We prove a formula, which, given a principally polarized abelian variety $(A,\lambda )$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron–Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,\lambda )$.

We compute the $g=1$, $n=1$ B-model Gromov–Witten invariant of an elliptic curve $E$ directly from the derived category $\mathsf{D}_{\mathsf{coh}}^{b}(E)$. More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic $\mathscr{A}_{\infty }$ model of $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ described by Polishchuk. This is the first non-trivial computation of a positive-genus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov–Witten invariants of a symplectic 2-torus computed by Dijkgraaf.

The Weierstrass function σ(u) associated with an elliptic curve can be generalized in a natural way to an entire function associated with a higher genus algebraic curve. This generalized multivariate sigma function has been investigated since the pioneering work of Felix Klein. The present paper shows Hurwitz integrality of the coefficients of the power series expansion around the origin of the higher genus sigma function associated with a certain plane curve, which is called an (n, s)-curve or a plane telescopic curve. For the prime (2), the expansion of the sigma function is not Hurwitz integral, but its square is. This paper clarifies the precise structure of this phenomenon. In Appendix A, computational examples for the trigonal genus 3 curve ((3, 4)-curve) y3 + (μ1x + μ4)y2 + (μ2x2 + μ5x + μ8)y = x4 + μ3x3 + μ6x2 + μ9x + μ12 (where μj are constants) are given.

The ‘Borcherds products everywhere’ construction [Gritsenko et al., ‘Borcherds products everywhere’, J. Number Theory148 (2015), 164–195] creates paramodular Borcherds products from certain theta blocks. We prove that the $q$-order of every such Borcherds product lies in a sequence $\{C_{\unicode[STIX]{x1D708}}\}$, depending only on the $q$-order $\unicode[STIX]{x1D708}$ of the theta block. Similarly, the $q$-order of the leading Fourier–Jacobi coefficient of every such Borcherds product lies in a sequence $\{A_{\unicode[STIX]{x1D708}}\}$, and this is the sequence $\{a_{n}\}$ from work of Newman and Shanks in connection with a family of series for $\unicode[STIX]{x1D70B}$. Our proofs use a combinatorial formula giving the Fourier expansion of any theta block in terms of its germ.

Let $A$ be an abelian variety of dimension $g$ together with a principal polarization ${\it\phi}:A\rightarrow \hat{A}$ defined over a field $k$. Let $\ell$ be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup of $A[\ell ]$ which is maximal isotropic for the Riemann form associated to ${\it\phi}$. We suppose that $K$ is defined over $k$ and let $B=A/K$ be the quotient abelian variety together with a polarization compatible with ${\it\phi}$. Then $B$, as a polarized abelian variety, and the isogeny $f:A\rightarrow B$ are also defined over $k$. In this paper, we describe an algorithm that takes as input a theta null point of $A$ and a polynomial system defining $K$ and outputs a theta null point of $B$ as well as formulas for the isogeny $f$. We obtain a complexity of $\tilde{O} (\ell ^{(rg)/2})$ operations in $k$ where $r=2$ (respectively, $r=4$) if $\ell$ is a sum of two (respectively, four) squares which constitutes an improvement over the algorithm described in Cosset and Robert (Math. Comput. (2013) accepted for publication). We note that the algorithm is quasi-optimal if $\ell$ is a sum of two squares since its complexity is quasi-linear in the degree of $f$.

We show that there is no stable Siegel modular form that vanishes on every moduli space of curves.

We describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let A be an abelian variety of dimension g defined over a field of odd characteristic. Our algorithm comprises two principal steps. First, given a theta null point for A and a subgroup K isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian variety A/K. Then, from the knowledge of a theta null point of A/K, we present an algorithm to obtain a rational expression for an isogeny from A to A/K. The algorithm that results from combining these two steps can be viewed as a higher-dimensional analog of the well-known algorithm of Vélu for computing isogenies between elliptic curves. In the case where K is isomorphic to (ℤ/ℓℤ)g for ℓ∈ℕ*, the overall time complexity of this algorithm is equivalent to O(log ℓ) additions in A and a constant number of ℓth root extractions in the base field of A. In order to improve the efficiency of our algorithms, we introduce a compressed representation that allows us to encode a point of level 4ℓ of a g-dimensional abelian variety using only g(g+1)/2⋅4g coordinates. We also give formulas for computing the Weil and commutator pairings given input points in theta coordinates.

We develop the theory of Abelian functions associated with algebraic curves. The growth in computer power and the advancement of efficient symbolic computation techniques have allowed for recent progress in this area. In this paper we focus on the genus three cases, comparing the two canonical classes of hyperelliptic and trigonal curves. We present new addition formulae, derive bases for the spaces of Abelian functions and discuss the differential equations such functions satisfy.

Supplementary materials are available with this article.

Let be an abelian scheme over a scheme S which is quasi-projective over an affine noetherian scheme and let be a symmetric, rigidified, relatively ample line bundle on . We show that there is an isomorphism of line bundles on S, where d is the rank of the (locally free) sheaf . We also show that the numbers 24 and 12d are sharp in the following sense: if N>1 is a common divisor of 12 and 24, then there are data as above such that

we study some automorphic cohomology classes of degree one on the griffiths–schmid varieties attached to some unitary groups in three variables. using partial compactifications of those varieties, constructed by kato and usui, we define for such a cohomology class some analogues of fourier–shimura coefficients, which are cohomology classes on certain elliptic curves. we show that a large space of such automorphic classes can be generated by those with rational ‘coefficients’. more precisely, we consider those cohomology classes that come from picard modular forms, via some penrose-like transform studied in a previous article: we prove that the coefficients of the classes thus obtained can be computed from the coefficients of the picard form by a similar transform defined at the level of the elliptic curve.

One way of realizing representations of the Heisenberg group is by using Fock representations, whose representation spaces are Hilbert spaces of functions on complex vector space with inner products associated to points on a Siegel upper half space. We generalize such Fock representations using inner products associated to points on a Hermitian symmetric domain that is mapped into a Seigel upper half space by an equivariant holomorphic map. The representations of the Heisenberg group are then given by an automorphy factor associated to a Kuga fiber variety. We introduce theta functions associated to an equivariant holomorphic map and study connections between such generalized theta functions and Fock representations described above. Furthermore, we discuss Jacobi forms on Hermitian symmetric domains in connection with twisted torus bundles over symmetric spaces.

Using a classical generalization of Jacobi’s derivative formula, we give an explicit expression for Gunning’s prime form in genus 2 in terms of theta functions and their derivatives.