neutrinos, diffuse supernova neutrino background, geoneutrinos, atmospheric neutrinos, and solar neutrinos. As a result of JUNO\'s large size, excellent energy resolution, and vertex reconstruction capability, interesting new data on these topics can be collected. For example, a neutrino burst from a typical core-collapse supernova at a distance of 10 kpc would lead to similar to 5000 inverse-beta-decay events and similar to 2000 all-flavor neutrino-proton ES events in JUNO, which are of crucial importance for understanding the mechanism of supernova explosion and for exploring novel phenomena such as collective neutrino oscillations. Detection of neutrinos from all past core-collapse supernova explosions in the visible universe with JUNO would further provide valuable information on the cosmic star-formation rate and the average core-collapse neutrino energy spectrum. Antineutrinos originating from the radioactive decay of uranium and thorium in the Earth can be detected in JUNO with a rate of similar to 400 events per year, significantly improving the statistics of existing geoneutrino event samples. Atmospheric neutrino events collected in JUNO can provide independent inputs for determining the MH and the octant of the theta(23) mixing angle. Detection of the Be-7 and B-8 solar neutrino events at JUNO would shed new light on the solar metallicity problem and examine the transition region between the vacuum and matter dominated neutrino oscillations. Regarding light sterile neutrino topics, sterile neutrinos with 10-(5) eV(2) < Delta m(41)(2) < 10(-2) and a sufficiently large mixing angle theta(14) could be identified through a precise measurement of the reactor antineutrino energy spectrum. Meanwhile, JUNO can also provide us excellent opportunities to test the eV-scale sterile neutrino hypothesis, using either the radioactive neutrino sources or a cyclotron-produced neutrino beam. The JUNO detector is also sensitive to several other beyondthe-standard-model physics. Examples include the search for proton decay via the p -> K++ <(v)over bar> decay channel, search for neutrinos resulting from dark-matter annihilation in the Sun, search for violation of Lorentz invariance via the sidereal modulation of the reactor neutrino event rate, and search for the effects of non-standard interactions. The proposed construction of the JUNO detector will provide a unique facility to address many outstanding crucial questions in particle and astrophysics in a timely and cost-effective fashion. It holds the great potential for further advancing our quest to understanding the fundamental properties of neutrinos, one of the building blocks of our Universe.
New analogues of Clausen's identities arising from the theory of modular forms
(2011-10-01)
Chan, Heng Huat
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Tanigawa, Yoshio
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Yang, Yifan
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Zudilin, Wadim
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應用數學系
Around 1828, T. Clausen discovered that the square of certain hypergeometric (2)F(1) function can be expressed as a hypergeometric (3)F(2) function. Special cases of Clausen's identities were later used by S. Ramanujan in his derivation of 17 series for 1/pi. Since then, there were several attempts to find new analogues of Clausen's identities with the hope to derive new classes of series for 1/pi. Unfortunately, none were successful. In this article, we will present three new analogues of Clausen's identities. Their discovery is motivated by the study of relations between modular forms of weight 2 and modular functions associated with modular groups of genus 0. (C) 2011 Elsevier Inc. All rights reserved.