Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil–Petersson form. Mirzakhani proved that Weil–Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as well as some consequences and applications.

1 Gross’ formula for special values of L-series . . . . . . . . . . . . . . . 4 2 Bad reduction of Shimura curves . . . . . . . . . . . . . . . . . . . 5 3 Heegner points and connected components . . . . . . . . . . . . . . . 7 4 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 A rigid analytic Gross-Zagier formula . . . . . . . . . . . . . . . . 11 6 Kolyvagin cohomolog...

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...

Weil restriction of scalars can be used to construct curves suitable for cryptography whose Jacobian has known group order. The description given by Hess, Seroussi and Smart shows how to construct hyperelliptic curves of genus two or three over nite elds of characteristic two. One drawback of their method is discussed, namely that only a small proportion of the set of all curves can be construc...

The Weil conjectures constitute one of the central landmarks of 20th century algebraic geometry: not only was their proof a dramatic triumph, but they served as a driving force behind a striking number of fundamental advances in the field. The conjectures treat a very elementary problem: how to count the number of solutions to systems of polynomial equations over finite fields. While one might ...

In pointfree topology, the notion of uniformity in the form of a system of covers was introduced by J. Isbell in [11], and later developed by A. Pultr in [14] and [15]. Another equivalent notion of locale uniformity was given by P. Fletcher and W. Hunsaker in [6], which they called “entourage uniformity”. The purpose of this paper is to formulate and investigate an alternative definition of ent...

The Hasse-Weil bound is a deep result in mathematics and has found wide applications in mathematics, theoretical computer science, information theory etc. In general, the bound is tight and cannot be improved. However, for some special families of curves the bound could be improved substantially. In this paper, we focus on the Hasse-Weil bound for the curve defined by y − y = f(x) over the fini...

This is the second of our two papers on the Siegel-Weil theorem for loop groups. In the first paper [3] we proved the Siegel-Weil theorem for (finite dimensional) snt-modules ([3], Theorem 8.1). In the present paper we use this result to obtain the Siegel-Weil theorem for loop groups, Theorem 7.5, below. In addition to the corresponding result for snt-modules, our proof depends on a convergence...

Let E/F be a finite and Galois extension of non-archimedean local fields. G connected reductive group defined over E let M:=RE/FG the F obtained by Weil restriction scalars. We investigate depth, enhanced Langlands correspondence, in transition from G(E) to M(F). obtain depth-comparison formula for Weil-restricted groups.

This expository paper gives an elementary proof, using the Weil Conjectures for curves, that an infinite algebraic extension of a finite field is PAC and large.