Minimal discriminant of an elliptic curve in terms of its Galois representation

From the Galois representation of an elliptic curve E we can read the conductor of E, and further some information about the minimal discriminant. So is there any more information about the minimal discriminant we can read from the Galois representation? For example, it is natural to consider the differences between isogenous (equivalently having the … Read more

What is the Artin invariant of an elliptic supersingular K3 surface?

Let $X$ be a supersingular K3 surface over an algebraically closed field $k$ of positive characteristic $\!p$. Artin proved in the paper that the determinant $\mathrm{disc}(X)$ of a matrix of the intersection pairing on the Néron-Severi group $NS(X)$ is equal to $-p^{2\sigma}$, where $1 \leqslant \sigma \leqslant 10$. If X has a quasi-elliptic fibration, … Read more

Height pairings of Heegner points of nontrivial conductor

I am studying Gross’s and Zagier’s proof of the BSD conjecture for elliptic curves of rank ≤1. Their calculation essentially boils down to the following ingredients: (1.) Finding a suitable imaginary-quadratic extension K of Q with no rank growth and calculating the Néron-Tate height pairings ⟨x,σ(x)⟩ of Heegner points x of conductor 1 with their … Read more

Expansion of Jacobi theta function at pp-torsion

I am aware of the formula Θ(z,q)=zexp(−2∑k≥1z2k(2k)!E2k(q)) for the Jacobi theta function at the origin z=0. The definition I am using for the theta function is Θ(z,q)=(ez/2−e−z/2)∏m≥1(1−ezqm)(1−e−zqm)(1−qm)2. Is there a similar formula for the expansion at a p-torsion point z=2πrip? EDIT: It isn’t hard to show that the qn coefficients in the expansion of lnΘ(z,q) … Read more

identifying components of points on elliptic curves with Kodaira symbol I∗2nI_{2n}^{*}

Let K be a local field that is complete with respect to a discrete valuation. When an elliptic curve, E/K, has reduction type represented by the Kodaira symbol I∗2n, its component group can be Z/2Z×Z/2Z. In this case, is there a criterion to determine which component a point is on? In particular, to tell if … Read more

Finding short linear combinations in abelian groups

Let M be a finitely generated abelian group. Assume we are given a presentation of M, that is M=⨁ri=1Zgi∑sj=1Zrj where the gi are the generators and the rj are the relations. Let x be an element of M, given as a linear combination of the generators. I want to express x as a linear combination … Read more

Weierstrass model of an elliptic curve: a line bundle over the base

Let S be a Weierstrass model of an elliptic surface (for me it works better to understand it as an elliptic fibration), that is a map π:S→C where C is a compact Riemann surface. Disclaimer: Physicist here. Intuitively I think of this as a compact Riemann surface (e.g. CP1) where over each point z we … Read more

Action of the Picard Scheme of an Elliptic Fibration

Suppose that we have a surface X defined over a field k (I am interested in k being a number field) and an elliptic fibration f:X→P1, i.e. f is proper and almost all fibres are smooth curves of genus 1. Let U⊂P1 be the subset of points above which the fibre of f is smooth. … Read more

Which proportion of elliptic curves over Q is proved to satisfy BSD conjecture?

In 2014, Bhargava and other authors proved that more than 66 % of all elliptic curves defined over Q satisfy the Birch and Swinnerton-Dyer conjecture. Tonight I stumbled upon an article of Kozuma Morita (arXiv:1803.11074) claiming to prove that all elliptic curves with complex multiplication satisfy this conjecture, entailing a solution of the congruent number … Read more