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Showing 1 to 20 of 45 for “"Modular forms"”.
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Modular group and modular forms
… of SL2(Z) and related groups. We define modular forms for this group and develop the basic theory. We then use the theory of lattices to construct examples of modular forms.
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On Hermitian theta series and modular forms
In their paper glqq Hermitian quadratic forms and Hermitian modular formsgrqq ~(1978) the authors D.M. Cohen und H.L. Resnikoff introduce the construction of Hermitian modular forms using theta series. One starts with lattices over imaginary quadratic fields, which are, when considered as …
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Modular forms for the orthogonal group O(2,5)
This thesis deals with modular forms for orthogonal groups of signature (2,n), n>2, with particular emphasis on the case n=5. First we determine suitable generators and the abelian characters of some orthogonal groups. Then we present a few methods for constructing orthogonal modular forms: by …
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Iwasawa theory for modular forms at supersingular primes
… other inclusion of the main conjecture for CM modular forms, generalising works of Pollack and Rubin on CM elliptic curves. As a key step of the proof, we generalise the reciprocity law of Coates-Wiles and Rubin. Next, we study Wach modules associated to positive crystalline p-adic …
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Congruence Properties of Fourier Coefficients of Modular Forms
Fourier coefficients of modular forms have profound connections with many areas of number theory. We will consider three different applications of these coefficients. First, we extend the Apery number supercongruence, proving an observation of Rodriguez-Villegas. Second, we prove an analogue of …
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Fourier Coefficients of Modular Forms and Their Applications
The theory of modular forms, as it has been developed over the past several decades, has highlighted deep connections between the areas of analytic and algebraic number theory and arithmetic geometry. In this thesis we explore some applications. First, we give some new and simpler proofs of recent …
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Modular forms, the Shimura correspondence, and arithmetic applications
Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-05-01
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Iwasawa theory for tensor products of Hilbert modular forms
… the Iwasawa Main Conjecture, applied to Hilbert modular forms and their tensor products. Greenberg and Vatsal developed an approach to study the main conjecture for a large class of elliptic curves simultaneously. They showed that if a given pair of elliptic curves share the same residual Galois …
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Congruences for the Coefficients of Weakly Holomorphic Modular Forms
Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. In each case, the values of the given arithmetic function appear as the Fourier coefficients of a weakly holomorphic modular form. We show that similar …
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p-adic modular forms over Shimura curves over Q
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999.
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Siegel modulator form (mod p) and algebraic modular forms
… that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions ... , where B is the endomorphism algebra of a supersingular elliptic curve. After giving a detailed exposition of Serre's result, we prove that the systems of Hecke …
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Congruences in modular, Jacobi, Siegel, and mock modular forms with applications
We study congruences in the coefficients of modular and other automorphic forms. Ramanujan famously found congruences for the partition function like p(5n+4) = 0 mod 5. For a wide class of modular forms, we classify the primes for which there can be analogous congruences in the coefficients of the …
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Effective estimates for sums of Kloosterman sums, modular forms, and applications
Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms
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Hilbert modular forms for the fields Q( 5), Q( 13) and Q( 17)
… had investigated both the theory of elliptic modular functions and the theory of algebraic number fields and came up with the idea to link these areas. Elliptic modular functions are meromorphic maps from the upper half plane H into the Riemannsphere, which are invariant under the …
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Comparison of Integral Structures on the Space of Modular Forms of Full Level N
… different integral structures on the space of modular forms over the rationals Q, one coming from arithmetic via q-expansions, the other coming from geometry via integral models of modular curves. Both structures are stable under the Hecke operators; furthermore, their quotient is finite …
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Ramanujan's formulas for the coefficients in the power series expansions of certain modular forms
… in the power series expansions of certain modular forms. We prove his formulas for the coefficients of 1/$E\sb4, E\sb4/E\sb6$ and other functions involving the Eisenstein series $E\sb4, E\sb6$ and $E\sbsp{2}{*}$. These formulas are stated, without proof, in a three-page manuscript published …
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SEMISTABLE MODELS OF HYPERELLIPTIC CURVES IN THE WILD CASE & DIFFERENTIAL OPERATORS ON P-ADIC MODULAR FORMS
… of differential operators on sheaves of p-adic modular forms.
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Lineare Unabhängigkeit von Hermiteschen Modulformen
In the theory of modular forms it is desired to be able to validate the linear independance of modular forms by calculation. Normally, this is done by the calculation of the Fourier coefficients of the modular forms. In general, no exact bounds are known for the number of Fourier coefficients that …
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Performance Engineering of Modular Symbols
… computes information about newform subspaces for modular forms of weight 2 and trivial character. Modular forms are certain functions in mathematics that appear in many different subfields of mathematics, including number theory and complex analysis; newform subspaces are spaces spanned by a …
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