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Is every abelian variety a subvariety of a Jacobian?

Embed the dual abelian variety into projective space. Take a smooth hyperplane section and interate until it's one-dimensional, obtaining a smooth curve $C$. By Lefschetz $C$ is irreducible, and the ...
• 116k
Accepted

An explicit equation for $X_1(13)$ and a computation using MAGMA

To get a model with good reduction at $2$, take $y = 2Y + x^3 + x^2 + 1$, subtract $(x^3+x^2+1)^2$ from both sides, and divide by $4$ to get $$Y^2 + (x^3+x^2+1) \, Y = -x^5-x^3+x^2-x.$$ (A similar ...
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Good lecture notes/books on Jacobian of hyperelliptic curve

Over $\mathbb C$, there's also Mumford's beautiful monograph Curves and their Jacobians, The University of Michigan Press, Ann Arbor, Mich., 1975. But a monograph covering all of the topics in your ...
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Is every abelian variety a subvariety of a Jacobian?

Let me give an answer for $k = \mathbb{C}$. By a theorem of Matsusaka, every abelian variety $A$ over an algebraic closed field $k$ is a quotient of a Jacobian. Now just apply Matsusaka's theorem to ...
Accepted

Embedding of a derived category into another derived category

Any fully faithful functor from $D^b(\mathcal{A})$ has adjoints (because $D^b(\mathcal{A})$ is a smooth and proper category), so its image is an admissible subcategory. A recent result from Dmitrii ...
• 31.8k
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functions with orthogonal Jacobian

Such maps are conformal. A theorem of Liouville says that if $n\geq 3$, the only conformal maps (defined in some region in $R^n$) are Mobius. A Mobius map is a composition of inversions in spheres. ...

Non-linear rational recurrence related to the Jacobian of hyperelliptic curve

NEW (Oct 27, 2017) I can now show that the recurrence given in the Question is correct. In fact, I can deduce a simpler and shorter recurrence relation. For this, we consider the Kummer Surface $K$ ...
• 10.6k
Accepted

• 34.5k
Accepted

Jacobians of genus 2 curves isogenous to a square of a supersingular elliptic curve over $\mathbb{F}_{p^2}$

Such curves are constructed in my paper "Familles de courbes et de variétés abéliennes sur $\mathbb{P}^1$, II", Astérisque vol. 86 (1981).
• 18.1k
Accepted

Abel-Jacobi map for Mumford curves analytically

This is done in Manin and Drinfeld's "Periods of p-adic Schottky groups." Journal für die reine und angewandte Mathematik 0262_0263 (1973): 239-247. You will also find it in Gerritzen and van der Put'...
• 3,697
Accepted

Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian

There is a duality between degree 2 coverings and two-torsion points on the Jacobian — i.e. both form elementary abelian 2-groups, and these groups are naturally dual. This is the Artin–Milne Poincaré ...
• 116k
Accepted

Perfectness of the Jacobian of a curve

No in general, for instance if $K$ is a number field : assuming that $C$ has a rational point, the group $\mathrm{Pic}(C)$ is naturally isomorphic to $JC(K)$, the group of rational points of the ...
• 34.5k