Related papers: Explicit Gross-Zagier and Waldspurger Formulae
We provide explicit equations and parameterizations of some new rational Gushel-Mukai fourfolds of special type.
We give explicit structure of the graded ring of modular forms with respect to Gamma(N) (N=1,2,3,4,5,6,7,8,9,10,12,16,18) and for some other congruence groups. We also study the modular forms of half-integer weight for certain groups.
This paper gives a fairly explicit formula for the Hilbert series of algebras of Veronese type.
We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of…
By constructing suitable Borcherds forms on Shimura curves and using Schofer's formula for norms of values of Borcherds forms at CM-points, we determine all the equations of hyperelliptic Shimura curves $X_0^D(N)$. As a byproduct, we also…
We give new explicit formulas for Grassmannian and Aomoto polylogarithms in terms of iterated integrals, for arbitrary weight. We also explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2.…
In the previous article, we classified the characters associated to algebraic points on Shimura curves of $\Gamma_0(p)$-type, and over a quadratic field we showed that there are at most elliptic points on such a Shimura curve for every…
Using an explicit version of the Mumford isomorphism on the moduli space of hyperelliptic curves we derive a closed formula for the Arakelov-Green function of a hyperelliptic Riemann surface evaluated at its Weierstrass points.
The intuitive notion of the Gromov invariant for maps from a Riemann surface to a Grassmannian is shown to agree with the definition in \cite{BDW}. Also, an induction on the genus is proved, which extends the results of \cite{BDW} to a…
Let f be a newform, as specified by its Hecke eigenvalues, on a Shimura curve X. We describe a method for evaluating f. The most interesting case is when X arises as a compact quotient of the hyperbolic plane, so that classical q-expansions…
We show an exact geometric mass formula for superspecial points in the reduction of any quaternionic Shimura variety modulo at a good prime $p$.
Using Fourier analysis, we derive Wirtinger-type inequalities of arbitrary high order. As applications, we prove various sharp geometric inequalities for closed curves on the Euclidean plane. In particular, we obtain both sharp lower and…
We formulate and prove the Siegel-Weil formula for loop groups.
We derive regularized contour dynamics equations for the motion of infinite sharp fronts in the two-dimensional incompressible Euler, surface quasi-geostrophic (SQG), and generalized surface quasi-geostrophic (gSQG) equations. We derive a…
We give a functional equation for the refined Herglotz-Zagier function. It is analogous to a result in the theory of modular forms.
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
New formulas for the construction of Pythagorean triples and generalizations to equations of higher powers. Application of formulas to some problems, in particular Fermat's equation with n=4.
A generalization of the affine-geometric Wirtinger inequality for curves to hypersurfaces is given.
We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell-Weil…
An integral representation for form-factors of exponential fields in the sine-Gordon model is proposed.