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when we want to find high rank curve over $\mathbb{Q}$ among an elliptic curve over $\mathbb{Q}(t)$  ,we give some scores for the curves (the curves with bigger points on them over various primes in reduction module p: #$E(F_p)$,i mean $S(E,N)=\sum_{p<N}\frac{2-a_{p}}{|E(F_p)|}*log(p)$,it is briefly called Mestre's sum) and the curves with bigger scores usually have bigger rank than the original curve over  $\mathbb{Q}(t)$.the specialization theorem help us to be sure that group does't becomes smaller.

it is interesting that when we do the same thing from $\mathbb{Q}(t,t')$ to $\mathbb{Q}(t)$ and try to compute rank of surfaces with conjectural limit ( proposed by Nagao,the special case of Tate's conjecture,proved by Rosen and silverman for rational surfaces) it gives the same rank always, not bigger rank in any cases( i tried an example and compute about 100000 surfaces and compute the limit, all of them was equal)

what's happening?

When we want to find a high rank curve over $\mathbb{Q}$ among an elliptic curve over $\mathbb{Q}(t)$, we give some scores for the curves (the curves with bigger points on them over various primes in reduction module p: #$E(F_p)$,i mean $S(E,N)=\sum_{p<N}\frac{2-a_{p}}{|E(F_p)|}*log(p)$, it is briefly called Mestre's sum) and the curves with bigger scores usually have bigger rank than the original curve over $\mathbb{Q}(t)$. The specialization theorem helps us to be sure that the group doesn't become smaller.

It is interesting that when we do the same thing from $\mathbb{Q}(t,t')$ to $\mathbb{Q}(t)$ and try to compute the rank of surfaces with conjectural limit (proposed by Nagao, the special case of Tate's conjecture, proved by Rosen and Silverman for rational surfaces) it always gives the same rank, not bigger rank in any cases  (I tried an example and computed about 100000 surfaces and computed the limit; all of them were equal).

What's happening?

when we want to find high rank curve over $\mathbb{Q}$ among an elliptic curve over $\mathbb{Q}(t)$ ,we give some scores for the curves (the curves with bigger points on them over various primes in reduction module p: #$E(F_p)$,i mean $S(E,N)=\sum_{p<N}\frac{-a_{p}+2}{p+1-a_p}*log(p)$,it is briefly called Mestre's sum) and the curves with bigger scores usually have bigger rank than the original curve over $\mathbb{Q}(t)$.the specialization theorem help us to be sure that group does't becomes smaller.

it is interesting that when we do the same thing from $\mathbb{Q}(t,t')$ to $\mathbb{Q}(t)$ and try to compute rank of surfaces with conjectural limit ( proposed by Nagao,the special case of Tate's conjecture,proved by Rosen and silverman for rational surfaces) it gives the same rank always, not bigger rank in any cases( i tried an example and compute about 100000 surfaces and compute the limit, all of them was equal)

what's happening?

when we want to find high rank curve over $\mathbb{Q}$ among an elliptic curve over $\mathbb{Q}(t)$ ,we give some scores for the curves (the curves with bigger points on them over various primes in reduction module p: #$E(F_p)$,i mean $S(E,N)=\sum_{p<N}\frac{2-a_{p}}{|E(F_p)|}*log(p)$,it is briefly called Mestre's sum) and the curves with bigger scores usually have bigger rank than the original curve over $\mathbb{Q}(t)$.the specialization theorem help us to be sure that group does't becomes smaller.

it is interesting that when we do the same thing from $\mathbb{Q}(t,t')$ to $\mathbb{Q}(t)$ and try to compute rank of surfaces with conjectural limit ( proposed by Nagao,the special case of Tate's conjecture,proved by Rosen and silverman for rational surfaces) it gives the same rank always, not bigger rank in any cases( i tried an example and compute about 100000 surfaces and compute the limit, all of them was equal)

what's happening?

when we want to find high rank curve over $\mathbb{Q}$ among an elliptic curve over $\mathbb{Q}(t)$ ,we give some scores for the curves (the curves with bigger points on them over various primes in reduction module p: #$E(F_p)$,it is briefly called Mestre's sum) and the curves with bigger scores usually have bigger rank than the original curve over $\mathbb{Q}(t)$.the specialization theorem help us to be sure that group does't becomes smaller.

it is interesting that when we do the same thing from $\mathbb{Q}(t,t')$ to $\mathbb{Q}(t)$ and try to compute rank of surfaces with conjectural limit ( proposed by Nagao,the special case of Tate's conjecture,proved by Rosen and silverman for rational surfaces) it gives the same rank always, not bigger rank in any cases( i tried an example and compute about 100000 surfaces and compute the limit, all of them was equal)

what's happening?

when we want to find high rank curve over $\mathbb{Q}$ among an elliptic curve over $\mathbb{Q}(t)$ ,we give some scores for the curves (the curves with bigger points on them over various primes in reduction module p: #$E(F_p)$,i mean $S(E,N)=\sum_{p<N}\frac{-a_{p}+2}{p+1-a_p}*log(p)$,it is briefly called Mestre's sum) and the curves with bigger scores usually have bigger rank than the original curve over $\mathbb{Q}(t)$.the specialization theorem help us to be sure that group does't becomes smaller.

it is interesting that when we do the same thing from $\mathbb{Q}(t,t')$ to $\mathbb{Q}(t)$ and try to compute rank of surfaces with conjectural limit ( proposed by Nagao,the special case of Tate's conjecture,proved by Rosen and silverman for rational surfaces) it gives the same rank always, not bigger rank in any cases( i tried an example and compute about 100000 surfaces and compute the limit, all of them was equal)

what's happening?