This is my senior honors thesis done in my final year as an undergraduate at Stanford University, under the direction of Professor Akshay Venkatesh. We will construct the Weil representation of SL2(R) through a natural action of the Heisenberg group Heis(R) on the space of square-integrable complex-valued functions L 2(R), together with the celebrated Stone-von Neumann theorem of functional ana...

Takeshi SAITO When he formulated an analogue of the Riemann hypothesis for congruence zeta functions of varieties over finite fields, Weil predicted that a reasonable cohomology theory should lead us to a proof of the Weil conjecture. The dream was realized when Grothendieck defined etale cohomology. Since then, -adic etale cohomology has been a fundamental object in arithmetic geometry. It ena...

We apply the Weil conjectures to the Hessenberg Varieties to obtain information about the combinatorics of descents in the symmetric group. Combining this with elementary linear algebra leads to elegant proofs of some identities from the theory of descents. 3

We answer a question asked by Hajdu and Tengely: The only arithmetic progression in coprime integers of the form (a, b, c, d) is (1, 1, 1, 1). For the proof, we first reduce the problem to that of determining the sets of rational points on three specific hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We find generat...

As stated above, (A) is aimed at establishing a Non-Abelian Class Field Theory. The starting point here is the following classical result: Over a compact Riemann surface, a line bundle is of degree zero if and only if it is flat, i.e., induced from a representation of fundamental group of the Riemann surface. Clearly, being a bridge connecting divisor classes and fundamental groups, this result...

The Segal–Shale–Weil representation associates to a symplectic transformation of the Heisenberg group an intertwining operator, called metaplectic operator. We develop an explicit construction of metaplectic operators for the Heisenberg group H(G) of a finite abelian group G, an important setting in finite time-frequency analysis. Our approach also yields a simple construction for the multivari...

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use...

In one of their early works, Miranda and Persson have classified all possible configurations of singular fibers for semistable extremal elliptic fibrations on K3 surfaces. They also obtained the Mordell-Weil groups in terms of the singular fibers except for 17 cases where the determination and the uniqueness of the groups were not settled. In this paper, we settle these problems completely. We ...

The Weil algebra of a semisimple Lie group and an exterior algebra of a sym-plectic manifold possess antibrackets. They are applied to formulate the models of non–abelian equivariant cohomologies.