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Dec
9
awarded  Caucus
Nov
10
comment How to find certain quadratic curves over $\mathbb{Q}$
Thanks. It is likely that there are solutions other than the unit circle. It is possible to generate a family of such curves?
Nov
10
asked How to find certain quadratic curves over $\mathbb{Q}$
Nov
6
comment Hasse's theorem on elliptic curves over finite fields
I think multiplicity-one theorem forbids such a construction. Wiles proves that $\lambda_p+\bar{\lambda_p}$ comes from the coeffecient of some modular form of weight 2, and multiplicity-one theorem asserts that coefficients from two modular (eigen)forms satisfy $a_p^2=b_p^2$ for almost all prime $p$ if and only if $a(p)=b(p)\chi(p)$, where $\chi$ is some quadratic character. Your construct forces $\chi(p)=-1$ for almost all prime $p$, which is not possible.
Nov
2
comment Identities related to hypergeometric functions
I use Clausen's formula and linear transformation formula for the deduction, and things might go bad for the hypergeometric functions are multi-valued around its singularities. A numerical computation on Mathematica suggest that A is true for $0<k<\sqrt{2}-1$ while B is true for $k>\sqrt{2}-1$.
Nov
2
comment Identities related to hypergeometric functions
Hmm. A and B cannot be simultaneously true for the same k. I am wondering whether a transformation formula exists. RHS has a singular point at $\sqrt{2}-1$, and I need such a formula for analytic continuation, and I hope this would help.
Nov
2
revised Identities related to hypergeometric functions
added 16 characters in body
Nov
2
comment Identities related to hypergeometric functions
Eh, $K^{\prime}$ does not represent the derivative of $K$ in elliptic integral theory. It is $K(\sqrt{1-k^2})$
Nov
2
asked Identities related to hypergeometric functions
Oct
7
comment Some questions on the formation of the BSD conjecture
Hmm. I have read a nice paper of Franz Lemmermeyer with the title "Conics: A poor man's elliptic curves", where such analog is discussed. But I am still wondering how Tate found the analog of regulator on the elliptic curves. I don't think numerical computations will help to establish the refined BSD conjecture.
Oct
5
asked Some questions on the formation of the BSD conjecture
Sep
8
comment Integral point on certain cubic surfaces and rational parametric solutions
@guy-in-seoul: He discuss about heights in the last chapter in his book.
Sep
8
comment Integral point on certain cubic surfaces and rational parametric solutions
@guy-in-seoul: I did take a look at Manin's book on cubic forms, and the construction is just the one he introduces in the introduction. But I am not sure if he ever talks about something about integral points in his book.
Sep
7
comment Integral point on certain cubic surfaces and rational parametric solutions
@GerryMyerson: Hmm. The denominators in the constructions are polynomials of (at least) four variables, and I'm not sure what an obstruction will be to prevent these polynomials from taking small values.
Sep
7
asked Integral point on certain cubic surfaces and rational parametric solutions
Aug
20
accepted How to bound the following sum
Aug
20
comment How to bound the following sum
@Assaultous2: Terence Tao once wrote on moebius function in his blog, and he stated that $\{x/i\}$ has bounded variation, which is key to the proof of the identity $\sum\mu(n)/n=0$. The statement of Tao is the motivation of this question.
Aug
20
asked How to bound the following sum
Jul
27
awarded  Popular Question
Jul
2
awarded  Curious