We obtain a lower bound on the number of quadratic Dirichlet L-functions over the rational
function field which vanish at the central point s= 1/2. This is in contrast with the situation …

We derive a local formula for the parity of the 2∞-Selmer rank of Jacobians of curves of
genus 2 or 3 which admit an unramified double cover. We give an explicit example to show …

We consider normalized newforms f∈ S k (Γ 0 (N), ε f) whose nonconstant term Fourier
coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a …

Let $ E: y^ 2= F (x) $ be an elliptic curve over $\mathbb {Q} $ defined by a monic irreducible
integral cubic polynomial $ F (x) $ with negative square-free discriminant $-D $. We …

We prove a $ p $-converse theorem for elliptic curves $ E/\mathbb {Q} $ with complex
multiplication by the ring of integers $\mathcal {O} _K $ of an imaginary quadratic field $ K …

The Birch–Swinnerton-Dyer Conjecture predicts that, given an abelian variety A over a
number field K, its rank, rk (A/K), is equal to the order of vanishing of its L-function L (A/K, s) …

We revisit the mathematics that Ramanujan developed in connection with the famous “taxi-
cab” number 1729. A study of his writings reveals that he had been studying Euler's …

The parity conjecture has a long and distinguished history. It gives a way of predicting the
existence of points of infinite order on elliptic curves without having to construct them, and is …

Caraiani and Newton have proven that if $ F $ is an imaginary quadratic number field such
that $ X_0 (15) $ has rank $0 $ over $ F $, then every elliptic curve over $ F $ is modular …

PhD dissertation consists in three lines of investigation involving rational elliptic surfaces,
namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible …