Ring-Theoretic Properties of Certain Hecke Algebras

Computing rational points on rank 1 elliptic curves via $L$-series and canonical heights

“…We mention that the validity of the output of this algorithm is dependent on knowing that the elliptic curve E/Q is modular. For example, this condition will be satisfied if E/Q has good or multiplicative reduction at 3 and 5 (see [20], [18], [8]). …”

“…By the work of Wiles [20], Taylor-Wiles [18], and Diamond [8], we know that most E/Q are modular; and assuming the modularity, the work of Gross-Zagier [10] and Kolyvagin [11] gives the desired value of L (E, 1). We will implictly be using these results when we apply the Canonical Height Search Algorthm to prove that E(Z S ) contains no non-torsion points without actually finding a generator for E(Q).…”

Perspectives on American mathematics

“…For the first time in history, the federal government appropriated large sums of money for the support of science and its infrastructure. Federal grants, primarily from the NSF, funded mathematical projects from the mammoth classification of the finite simple groups in the 1980s, long under the direction of Daniel Gorenstein at Rutgers University [31], to the solution of the four-color problem in 1977 by Kenneth Appel and Wolfgang Haken, both then of the University of Illinois at Urbana-Champaign [3], [4], to the proof of Fermat's Last Theorem in 1995 by Princeton's Andrew Wiles [81], [72]. With this sort of support, however, came greater responsibilities.…”

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

“…Thanks to the work of Wiles [51] and Taylor-Wiles [46], successively improved in a series of papers [17], [14] and [8], it is known that all elliptic curves over Q are modular. Set N := n. For such curves, there is a parametrization over Q:…”