The classical Pell equation $X^2-DY^2=1$, to be solved in
integers $X,Y\neq 0$, has a variant for function fields (studied
already by Abel), where now $D=D(t)$ is a complex polynomial of
even degree and we seek solutions in nonzero complex...
Calibrated currents naturally appear when dealing with several
geometric questions, some aspects of which require a deep
understanding of regularity properties of calibrated currents. We
will review some of these issues, then focusing on the two...
We all know Shannon's entropy of a discrete probability
distribution. Physicists define entropy in thermodynamics and in
statistical mechanics (there are several competing schools), and
want to prove the Second Law, but they didn't succeed yet
(very...
he Weil height measures the “complexity” of an algebraic number.
It vanishes precisely at 0 and at the roots of unity. Moreover, a
finite field extension of the rationals contains no elements of
arbitrarily small, positive heights. Amoroso, Bombieri...
The theorem of the title is that if the L-function L(E,s) of an
elliptic curve E over the rationals vanishes to order r=0 or 1 at
s=1 then the rank of the group of rational rational points of E
equals r and the Tate-Shafarevich group of E is finite...
Resonances are complex analogs of eigenvalues for Laplacians on
noncompact manifolds, arising in long time resonance expansions of
linear waves. We prove a Weyl type asymptotic formula for the
number of resonances in a strip, provided that the set...
We give an arithmetic version of the recent proof of the
improved triangle removal lemma by Fox [Fox11], for the group
$F_2^n$.
A triangle in $F_2^n$ is a tuple (x,y,z) such that $x+y+z = 0$. The
triangle removal lemma for $F_2^n$ states that for...
