In this paper we determine the quadratic points on the modular curves 𝑋₀(𝑁), where the curve is non-hyperelliptic, the genus is 3, 4, or 5, and the Mordell–Weil group of 𝐽₀(𝑁) is finite. The ...

Let f be a newform, as specified by its Hecke eigenvalues, on a Shimura curve X. We describe a method for evaluating f. The most interesting case is when X arises as a compact quotient of the ...

We all know the usual jokes about the ‘S’ in ‘IoT’ standing for ‘Security’. It’s hardly a secret that security in embedded, networked devices (‘IoT devices’) is all too often a last-minute task that ...

Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" ...

Post-quantum cryptography is rapidly evolving to counter threats posed by quantum computing, and elliptic curves combined with isogeny methodologies offer a promising avenue. This approach leverages ...

In August, a pair of mathematicians discovered an exotic, record-breaking curve. In doing so, they tapped into a major open question about one of the oldest and most fundamental kinds of equations in ...

Forbes contributors publish independent expert analyses and insights. Ewan Spence covers the digital worlds of mobile technology. This article is more than 10 years old. The Taniyama-Shimura ...

“Elliptic curve cryptography (ECC), as one of the public key cryptography systems, has been widely applied to many security applications. It is challenging to implement a scalar multiplication (SM) ...

Quantum modular forms have emerged as a versatile framework that bridges classical analytic number theory with quantum topology and mathematical physics. Initially inspired by the pioneering work on ...

Manjul Bhargava was warned long ago never to think about math while driving. "I find doing mathematical research requires very deep concentration," said Bhargava, the Brandon Fradd, Class of 1983, ...