# Lawrence Vu

I am a PhD student at the Graduate Center, City University of New York (CUNY). My advisor is Krzysztof Klosin.

My research interest is in the field of number theory, specifically on applications of automorphic forms to number theory. Traditionally, number theory revolves around the problem of solving Diophantine equations i.e. equations given by polynomials (typically in multiple variables) such as the famous Pythagorean equation $X^2 + Y^2 = Z^2.$ We are mainly interested in the solutions in the rational numbers $$\mathbb{Q}$$, in the integers $$\mathbb{Z}$$ and more generally, in algebraic number fields as well as finite fields. For instance, $$X = 3, Y = 4, Z = 5$$ is an integral solution to the above equation. There are various questions one can ask regarding such an equation:

1. Does the equation have a solution?
2. How many solutions does it have?
3. Is there a simple description (for example, an explicit formula) for all/most of the solutions?
In case of the Pythagorean equation, the answer to 1 is obviously "Yes", the answer to 2 is "Infinitely many" and the answer to 3 is that every solutions in the integers can be obtained via the formula $$X = m^2 - n^2, Y = 2mn, Z = m^2 + n^2$$ where $$m$$ and $$n$$ are arbitrary integers. (Such a formula is typically called a parametrization.)

It turns out that these questions in general are very hard.

For question 1, it is now well-known (see Hilbert's 10th problem) that there is no effective decision procedure (i.e. algorithm) to decide whether a Diophantine equation has an integral solution. Nevertheless, mathematicians have developed many methods and heuristics to solve Diophantine equations; for instance, one can prove unsolvability of such equations using congruences. The first significant method, infinite descent, is introduced by Fermat to prove that $$X^4 + Y^4 = Z^4$$, a slight generalization of the above Pythagorean equation, has no non-trivial integral solution (solution for which none of the $$X, Y, Z$$ is zero). Fermat famously claimed that the generalized equations $$X^n + Y^n = Z^n$$ (generally referred to as Fermat's equation for exponent $$n$$) has no non-trivial integral solution for all natural number $$n \geq 3$$, a fact now known as Fermat Last Theorem, established by Andrew Wiles. Before Wiles, in the pursuit of Fermat Last Theorem, Kummer and many others created algebraic number theory which engenders an abundant amount of new mathematics such as class field theory, reciprocity laws. Kummer's famous result on unsolvability of Fermat's equation for exponent $$n = \ell$$ whenever $$\ell$$ is a regular prime is the first significant progress on FLT. (Kummer also discovered an easy criterion to determine regularity via Bernoulli numbers. These numbers play an essential role in Iwasawa theory.)

Fermat also proved another notable fact that a prime number can be written as a sum of two squares if and only if it is congruent to 1 modulo 4. In other words, if $$p$$ is a prime number then the equation $$X^2 + Y^2 = p$$ has an (unique up to interchanging $$X$$ and $$Y$$ and their signs) integral solution if and only if $$p \equiv 1 \bmod 4$$. My curious self computed the exact number of solutions to $$X^2 + Y^2 = n$$ for any natural number $$n$$ by an adhoc method. A fairly general method is obtained by Jacobi to compute the exact number of solutions to $$X^2 + Y^2 + Z^2 + T^2 = n$$ via modular forms, special complex valued functions on the upper half plane. There is a generalization of Jacobi's method to the problem of finding representations by general quadratic forms via Siegel modular forms. Modular forms play an essential role in Wiles' proof of Fermat Last Theorem.

As for question 3, its study (for polynomial/rational parametrization) leads to the field of algebraic geometry.