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Results tagged with abelian-varieties
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user 26522
Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic geometry and number theory.
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Average ranks of abelian surfaces
Most people nowadays believe that over a fixed global field, $50$% of the elliptic curves have $0$ rank, $50$% have rank $1$, and $0$% have higher rank. A significant advance in this direction has bee …
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The torsion point count in higher dimension
It is an easy consequence of the Serre open image theorem that for the torsion point count on elliptic curves, the following possibilities arise.
If $E/\bar{\mathbb{Q}}$ is an elliptic curve without …
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Nakai-Moishezon theorem for abelian varieties
On an abelian variety (regardless of the characteristic), an effective divisor with positive self-intersection is ample. To be more precise, it suffices here to recall that on any simple abelian varie …
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What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal e...
Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added t …
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Points of minimum Arakelov height and harmonic arithmetical varieties
Added. (28/2) To put it less pompously (and more vaguely, less concretely), I wanted to relate the impression that it is the general rule that an Arakelov (i.e., geometric) height on an arithmetical v …
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Orders of reductions of rational points on elliptic curves
For example, is it known that there is an infinite sequence of rational primes $p_i$ and primes $P_i$ of (the ring of integers of) $K$ such that $p_i$ divides the order of the reduction of $x$ modu …
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The uniform boundedness of rational torsion for traceless abelian surfaces over a function f...
The function field analog of the theorem of Mazur-Kamienny-Merel (giving a universal bound in $[K:\mathbb{Q}]$ for the size of the $K$-rational torsion of an elliptic curve) is immediate just from the …
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Purely additive reduction of Jacobian of Hyperelliptic curve
It is all about the $g$-cuspidal singularity $y^2 = x^{2g+1}$ of the special fibre, and how this singularity affects the Picard group in terms of the Picard group of the normalization (which is an abe …
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Must the coordinates of a polynomial iteration have about the same size?
Original post. The following statements seem plausible (not to say intuitively obvious), but I do not see how to prove them.
Let us say that a polynomial mapping of $\mathbb{C}^2$ is reducible if ei …
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Is the canonical height of a totally p-adic point on an abelian variety bounded away from zero?
Inspired by the result of Schinzel and Smyth that a totally real number other than $0$ and $\pm 1$ has height at least $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.240659\ldots$, Bombieri and …
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