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.

Let $ K = \mathbb{Q}(\theta) $ be an algebraic number field with $ f(x) $ as the minimal polynomial of the algebraic integer $ \theta $ over $ \mathbb{Q} $. Let $ p $ be a rational prime. Let \[ \bar{f}(x) = \bar{g}_{1}(x)^{e_{1}} \ldots \bar{g}_{r}(x)^{e_{r}} \] be the factorization of $ \bar{f}(x) $ as a product of powers of distinct irreducible polynomials over $ \mathbb{Z}/ p\mathbb{Z} $, with $ g_{i}(x) $ monic polynomials belonging to $ \mathbb{Z}[x] $. In 1878, Dedekind proved if $ p $ does not divide the index of the subgroup $ \mathbb{Z}[\theta] $ in $ A_{K} $, then $ pA_{K} = \wp_{1}^{e_{1}} \ldots \wp_{r}^{e_{r}} $, where $ \wp_{1}, \ldots, \wp_{r} $ are distinct prime ideals of $ A_{K} $, \wp_{i} = pA_{K} + g_{i}(\theta)A_{K} $ with residual degree of $ \wp_i/p $ equal to $ \deg {g}_{i}(x) $ for all $ i$. In 2008, we proved that converse of Dedekind's theorem holds, i.e. if for a rational prime $ p $, the decomposition of $ pA_K $ satisfies the above three properties, then $ p $ does not divide $ [A_K:\mathbb Z[\theta]] $. Dedekind also gave a simple criterion known as Dedekind Criterion to verify when $ p $ does not divide $ [A_K:\mathbb{Z}[\theta]] $. We will also discuss the Dedekind Criterion and its generalization. In 2014, we have proved the analogue of Dedekind's theorem for finite extensions of valued fields of arbitrary rank as well as of its converse.

According to http://www.ams.org/news/math-in-the-media/mathdigest-md-201209-toc#201210-numbers Nature News, 10 September 2012, quoting Dorian Goldfeld, the abc Conjecture is "the most important unsolved problem in Diophantine analysis". It is a kind of grand unified theory of Diophantine curves: "The remarkable thing about the abc Conjecture is that it provides a way of reformulating an infinite number of Diophantine problems," says Goldfeld, "and, if it is true, of solving them." Proposed independently in the mid-80s by David Masser of the University of Basel and Joseph Oesterl� of Pierre et Marie Curie University (Paris 6), the abc Conjecture describes a kind of balance or tension between addition and multiplication, formalizing the observation that when two numbers a and b are divisible by large powers of small primes, a + b tends to be divisible by small powers of large primes. The abc Conjecture implies -- in a few lines -- the proofs of many difficult theorems and outstanding conjectures in Diophantine equations-- including Fermat's Last Theorem. This talk will be at an elementary level, giving a collection of consequences of the abc Conjecture. It will not include an introduction to the Inter-universal Teichm�ller Theory of Shinichi Mochizuki.