Number Theory

Let $E_1$ and $E_2$ be two non-CM elliptic curves defined over a number field $K$. By an isogeny theorem due to Kulkarni, Patankar, and Rajan, the two curves are geometrically isogenous if and only if the density of primes for which their Frobenius field coincide is positive. In this talk, we present a quantitative upper bounds of this criterion that improves the result of Baier–Patankar and Wong. The strategy relies on effective versions of the Chebotarev Density Theorem. This is joint work with Alina Cojocaru and Auden Hinz.

We prove explicit estimates of $1/\zeta(s)$ of various orders, and use an improved version of the Perron formula to get explicit estimates for the Mertens function $M(x)$ of order $O(x)$, $O(x/\log^k x)$, and $O(x\log x exp(−\sqrt{\log{x}})$. These estimates are good for small, medium, and large ranges of $x$, respectively.

Let f be a self-dual Maass form for $SL(n, Z)$. We write $L_f (s)$ for the Godement–Jacquet L-function associated to $f$ and $L_{f\times f} (s)$ for the Rankin–Selberg L-function of $f$ with itself. The inverse of $L_{f\times f} (s)$ is defined by
$$
\frac{1}{L_{f\times f}(s)} := \sum_{m=1}^\infty \frac{c(m)}{m^s}, \mathfrak{R}(s) > 1.
$$
It is well known that the classical Mertens function $M(x) := \sum_{m\leq x} \mu(m)$ is related to
$$
\frac{1}{\zeta(s)} = \sum_{m=1}^\infty \frac{\mu(m)}{m^s}, \mathfrak{R}(s) > 1.
$$
We define the analogue of the Mertens function for $L_{f\times f} (s)$ as $\widetilde{M}(x) := \sum_{m\leq x} c(m)$ and obtain an upper bound for this analogue $\widetilde{M}(x)$, similar to what is known for the Mertens function $M(x)$. In particular, we prove that $\widetilde{M}(x) \ll_f x \exp(−A\sqrt{\log{x}}$ for sufficiently large $x$ and for some positive constant $A$. This is a joint work with my Ph.D. supervisor Prof. A. Sankaranarayanan.

In 2020, Fiorilli and Jouve proved an unconditional Chebyshev bias result for a Galois extension of number fields under a group theoretic condition on its Galois group. We extend their result to a larger family of groups. This leads us to characterize abelian groups enabling extreme biases. In the case of prime power degree extensions, we give a simple criterion implying extreme biases and we also investigate the corresponding Linnik-type question.

As a refinement of Goldfeld’s conjecture, there is a conjecture of Keating–Snaith asserting that $\log L(1/2,E_d)$ for certain quadratic twists $E_d$ of an elliptic curve $E$ behaves like a normal random variable. In light of this, Radziwill and Soundararajan conjectured that the distribution of $\log(|Sha(E_d)|/\sqrt{|d|}$ is approximately Gaussian for these $E_d$, and proved that the conjectures of Keating–Snaith and theirs are both valid “from above”. More recently, under GRH, they further established a lower bound for the involving distribution towards Keating–Snaith’s conjecture. In this talk, we shall discuss the joint distribution of central values and orders of Sha groups of $E_d$ and how to adapt Radziwill–Soundararajan’s methods to study upper bound and lower bounds for such a joint distribution if time allows.