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If the jacobian of a curve splits then the L-function splits as well simply because isogenous Abelian varieties have the same L-function. Is the converse true? Or under what condition is it true?

Edit: According to Faltings' theorem pointed out by Francois, if the L-function split into product of L functions of Abelian variety then the Jacobian of the curve splits. But what if I only know the L function splits (not necessarily into product of L-function, this can happen according to the reference pointed out by Felipe). By the L function splits I mean the reduction of jacobian splits at all places with good reduction. Is there any way to see if the L-function splits into product of L function or not? If it splits into product of L functions, is it possible to construct the curve which give the factors in the L function?

More specifically, look at the genus 4 curve $C:y^3=x^6+1$, the L-function of the curve splits in the sense above, and $C$ maps to $E:y^3=x^3+1$ and $E′:y^3=x^2+1$ , so the Jacobian of $C$ splits to $E×E′×F$, how could I know if $F$ splits or not?

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Faltings proved that two abelian varieties $A$ and $B$ defined over a number field are isogenous if and only if they have the same $L$-function. See Korollar 2 p. 361 in

Faltings, G. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (1983), no. 3, 349--366.

So if you assume $L(A,s)=L(B,s) L(C,s)$ for some abelian varieties $A,B,C$ defined over a number field $K$, you can deduce that $A$ is $K$-isogenous to $B \times C$.

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    $\begingroup$ Thanks, Francois. But what if I only know the L function splits (not necessarily into product of L-functions, this can happen according to the reference pointed out by Felipe). $\endgroup$
    – Liu Hang
    Mar 13, 2013 at 12:21
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    $\begingroup$ I don't understand your new question. What do you mean exactly by "The L-function splits" then ? $\endgroup$
    – François Brunault
    Mar 13, 2013 at 13:14
  • $\begingroup$ The reduction of jacobian can split at all places with good reduction, so the local L factor splits everywhere, this is what I mean by "the L-function splits", sorry for the confusing. So could we know if the jacobian of the curve split or not from the information of the L function of the curve. More specifically, look at the genus 4 curve $C:y^3=x^6+1$, the L-function of the curve splits in the sense above, and $C$ maps to $E:y^3=x^3+1$ and $E′:y^3=x^2+1$ , so the Jacobian of $C$ splits to $E×E′×F$, how could I know if $F$ splits or not? $\endgroup$
    – Liu Hang
    Mar 13, 2013 at 13:37
  • $\begingroup$ I don't know the answer to this new question but the reference given by Felipe is certainly relevant. About your particular curve, you have an abelian surface $F$ and you want to know whether it splits. There is an algorithm due to Ruppert to test that over the complex numbers, see e.g. Birkenhake-Lange, Complex abelian varieties. If you have a decomposition of $F$ over $\mathbf{C}$, then you can check over which number field the elliptic factors are defined by computing the $c_4$ and $c_6$ invariants. Of course, there might be smarter methods in this particular case. $\endgroup$
    – François Brunault
    Mar 13, 2013 at 13:54

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