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Calculation of de Rham cohomology of abelian varieties/ jacobian varieties

It's known that for a elliptic curve like $E:y^2=x(x-1)(x-t)$ we have a basis $\frac{dx}{y}, \frac{xdx}{y}$ for $H_{dR}^1(E)$. But find such a basis is not an easy thing. I wonder for a general ...
Richard's user avatar
  • 785
3 votes
0 answers
301 views

Galois invariant of Tate module

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $V$ be a de Rham representation of $G_K=\operatorname{Gal}(\overline{K}/K)$. By Corollary 3.8.4 of Bloch and Kato - L-functions and Tamagawa ...
Desunkid's user avatar
  • 247
9 votes
1 answer
731 views

Tamagawa numbers

Let $K$ be a finite extension of $\mathbb{Q}_p$ with absolute Galois group $G_K$. Let $A$ be an abelian variety defined over $K$. The (geometric) Tamagawa number is defined as the order of the ...
Lukas's user avatar
  • 323
9 votes
1 answer
546 views

Morphisms for good reduction are maps respecting filtration

Please see edits below! So, let $A,A'/K$ be abelian varieties where $K$ is a $p$-adic local field with residue field $k$. Suppose further that they have good reduction with models $\mathscr{A},\...
Alex Youcis's user avatar
7 votes
1 answer
425 views

showing that abelian varieties are de Rham *without* showing that they are crystalline

If $X$ is a smooth projective variety over a $p$-adic field $K$, then Faltings' Theorem says that the etale cohomology of $X_{\overline{K}}$ is crystalline. There have been various steps towards this ...
Tony's user avatar
  • 287