The Langlands program, a set of conjectures relating objects from arithmetic algebraic geometry with modular and automorphic forms via Galois representations and Lfunctions, is the core of the LMFDB, a database of mathematical objects and the connections between them.Expand

This approach uses the graph of l-isogenies to eciently compute l mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT).Expand

Abstract For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to… Expand

Let $E$ be an elliptic curve without complex multiplication (CM) over a number field $K$ , and let $G_{E}(\ell )$ be the image of the Galois representation induced by the action of the absolute… Expand

This tutorial review describes the use of PET and SPECT for molecular imaging and highlights key strategies for the development of molecular probes for the imaging of both cancer and neurological diseases.Expand

We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese remainder theorem conditions, obtaining an… Expand

Let K be a number field. We consider a local-global principle for elliptic curves E/K that admit (or do not admit) a rational isogeny of prime degree n. For suitable K (including K=Q), we prove that… Expand

Practical optimizations are described that allow us to handle larger discriminants than other methods, with |D| as large as 10^13 and h(D) up to 10^6, and to construct pairing-friendly elliptic curves of prime order, using the CM method.Expand

This work analyzes the complexity of several existing algorithms and presents a new approach that exploits structural differences between ordinary and supersingular isogeny graphs, resulting in a simple algorithm that determines the supersingularity of E in O time and space.Expand