Publications/Preprints   |  Link to Page 3   |  Link to Page 4

Rational Points on Twists of Modular Curves: Let $X^d(N)$ be the modular curve described as qudratic twist of $X_0(N)$ by a quadratic field $K=\mathbb{Q}(\sqrt{d})$ and $w_N$. Rational points on this twist are $K$-rational points of $X_0(N)$ that are fixed by $\sigma$ composed with $w_N$ where $\sigma$ is the generator of $Gal(K/\mathbb{Q})$. Unlike $X_0(N)$, it's not immediate to say that there are points (global or local) on $X^d(N)$. Given $(N,d,p)$ we give necessary and sufficient conditions for existence of a $\mathbb{Q}_p$-rational point on $X^d(N)$, answering the following question of Ellenberg: For which $d$ and $N$ there exists points on $X^d(N)$ for every completion of $\mathbb{Q}$? This is a work in progress. Semi-direct product Galois covers of curves in characteristic p: Let k be an algebraically closed field with characteristicp>0. Raynaud showed in that the finite quotients of the algebraic fundamental group of the affine line over k is equal to the set of quasi p-groups. In this paper, we consider (Z/lZ)^b semi-direct Z/pZ Galois covers of the affine line ramified only at infinity, where l is a prime different from p. We show that the minimal genus of such a cover depends only on l, p, and and the order a of l modulo p. Moreover, we show that the number of such minimal genus covers equals (p-1)/a. The results of this paper, which are based on a project which took place in BANFF as a part of the conference WIN, are joint work with Linda Gruendken, Laura Hall-Seelig, Bo-Hae Im, Rachel Pries and Katherine Stevenson. Submitted for publication.

Math Notes   |  Link to Page 3   |  Link to Page 4

A Note on Quadratic Twists of Cartan Modular Curves: Let $\ell$ be an odd prime, and $E$ be an elliptic curve defined over $\Q$. Denote by $\rho_{E,\ell}: G_{\Q} \rightarrow \Aut(E[\ell])$ the mod-$ell$ Galois representation. Let $G_\ell$ denote the image of $\rho_{E,\ell}$. In this paper we are dealing with the the case that $G_\ell$ is in the normalizer of non-split Cartan and strictly containing non-split Cartan. Let $\epsilon_\ell$ be the character from $G_{\Q}$ to $N_\ell/C_l$ then $\epsilon_\ell$ is unramified away from a finite set of primes, the primes that $E$ has additive reduction. The associated moduli space to this specific class of elliptic curves is the twist of $X_0(N)$ by $\epsilon_\ell$ and we aim to count the exceptional points on this moduli space along the lines of Mazur. However, it turns out that whenever the character is real, there are no real points on the twist and when the character is imaginary, the correspondign L-function vanishes, hence there is no hope applying Mazur's techniques. Group Cohomology : A survey about the basic notions in group cohomology. On Symmetric Bilinear Forms over Vector Bundles: A short, introductory note about the topic. Some Questions Related with Hilbert Symbol    Partial Translation of Serre's 1972 Paper Notes of some of the talks that I gave: Basics of Non-commutative Iwasawa Theory , Images of Galois Representations and Elliptic Curves, Beyond Mazur   (some selection of theorems which are of the same flavour with Mazur's theorem on torsion points of elliptic curves), Speciality Exam talk, Dirichlet Characters and Cyclotomic Fields