I am studying the congruent number problem 
and I heard that there is a paper by Kazuma Morita
which claims to solve this problem from my colleague.
I saw the paper on his homepage but it is very short 
and I cannot belive it is true because it is too short.
While I try to find his mistakes, I don't have the knowledge 
about the Hodge-Tate weight (p,q). Although there is an explanation 
on the Hodge-Tate weight (p,q), he writes that it is based on the 
Japanese article by T.Yoshida. I know the notion about Hodge-Tate 
weight p but not Hodge-Tate weight(p,q) well. So, let me know the 
litaratures in English on Hodge-Tate weight (p,q).



congruent number problem
Objection to nfdc23. Sorry if it is wrong! 


His $p$-adic avatar carries the uniformizer to Frob 
and one gets the equality of L-functions but 
this is related just by $\chi=\iota\circ \sigma\circ Art$ which 
is not a continuous character in general because of a field isomorphism $\iota:\bar{Q}\_{p}\simeq C$. He looked at only the finite components but not infinite or $p$-adic components. So, I think that 
his proof is incomplete. The effect of the complex conjugation does 
not necessarily swap two Galois characters (as a continuous character). 
On the other hand, as I wrote in the comment, Morita takes care of this issue. 


