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1.
In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein
series of weight 2, 4 and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein series
and obtain them in a natural way as coefficients of a family of elliptic curves. The fact that a complex manifold over the
moduli of polarized Hodge structures in the case h
10= h
01=1 has an algebraic structure with an action of an algebraic group plays a basic role in all of the proofs.
相似文献
2.
We show that certain p-adic Eisenstein series for quaternionic modular groups of degree 2 become “real” modular forms of level p in almost all cases. To prove this, we introduce a U( p) type operator. We also show that there exists a p-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of p-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic). 相似文献
3.
Following Wolpert, we find a set of asymptotic relations among the Fourier coefficients of real-analytic Eisenstein series. The relations are found by evaluating the integral of the product of an Eisenstein series with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of ; the first of these evaluations immediately gives us one coefficient, while the other evaluation provides us with a sum of Fourier coefficients. The second evaluation of the integral is done using stationary phase asymptotics in the parameter is the eigenvalue of ) for a cubic phase. As applications we find sets of asymptotic relations for divisor functions. 相似文献
4.
An asymptotic series in Ramanujan’s second notebook (Entry 10, Chap. 3) is concerned with the behavior of the expected value of φ( X) for large λ where X is a Poisson random variable with mean λ and φ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of φ( X) when the distribution of X belongs to a suitable family indexed by a convolution parameter. Examples include the binomial, negative binomial, and gamma families. Some formulas associated with the negative binomial appear new. 相似文献
5.
The known WZ-proofs for Ramanujan-type series related to 1/ π gave us the insight to develop a new proof strategy based on the WZ-method. Using this approach we are able to find more generalizations and discover first WZ-proofs for certain series of this type. 相似文献
7.
The aim of this paper is to study certain multiple series which can be regarded as multiple analogues of Eisenstein series. As part of a prior research, the second-named author considered double analogues of Eisenstein series and expressed them as polynomials in terms of ordinary Eisenstein series. This fact was derived from the analytic observation of infinite series involving hyperbolic functions which were based on the study of Cauchy, and also Ramanujan. In this paper, we prove an explicit relation formula among these series. This gives an alternative proof of this fact by using the technique of partial fraction decompositions of multiple series which was introduced by Gangl, Kaneko and Zagier. By the same method, we further show a certain multiple analogue of this fact and give some examples of explicit formulas. Finally we give several remarks about the relation between the results of the present and the previous works for infinite series involving hyperbolic functions. 相似文献
9.
S. Ramanujan gave fourteen families of series in his Second Notebook in Chap. 17, Entries 13–17. In each case he gave only
the first few examples, giving us the motivation to find and prove a general formula for each family of series. The aim of
this paper is to develop a powerful tool (four versatile functions f
0, f
1, f
2, and f
3) to collect all of Ramanujan’s examples together.
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11.
The power series spaces of finite type, A1( α), and infinite type, A∞( α), are the most known and important examples of non-Archimedean nuclear Fréchet spaces. We study when Aπ ( α) has a subspace (or quotient) isomorphic to Aq( b). 相似文献
13.
We describe the span of Hecke eigenforms of weight four with nonzero central value of L-function in terms of Wronskians of certain weight one Eisenstein series. 相似文献
15.
In their last joint paper, Hardy and Ramanujan examined the coefficients of modular forms with a simple pole in a fundamental region. In particular, they focused on the reciprocal of the Eisenstein series . In letters written to Hardy from nursing homes, Ramanujan stated without proof several more results of this sort. The purpose of this paper is to prove most of these claims. 相似文献
16.
In this paper we introduce some modular forms of half-integral weight on congruence group Г o(4N) with N an odd positive integer which can be viewed as a natural generalization of Cohen-Eisenstein series. Using these series, we
can prove that the restriction of Shimura lifting on Eisenstein space E
k+1/2
+
(4 N,χ l) gives an isomorphism from E
k+1/2
+
(4 N,χ l) to E
2k( N). We consider some congruence relationships between modular forms in use of Shimura lifting. 相似文献
17.
Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas. 相似文献
18.
In this article, we determine the spectral expansion, meromorphic continuation, and location of poles with identifiable singularities
for the scalar-valued hyperbolic Eisenstein series. Similar to the form-valued hyperbolic Eisenstein series studied in Kudla
and Millson (Invent Math 54:193–211, 1979), the scalar-valued hyperbolic Eisenstein series is defined for each primitive,
hyperbolic conjugacy class within the uniformizing group associated to any finite volume hyperbolic Riemann surface. Going
beyond the results in Kudla and Millson (Invent Math 54:193–211, 1979) and Risager (Int Math Res Not 41:2125–2146, 2004),
we establish a precise spectral expansion for the hyperbolic Eisenstein series for any finite volume hyperbolic Riemann surface
by first proving that the hyperbolic Eisenstein series is in L
2. Our other results, such as meromorphic continuation and determination of singularities, are derived from the spectral expansion. 相似文献
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