Title: Arakelov Geometry of Modular Curves X_0(p^2)
Abstract: We shall explore the geometry of the Modular curve X_0(p^2) and it's regular minimal model over the ring of integers, which is an arithmetic surface. Arakelov has introduced an intersection pairing for divisors on arithmetic surfaces. We shall derive an expression for the Arakelov self-intersection of the relative dualising sheaf on the regular minimal model of X_0(p^2). As a consequence, we shall give s number theoretic applications for this computation. This is joint work with Debargha Banerjee and Diganta Borah.
Title: Self-dual Cupidal Representations
Abstract:
Let G be a connected reductive group over a finite field f of order q. When q is small, we make further assumptions on G. Then we determine precisely when G(f) admits irreducible, cuspidal representations that are self- dual, of Deligne-Lusztig type, or both. Finally, we outline some consequences for the existence of self-dual supercuspidal representations of reductive p-adic groups. This is a joint work with Jeffrey Adler.
Title: Reductions of Galois representations and the Theta operator.
Abstract: Let p ≥ 5 be a prime and f ∈ S_k(N, χ) be an eigenform of slope α such that p \mid N. Suppose that $\overline{\rho_f}$ denotes the reduction of the Galois representation associated with f and ω is the mod p cyclotomic character. Then under a mild assumption on f, we shall prove that there exists an eigenform g ∈ S_l(N, χ) of slope α + 1 such that $overline{\rho_f} ⊗ ω \simeq \overline{\rho}_g$. Now, the structure of the reductions of the local Galois representations associated to modular forms of large weights are determined by Deligne, Buzzard and Gee for slopes in the interval [0, 1) and by Ghate and his coauthors for slopes in [1, 2). As an application, we show that these results are compatible under the Theta operator. Furthermore, using our result, we extrapolate some of the images of the reductions of local Galois representations for eigenforms of slopes in [2, 3) and also get a lower bound of the radii of certain Coleman families. This talk is based on an ongoing project with Prof. E. Ghate.
Title: Lifting Galois representations
Abstract: A question originating in Serre's modularity conjecture is whether any continuous homomorphism of the absolute Galois group of a number field $F$ into $G(k)$, where $G$ is a split reductive group and $k$ is a finite field, lifts to $G(W(k))$. This was answered affirmatively by Ravi Ramakrishna for $G = GL_2$ and $F = \mathbb{Q}$ using a purely Galois theoretic method. Ramakrishna's method was extended to $GL_n$, $n > 2$, by Clozel--Harris--Taylor and to general $G$ (independently) by Booher and Patrikis. Their results require that the image of the homomorphism be relatively large. In this talk, based on joint work with Chandrashekhar Khare and Stefan Patrikis, I will explain some refinements of Ramakrishna's method that can be applied to construct (geometric) lifts even when the image is relatively small.
Title: Ternary Goldbach problem
Abstract . Recently Harold Helfgott settled Ternary Goldbach problem that every odd integer is a sum of atmost three primes . In this proof , he uses the usual Vaughan's identity and suggested that the alternative Vaughan's identity may be more effective . In this lecture , we explore the alternative identity. This is a joint work with Priyamvad Srivastav.
Title: ON THE MOD-p REPRESENTATION THEORY OF GL2(F)
Abstract. Supersingular representations are the building blocks in mod-p representation theory of $GL_2(F)$ where $F$ is a finite extension of $Q_p$. The supersingular representations of $GL_2(Q_p)$ are completely classified by Breuil. Here the central idea was to calculate the pro-p-Iwahori invariant space of some standard module. In this talk, we calculate this invariant space for general F using the so called Iwahori-Hecke model.