Elliptic curves are a type of mathematical object that have been extensively studied in number theory and cryptography. They are algebraic curves defined by equations of the form y^2 = x^3 + ax + b, where a and b are constants. Elliptic curves have a rich mathematical structure and properties that make them useful for various applications. In number theory, elliptic curves are used to study the properties of rational points on the curve and their connection to Diophantine equations. In cryptography, elliptic curve cryptography (ECC) is a popular method for securing communications and digital signatures. ECC is considered to be more secure and efficient than other encryption methods, such as RSA, for the same level of security. Overall, research in elliptic curves involves studying their properties, algorithms for computing with them, and applications in various areas of mathematics and computer science.