Numerical Computation of a Certain Dirichlet Series Attached to Siegel Modular Forms of Degree Two

The Rankin convolution type Dirichlet series D-F,D-G(s) of Siegel modular forms F and G of degree two, which was introduced by Kohnen and the second author, is computed numerically for various F and G. In particular, we prove that the series D-F,D-G(s), which shares the same functional equation and analytic behavior with the spinor L-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail.

Pond-Women Revelations: The Subaltern Registers in Maithil Women's Expressive Forms

Ponds are ubiquitous in the Maithil region of Nepal, and they figure prominently in folk narratives and ceremonial paintings produced by women there. I argue that in Maithil women's folktales, as in their paintings, the trope of ponds shifts the imaginative register toward women's perspectives and the importance of women's knowledge and influence in shaping Maithil society, even as this register shift occurs within plots featuring male protagonists. I argue further that in the absence of a habit of exegesis in their expressive arts, and given the cross-referential, dialogic nature of expressive practices, a methodology that draws into interpretive conversation the multitude of expressive forms exercised by Maithil women enhances analytical access to Maithil women's collective perspectives on their social and cosmological worlds.

Verifying Harder's Conjecture for Classical and Siegel Modular Forms

A conjecture by Harder shows a surprising congruence between the coefficients of “classical” modular forms and the Hecke eigenvalues of corresponding Siegel modular forms, contigent upon “large primes” dividing the critical values of the given classical modular form. Harder’s Conjecture has already been verified for one-dimensional spaces of classical and Siegel modular forms (along with some two-dimensional cases), and for primes p 37. We verify the conjecture for higher-dimensional spaces, and up to a comparable prime p.

Plato's Doctrine Of Forms: Modern Misunderstandings

Among the philosophical ideas of Plato, perhaps the most famous is his doctrine of forms. This doctrine has faced harsh criticism due, in large part, to the interpretations of this position by modern philosophers such as René Descartes, John Locke, and Immanuel Kant. For example, Plato has been interpreted as presenting a ¿two-worlds¿ approach to form and thing and as advancing a rationalist approach to epistemology. His forms have often been interpreted as ideas and as perfect copies of the things of the visible world. In this thesis, I argue that these, along with other interpretations of Plato presented by the moderns, are based on misunderstandings of Plato¿s overall philosophy. In so doing, I attempt to show that the doctrine of forms cannot be directly interpreted into the language of Cartesian, Lockean, and Kantian metaphysics and epistemology, and thus should not be prematurely dismissed because of these modern Platonic interpretations. By analyzing the Platonic dialogues beside the writings of the modern philosophers, I conclude that three of the most prominent modern philosophers, as representatives of their respective philosophical frameworks, have fundamentally misunderstood the nature of Plato¿s famous doctrine of forms. This could have significant implications for the future of metaphysics and epistemology by providing an interpretation of Plato which adds to, instead of contradicts, the developments of modern philosophy.