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? 



Does split L-function imply split jacobian
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.](http://dx.doi.org/10.1007/BF01388432)


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 $B \times C$.


