This will be for the most part a survey talk. I will define Siegel modular forms, their number theoretic significance, and explain the link between Siegel modular forms of degree 2 and automorphic representations of GSp(4). I will talk about the significance of their Fourier coefficients, and describe several known results and still unproven conjectures. I will also talk about certain lifts to and from spaces of Siegel modular forms that are special cases of Langlands' general conjectures.

This will be a description of precise conjectures (due mostly to K. Prasanna) about the p-adic valuation of quadratic twisted L-functions at the centre of the critical strip, and an approach we propose to prove them, based on work of Wei Zhang.

Let F(X,Y) \in \mathbb{Z}(X,Y) be a form of degree r >=3, irreducible over irrationals and having at most s+1 non-zero coefficients. Let h be a non-zero integer. Siegel proposed that the number N_F(h) of integer solutions to the Thue inequality |F(X,Y)|<=h may be bounded only in terms of s and h. In this talk, we present some contributions in this direction.

The order bound is a general method to obtain a lower bound for the minimum distance of an evaluation code. It is a very good bound in case the code is defined using Goppa's construction of codes from curves. In my talk I will outline the main ideas behind the order bound and make them more explicit in the case of one-point algebraic-geometry codes

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.

The classical form of Weyl's law is concerned with counting the eigenvalues of the Laplacian operator on a bounded domain. Atle Selberg generalized it in the 50s to show the existence of non-holomorphic modular forms. I will explain the Weyl's law with examples and how it counts automorphic forms, which are objects of number-theoretic interest.

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.

Unlike integer factorization, a reducible holomorphic eta quotient may not factorize uniquely as a product of irreducible holomorphic eta quotients. But whenever such an eta quotient is reducible, the occurrence of a certain type of factor could be observed: We conjecture that if a holomorphic eta quotient f of level M is reducible, then f has a factor of level M. In particular, it implies that rescalings and Atkin-Lehner involutions of irreducible holomorphic eta quotients are irreducible. We prove a number of results towards this conjecture: For example, we show that a reducible holomorphic eta quotient of level M always factorizes nontrivially at some level N which is a multiple of M such that rad(N) = rad(M) and moreover, N is bounded from above by an explicit function of M. This implies a new and much faster algorithm to check the irreducibility of holomorphic eta quotients. In particular, we show that our conjecture holds if M is a prime power. We also show that the level of any factor of a holomorphic eta quotient f of level M and weight k is bounded w.r.t. M and k. Further, we show that there are only finitely many irreducible holomorphic eta quotients of a given level and provide a bound on the weights of such eta quotients. Finally, we give an example of an infinite family of irreducible holomorphic eta quotients of prime power levels.