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Showing 1 to 20 of 40 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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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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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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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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An Arithmetic-geometric Reciprocity between Theta Functions Attached to Real and Imaginary Quadratic Fields
… to construct classical holomorphic modular forms associated to ideal classes in quadratic number fields. These modular forms are theta functions that were originally introduced by Hecke in the 1920s andhave been investigated by several authors since. Our framework allows us to prove …
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Arithmetic of partition functions and q-combinatorics
… of mathematics such as q-series, the theory of modular forms, representation theory, symmetric functions and mathematical physics. Among these, we study the arithmetic of partition functions and q-combinatorics via bijective methods, q-series and modular forms. In particular, regarding …
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Hermitian Maass Lift for General Level
… show that the space of special hermitian Jacobi forms of level $N$ is isomorphic to the space of plus forms of level $DN$ and nebentypus $\chi$ (the hermitian analogue of Kohnen's plus space) for any integer $N$ prime to $D$. This generalizes the results of Krieg from $N = 1$ to arbitrary level. …
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Proven Cases of a Generalization of Serre's Conjecture
… Serre conjectured a correspondence between modular forms and two-dimensional Galois representations. Ash, Doud, and Pollack have extended this conjecture to a correspondence between Hecke eigenclasses in arithmetic cohomology and n-dimensional Galois representations. We present some of the …
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