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Articles 1 - 6 of 6
Full-Text Articles in Number Theory
Monomial Quadratic Identities Of Hecke Eigenforms, Trevor Vilardi
Monomial Quadratic Identities Of Hecke Eigenforms, Trevor Vilardi
All Dissertations
Duke and Ghate independently studied the question of when it is possible for the product of two eigenforms to be an eigenform. In this dissertation, we take up a generalization of that question, namely is it possible for the product of two eigenforms to be equal to a different product of two eigenforms? Under this formulation, the question becomes closer to one about unique factorization, i.e., how closely do eigenforms work like irreducible elements? Our conjecture is that there are only finitely many cases where the product of two eigenforms is equal to a different product of two eigenforms, and …
Finite Monodromy And Artin Representations, Emma Lien
Finite Monodromy And Artin Representations, Emma Lien
LSU Doctoral Dissertations
Artin representations, which are complex representations of finite Galois groups, appear in many contexts in number theory. The Langlands program predicts that Galois representations like these should arise from automorphic representations and many examples of this correspondence have been found such as in the proof of Fermat's Last Theorem. This dissertation aims to make an analysis of explicitly computable examples of Artin representations from both sides of this correspondence. On the automorphic side, certain weight 1 modular forms have been shown to be related to Artin representations and an explicit analysis of their Fourier coefficients allows us to identify the …
On The Equality Case Of The Ramanujan Conjecture For Hilbert Modular Forms, Liubomir Chiriac
On The Equality Case Of The Ramanujan Conjecture For Hilbert Modular Forms, Liubomir Chiriac
Mathematics and Statistics Faculty Publications and Presentations
The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations π on GL(2) asserts that |av(π)| ≤ 2. We prove that this inequality is strict if π is generated by a CM Hilbert modular form of parallel weight two and v is a finite place of degree one. Equivalently, the Satake parameters of πv are necessarily distinct. We also give examples where the equality case does occur for primes of degree two.
Hermitian Maass Lift For General Level, An Hoa Vu
Hermitian Maass Lift For General Level, An Hoa Vu
Dissertations, Theses, and Capstone Projects
For an imaginary quadratic field $K$ of discriminant $-D$, let $\chi = \chi_K$ be the associated quadratic character. We will 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. Combining this isomorphism with the recent work of Berger and Klosin and a modification of Ikeda's construction we prove the existence of a lift from the space …
Congruence Relations Mod 2 For (2 X 4^T + 1)-Colored Partitions, Nicholas Torello
Congruence Relations Mod 2 For (2 X 4^T + 1)-Colored Partitions, Nicholas Torello
Senior Theses
Let p_r(n) denote the difference between the number of r-colored partitions of n into an even number of distinct parts and into an odd number of distinct parts. Inspired by proofs involving modular forms of the Hirschhorn-Sellers Conjecture, we prove a similar congruence for p_r(n). Using the Jacobi Triple Product identity, we discover a much stricter congruence for p_3(n).
On Sums Of Binary Hermitian Forms, Cihan Karabulut
On Sums Of Binary Hermitian Forms, Cihan Karabulut
Dissertations, Theses, and Capstone Projects
In one of his papers, Zagier defined a family of functions as sums of powers of quadratic polynomials. He showed that these functions have many surprising properties and are related to modular forms of integral weight and half integral weight, certain values of Dedekind zeta functions, Diophantine approximation, continued fractions, and Dedekind sums. He used the theory of periods of modular forms to explain the behavior of these functions. We study a similar family of functions, defining them using binary Hermitian forms. We show that this family of functions also have similar properties.