Related papers: Elliptic gamma functions, triptic curves and SL_3(…
Given a Diophantine triple $\{c_1(t),c_2(t),c_3(t)\}$, the elliptic curve over Q(t) induced by this triple, i.e. $y^2=(c_1(t) x+1) (c_2(t) x+1) (c_3(t) x+1)$, can have as torsion group one of the non-cyclic groups in Mazur's theorem, i.e.…
In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good…
Notes for a short lecture series, covering exploded manifolds, the moduli stack of curves in exploded manifolds, and a tropical gluing formula for Gromov-Witten invariants: a gluing formula providing a degeneration formula for Gromov-Witten…
We present expressions for the Weierstrass zeta-function and related elliptic functions by rapidly converging series. These series arise as triple products in the A-infinity category of an elliptic curve.
This paper presents empirical evidence supporting Goldfeld's conjecture on the average analytic rank of a family of quadratic twists of a fixed elliptic curve in the function field setting. In particular, we consider representatives of the…
Elliptic functions are largely studied and standardized mathematical objects. The two usual approaches are due to Jacobi and Weierstrass. From a contour integral which allowed us to unify many summation formulae (Euler-MacLaurin, Poisson,…
Contents: Rational functions with given monodromy on generic curves (I. Bouw & S. Wewers); Can deformation rings of group representations not be local complete intersections? (T. Chinburg); Lifting an automorphism group to finite…
Various integrals over elliptic integrals are evaluated as couplings on spheres, resulting in some integral and series representations for the mathematical constants $\pi$, $G$ and $\zeta(3)$.
We calculate the Tate-Shafarevich group of an elliptic three-fold $f:X\rightarrow S$ when $X$ and $S$ are regular and $f$ is flat, relating it to the Brauer group of $X$ and $S$. We show that given certain hypotheses on $f$, the…
We give an asymptotic formula with power saving error term for the twisted first moment of symmetric square L-functions on GL(3) in the level aspect. As applications, we obtain non-vanishing results as well as lower bounds of the expected…
The closed symmetrized tridisc $\Gamma_3$ and its distinguished boundary $b\Gamma_3$ are the sets $\Gamma_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|\leq 1, i=1,2,3 \}\subseteq \mathbb C^3$ $b\Gamma_3=\{…
In this paper, we study modular transformation properties of a certain class of functions with indefinite quadratic forms.
We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus. The first invariant is of topological nature and is related to…
This is a survey of current and recent works on deformation quantization and index theorems.
We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.
In this work, we study symplectic structures on graded manifolds and their global counterparts, higher Lie groupoids. We begin by introducing the concept of graded manifold, starting with the degree 1 case, and translating key geometric…
This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.
We study Galois embedding problems arising from the 3-torsion of elliptic curves defined over $\mathbb{Q}$, extending the correspondence to all possible images of mod 3 Galois representations; namely,…
After the appearance of my preprint [T3] (Special Lagrangian geometry and slightly deformed algebraic geometry (spLag and sdAG), Warwick preprint 22/1998, alg-geom/9806006, 54 pp.). I received an e-mail from Cumrun Vafa, who recognized that…
We construct a class of companion elliptic functions associated with the even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function $\wp(z|\tau)$ as the blueprint, we will derive their…