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6 papers · 1 filter
Large character sums: Pretentious characters and the Polya-Vinogradov Theorem
Andrew Granville, K. Soundararajan
In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums which still remains the best known general estimate. One of the main results of this paper p…
Poisson statistics via the Chinese remainder theorem
A. Granville, P. Kurlberg
We consider the distribution of spacings between consecutive elements in subsets of Z/qZ where q is highly composite and the subsets are defined via the Chinese remainder theorem.…
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.
An uncertainty principle for arithmetic sequences
Andrew Granville, K. Soundararajan
Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are ``well-distributed'' in some appropriate sense. In various discrepancy proble…
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…