In 1986, Hooley applied (what practically amounts to) the
general Langlands reciprocity (modularity) conjecture and GRH in a
fresh new way, over certain families of cubic 3-folds. This
eventually led to conditional near-optimal bounds for the
number...
The unbounded denominators conjecture, first raised by Atkin and
Swinnerton-Dyer, asserts that a modular form for a finite index
subgroup of SL2(ℤ) whose Fourier coefficients have bounded
denominators must be a modular form for some congruence...
In this talk, we prove an upper bound on the average number of
2-torsion elements in the class group of monogenised fields of any
degree n≥3 and, conditional on a widely expected tail estimate,
compute this average exactly. As an application, we...
For a polynomial f∈ℚ[x], Hilbert's irreducibility theorem
asserts that the fiber f−1(a) is irreducible over ℚ for all values
a∈ℚ outside a "thin" set of exceptions Rf. The problem of
describing Rf is closely related to determining the
monodromy...
We study CM cycles on Kuga-Sato varieties over X(N) via theta
lifting and relative trace formula. Our first result is the
modularity of CM cycles, in the sense that the Hecke modules they
generate are semisimple whose irreducible components are...
We consider the standard L-function attached to a cuspidal
automorphic representation of a general linear group. We present a
proof of a subconvex bound in the t-aspect. More generally, we
address the spectral aspect in the case of uniform parameter...
I will discuss some recent progress in analytic number theory
for polynomials over finite fields, giving strong new estimates for
the number of primes in arithmetic progressions, as well as for
sums of some arithmetic functions in arithmetic...
In the recent breakthrough on the uniform Mordell-Lang problem
by Dimitrov-Gao-Habegger and Kuhne, their key result is a uniform
Bogomolov type of theorem for curves over number fields. In this
talk, we introduce a refinement and generalization of...
This is intended to complement the recent talk of Pham Huu Tiep
in this seminar but will not assume familiarity with that talk. The
estimates in the title are upper bounds of the form |χ(g)|≤χ(1)α,
where χ is irreducible and α depends on the size of...
I will discuss recent work with Harald Helfgott in which we
establish roughly speaking that the graph
connecting nn to n±pn±p with pp a
prime dividing nn is almost "locally Ramanujan". As a
result we obtain improvements of results of Tao and Tao...
