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Full-Text Articles in Physical Sciences and Mathematics
An Analysis Of Antichimeral Ramanujan Type Congruences For Quotients Of The Rogers-Ramanujan Functions, Ryan A. Mowers
An Analysis Of Antichimeral Ramanujan Type Congruences For Quotients Of The Rogers-Ramanujan Functions, Ryan A. Mowers
Theses and Dissertations
This paper proves the existence of antichimeral Ramanujan type congruences for certain modular forms These modular forms can be represented in terms of Klein forms and the Dedekind eta function. The main focus of this thesis is to introduce the necessary theory to characterize these specific Ramanujan type congruences and prove their antichimerality.
Congruences For Quotients Of Rogers-Ramanujan Functions, Maria Del Rosario Valencia Arevalo
Congruences For Quotients Of Rogers-Ramanujan Functions, Maria Del Rosario Valencia Arevalo
Theses and Dissertations
In 1919 the mathematician Srinivasa Ramanujan conjectured congruences for the partition function p(n) modulo powers of the primes 5,7,11. In this work, we study Ramanujan type congruences modulo powers of primes p = 7,11,13,17,19,23 satisfied by the Fourier coefficients of quotients the Rogers-Ramanujan Functions G(τ) and H(τ) and the Dedekind eta function η(5τ). In addition to deriving new congruences, we develop the foundational theory of modular forms to motivate and prove the results. The work includes proofs of congruences facilitated by Python/SageMath code.
Some Properties And Applications Of Spaces Of Modular Forms With Eta-Multiplier, Cuyler Daniel Warnock
Some Properties And Applications Of Spaces Of Modular Forms With Eta-Multiplier, Cuyler Daniel Warnock
Theses and Dissertations
This dissertation considers two topics. In the first part of the dissertation, we prove the existence of fourteen congruences for the $p$-core partition function of the form given by Garvan in \cite{G1}. Different from the congruences given by Garvan, each of the congruences we give yield infinitely many congruences of the form $$a_p(\ell\cdot p^{t+1} \cdot n + p^t \cdot k - \delta_p) \equiv 0 \pmod \ell.$$ For example, if $t \geq 0$ and $\sfrac{m}{n}$ is the Jacobi symbol, then we prove $$a_7(7^t \cdot n - 2) \equiv 0 \pmod 5, \text{ \ \ if $\bfrac{n}{5} = 1$ and $\bfrac{n}{7} = …
Congruences For Coefficients Of Modular Functions In Levels 3, 5, And 7 With Poles At 0, Ryan Austin Keck
Congruences For Coefficients Of Modular Functions In Levels 3, 5, And 7 With Poles At 0, Ryan Austin Keck
Theses and Dissertations
We give congruences modulo powers of p in {3, 5, 7} for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and Jenkins and continuing work done by the Jenkins, the author, and Moss. The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at infinity.
Spaces Of Weakly Holomorphic Modular Forms In Level 52, Daniel Meade Adams
Spaces Of Weakly Holomorphic Modular Forms In Level 52, Daniel Meade Adams
Theses and Dissertations
Let M#k(52) be the space of weight k level 52 weakly holomorphic modular forms with poles only at infinity, and S#k(52) the subspace of forms which vanish at all cusps other than infinity. For these spaces we construct canonical bases, indexed by the order of vanishing at infinity. We prove that the coefficients of the canonical basis elements satisfy a duality property. Further, we give closed forms for the generating functions of these basis elements.
Weakly Holomorphic Modular Forms In Prime Power Levels Of Genus Zero, David Joshua Thornton
Weakly Holomorphic Modular Forms In Prime Power Levels Of Genus Zero, David Joshua Thornton
Theses and Dissertations
Let N ∈ {8,9,16,25} and let M#0(N) be the space of level N weakly holomorphic modular functions with poles only at the cusp at infinity. We explicitly construct a canonical basis for M#0(N) indexed by the order of the pole at infinity and show that many of the coefficients of the elements of these bases are divisible by high powers of the prime dividing the level N. Additionally, we show that these basis elements satisfy an interesting duality property. We also give an argument that extends level 1 results …
Explicit Computations Supporting A Generalization Of Serre's Conjecture, Brian Francis Hansen
Explicit Computations Supporting A Generalization Of Serre's Conjecture, Brian Francis Hansen
Theses and Dissertations
Serre's conjecture on the modularity of Galois representations makes a connection between two-dimensional Galois representations and modular forms. A conjecture by Ash, Doud, and Pollack generalizes Serre's to higher-dimensional Galois representations. In this paper we discuss an explicit computational example supporting the generalized claim. An ambiguity in a calculation within the example is resolved using a method of complex approximation.