Jacobian Nullwerte, Periods and Symmetric Equations for Hyperelliptic Curves

We propose a solution to the hyperelliptic Schottky problem, based on the use of Jacobian Nullwerte and symmetric models for hyperelliptic curves. Both ingredients are interesting on its own, since the first provide period matrices which can be geometrically described, and the second have remarkable arithmetic properties.Comment: To appear in "Annales de l'Institut Fourier

A database of genus 2 curves over the rational numbers

We describe the construction of a database of genus 2 curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated L-function. This data has been incorporated into the L-Functions and Modular Forms Database (LMFDB).Comment: 15 pages, 7 tables; bibliography formatting and typos fixe

Examples of abelian surfaces with everywhere good reduction

We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler-Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms with paramodular level structures.Comment: 26 pages. Final version (to appear in Mathematische Annalen

Rational points on twists of X0(63)

Let $\varrho\colon G_\mathbb{Q}\longrightarrow PGL_2(\mathbb{F}_p)$ be a Galois representation with cyclotomic determinant, and let $N>1$ be an integer that is square mod $p$. There exist two twisted modular curves $X^+(N,p)_\varrho$ and $X^+(N,p)'_\varrho$\, defined over~$\mathbb{Q}$ whose rational points classify the quadratic $\mathbb{Q}$-curves of degree $N$ realizing $\varrho$. The paper focuses on the only genus-three instance: the case $N\!=7,\,p=3$. From an explicit description of the automorphism group of the modular curve $X_0(63)$, it follows that the twisted curves are isomorphic over $\mathbb{Q}$ in this case. We also obtain a plane quartic equation for the twists and then produce the desired $\mathbb{Q}$-curves, provided that the set of rational points on this quartic can be determined. The existence of elliptic quotients and of an unramified double cover $X(7,3)_\varrho$ having a genus-two quotient permits a variety of combinations of covers and Prym-Chabauty methods to determine these rational points. We include two examples where these methods apply

Computations on Modular Jacobian Surfaces Enrique González-Jiménez1, ⋆ , Josep González2,⋆ ⋆ 2, ⋆ ⋆ ⋆

Abstract. We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties Af attached by Shimura to normalized newforms f ∈ S2(Γ0(N)). We present all the curves corresponding to principally polarized surfaces Af for N ≤ 500.