Related papers: The Sato-Tate Conjecture on Average for Small Angl…

We make explicit a construction of Serre giving a definition of an algebraic Sato-Tate group associated to an abelian variety over a number field, which is conjecturally linked to the distribution of normalized L-factors as in the usual…

Generalizing the problem of counting rational points on curves and surfaces over finite fields, we consider the setting of $n \times n$ matrix points with finite field entries. We obtain exact formulas for matrix point counts on elliptic…

We show that the reductions modulo primes $p\le x$ of the elliptic curve $$ Y^2 = X^3 + f(a)X + g(b), $$ behave as predicted by the Lang-Trotter and Sato-Tate conjectures, on average over integers $a \in [-A,A]$ and $b \in [-B,B]$ for $A$…

We formulate several conjectures on mean convex domains in the Euclidean spaces, as well as in more general spaces with lower bonds on their scalar curvatures, and prove a few theorems motivating these conjectures.

We prove a few new cases of the Sato-Tate conjecture for abelian surfaces, using a new automorphy theorem of Allen et al. Then in the unproven cases, we use partial results to describe nontrivial asymptotics on the trace of Frobenius, and…

We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form $$ y^2=x^p-1 \text{ and } y^2=x^{2p}-1,$$ where $p$ is an odd prime. Our results rely on the fact the Jacobians of…

We remark that Tate's algorithm to determine the minimal model of an elliptic curve can be stated in a way that characterises Kodaira types from the minimum of v(a_i)/i. As an application, we deduce the behaviour of Kodaira types in tame…

For an elliptic curve $E$ over $\ratq$ and an integer $r$ let $\pi_E^r(x)$ be the number of primes $p\le x$ of good reduction such that the trace of the Frobenius morphism of $E/\fie_p$ equals $r$. We consider the quantity $\pi_E^r(x)$ on…

We provide a new interpretation of the Mazur-Tate Conjecture and then use it to obtain the first (unconditional) theoretical evidence in support of the conjecture for elliptic curves of strictly positive rank.

We prove the generalised Tate conjecture for H^3 of products of elliptic curves over finite fields, by slightly modifying an argument of M. Spiess concerning the Tate conjecture. We prove it fully if the elliptic curves run among at most 3…

We prove under some assumptions that the Tate conjecture holds for products of Fermat varieties of different degrees.

In this article, we aim to largely complete the program of proving the Tate conjecture for surfaces of geometric genus one, by introducing techniques to analyze those surfaces whose "natural models" are singular. As an application, we show…

Under a hypothesis which is slightly stronger than the Riemann Hypothesis for elliptic curve $L$-functions, we show that both the average analytic rank and the average algebraic rank of elliptic curves in families of quadratic twists are…

In this mostly expository note, we explain a proof of Tate's two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.

The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito-Ito-Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between…

We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell \neq p$.

For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjectured that the number of primes $p <X$ at which $E$ has a supersingular reduction is asymptotically equal to $c\sqrt{X}/\log X$, where $c>0$…

We prove many non-generic cases of the Sato-Tate conjecture for abelian surfaces as formulated by Fite, Kedlaya, Rotger and Sutherland, using the potential automorphy theorems of Barnet-Lamb, Gee, Geraghty and Taylor.

This is an informal paper presenting historical results around the recent paper of the author about Lang's Conjecture and torsion of elliptic curves. This paper also discusses a few aspects of the proof.

We obtain, under an additional assumption on the subanalytic abnormal distribution constructed in [4], a proof of the minimal rank Sard conjecture in the analytic category. It establishes that from a given point the set of points accessible…