50 citations · 353 across the 37 of their papers we have counts for
5 papers · 1 filter
Power-free values, large deviations, and integer points on irrational curves
H. A. Helfgott
Let be a polynomial of degree without roots of multiplicity or . Erdős conjectured that, if satisfies the necessary local…
Root numbers and ranks in positive characteristic
B. Conrad, K. Conrad, H. Helfgott
For a global field K and an elliptic curve E_eta over K(T), Silverman's specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all but finitely many t in P^1(K).…
The parity problem for reducible cubic forms
H. A. Helfgott
Let f in Z[x,y] be a reducible homoegeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.
On the behaviour of root numbers in families of elliptic curves
H. A. Helfgott
Let E be a one-parameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Fu…
Integral points on elliptic curves and 3-torsion in class groups
H. A. Helfgott, A. Venkatesh
We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and metho…