Related papers: Norm relations for CM points on modular curves
We investigate the elliptic analogs of multi-indexed polylogarithms that appear in the theory of the fundamental group of the projective line minus three points as sections of a universal nilpotent bundle with regular singular connection.…
In this paper we prove some divisibility properties of the cardinality of elliptic curves modulo primes. These proofs explain the good behavior of certain parameters when using Montgomery or Edwards curves in the setting of the elliptic…
The parity of Selmer ranks for elliptic curves defined over the rational numbers $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$ has been studied by Shekhar. The proof of Shekhar relies on proving a parity result for the…
In this note, we will apply the results of Gross-Zagier, Gross-Kohnen-Zagier and their generalizations to give a short proof that the differences of singular moduli are not units. As a consequence, we obtain a result on isogenies between…
It is well-known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular…
A class of Hamiltonian stochastic differential equations with multiplicative L\'{e}vy noise in the sense of Marcus, and the construction and numerical implementation methods of symplectic Euler scheme, are considered. A general symplectic…
In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements, utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. We show how to apply this construction to prove results on the…
We calculate the constant term of Coleman power series and use it to prove an analogue of Iwasawa Main Conjecture in function fields of characteristic p>0 using Euler systems. This result is proved by a similar method of classical proof of…
Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant K-homology of M. The…
In this paper, certain natural and elementary polygonal objects in Euclidean space, {\it the stable polygons}, are introduced, and the novel moduli spaces ${\bfmit M}_{{\bf r}, \epsilon}$ of stable polygons are constructed as complex…
In this paper, we use the central configuration coordinate decomposition to study the linearized Hamiltonian system near the elliptic Euler solutions. Then using the Maslov-type \omega-index theory of symplectic paths and the theory of…
At a prime of ordinary reduction, the Iwasawa ``main conjecture'' for elliptic curves relates a Selmer group to a $p$-adic $L$-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the…
This master thesis describes how Selmer groups can be used to determine the Mordell-Weil group of elliptic curves over a number field K. The Mordell-Weil Theorem states that $E(K) = E(K)_{tors} \times Z^r$, where $r$ is the rank of $E$, and…
We discuss refined applications of Kato's Euler systems for modular forms of higher weight at good primes (with more emphasis on the non-ordinary ones) beyond the one-sided divisibility of the main conjecture and the finiteness of Selmer…
We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert…
Equivalence classes of $n$-point configurations in Euclidean, Hermitian, and quaternionic spaces are related, respectively, to classical determinantal varieties of symmetric, general, and skew-symmetric bilinear forms. Cayley-Menger…
In this article, we propose a new probabilistic model for the distribution of ranks of elliptic curves in families of fixed Selmer rank, and compare the predictions with previous results, and with the databases of curves over the rationals…
We show existence of self-similar solutions satisfying Kolmogorov's scaling for generalized dyadic models of the Euler equations, extending a result of Barbato, Flandoli, and Morandin. The proof is based on the analysis of certain dynamical…
We introduce a method for solving Calder\'on type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain…
A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding function satisfies WDVV equations are found…