Chris Wuthrich
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Registered User
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1d |
answered | von Staudt-Clausen for other special values |
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Jun 9 |
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Textbooks on Algorithmic Number Theory Definitely Henri Cohen's "A course in computational algebraic number theory" is the best place to start. At a lower level David M. Bressoud "Factorization and Primality Testing" is a nice introduction. |
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Jun 7 |
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Behavior of a quantity related to Fermat’s 4n + 1 Theorem You meant $\sqrt{2}$ not $2$, no? |
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Jun 3 |
accepted | Specialization of sections in an elliptic fibration |
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Jun 3 |
answered | Specialization of sections in an elliptic fibration |
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May 23 |
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can we say that $(p^2+1)/2\ne p_0^2$ where $p$ is a Mersenne prime You are absolutely right. A stupid sign error. I delete my comment, so ashamed I am of it. |
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May 13 |
accepted | Best bounds toward Serre’s uniformity conjecture |
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May 13 |
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Best bounds toward Serre’s uniformity conjecture I should add that I don't know about GRH-conditional bounds. But it feels a bit awkward to assume such a strong conjecture when instead you could assmue the conjecture $M(E)\leq 37$; which may well be proven one day. |
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May 13 |
answered | Best bounds toward Serre’s uniformity conjecture |
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May 7 |
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Are Kato’s zeta elements integral? @Francois. Integrality is true only for the lattice $T$ above and any larger ones. And I happen to be able to get around doing explicit computations of the zeta elements; their properties are enough. |
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May 7 |
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Are Kato’s zeta elements integral? I edited all of the answer now I actually know the full answer. |
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May 2 |
awarded | ● Nice Answer |
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May 2 |
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Torsion subgroups in families of twists of elliptic curves I think you are not meant to ask questions as answers. The non-trivial Galois element $\sigma$ of $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$ acts on the right hand side. The subgroup fixed by $\sigma$ is clearly the $n$-torsion over $\mathbb{Q}$. Now if $\sigma$ sends a point $P$ to $-P$, then its $y$-coordinate is of the form $z\cdot\sqrt{d}$ for some rational number $z$. Now writing out the equation between $z$ and the $x$-coordinate of $P$ gives you exactly the Weierstrass equation of the twist of $E$ by $d$. |
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Mar 30 |
awarded | ● Yearling |
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Mar 11 |
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Deducing BSD from Gross-Zagier and Kolyvagin edited body |
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Mar 11 |
answered | Deducing BSD from Gross-Zagier and Kolyvagin |
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Mar 7 |
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Inequality relating rank and analytic rank @ACL: Yes I do. It can already be deduced from Tate's Boubaki talk on the geometric analogue of BSD. |
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Mar 7 |
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Inequality relating rank and analytic rank In the function field case, there is a known inequality, but it is the other way around than yours: the analytic rank is an upper bound to the algebraic rank. Iwasawa theory tries to mimic this for number fields, but all we currently know is that the $p$-adic analytic rank is an upper bound to the alebraic rank. Here the $p$-adic analytic rank is the order of vanishing of the $p$-adic L-function. |
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Mar 3 |
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Do isogenies with rational kernels tend to be surjective? @Dror: Unless the reudction is additive at $p=5$, the modular parametrisation of minimal degree $X_1(N)\to E'_d$ should factor through $\eta$. So your Heegner point argument gives again the link from the question to the quotient of Tamagawa numbers and Shas via the formula for their index. Nice remark. |
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Mar 2 |
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Do isogenies with rational kernels tend to be surjective? Sorry, René, when you said "via Galois cohomology" I though you used the map from $E(\mathbb{Q})/\hat\eta$ into $H^1(\mathbb{Q},\mu_5)=\mathbb{Q}^{\times}/5$. So now, I don't see how you get your formula. Don't you want to write it up as an answer ? |
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Mar 1 |
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Do isogenies with rational kernels tend to be surjective? Thanks, Stefan, for your answer. You have thought about it more than I have. See my comments below your answer. |
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Mar 1 |
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Do isogenies with rational kernels tend to be surjective? @René: I agree with you that the formula (if it is this or anything similar) is hardly ever a $5$-th power in $\mathbb{Q}^{\times}$ if the coordinates $x_0,y_0$ of the point are random rationals. However, they are far from random. For instance, you know from descent that the formula will be a $5$-th power in $\mathbb{Q}_v$ at all good places $v$ (and many more), which translates into stringent conditions on $x_0$ and $y_0$. So I am not sure one can say anything through this approach... |
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Mar 1 |
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Do isogenies with rational kernels tend to be surjective? Your formula shows well that $c$ which is my $-(a-b)$ has a tendancy to be negative. Do you also have conditions from the descent on $c$ implied by the fact that your curve has rank $1$ ? |
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Mar 1 |
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Do isogenies with rational kernels tend to be surjective? Of course I agree with you that the local information translates your question into one about the quotient of Shas above. My assumption was that this quotient $s$ is 1 and I was not following the curves in the order they appear in your family. So I - probably wrongly - would have guessed that this quotient is very often 1 for curves of small conductor. If you restrict your attention to those with $s=1$, do you get the same phenomenon as I ? How many have $s\neq 1$ in your list ? |
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Feb 28 |
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Do isogenies with rational kernels tend to be surjective? @Maarten: For $p=2$ you should not only assume that $E$ has a two-torsion point in the kernel, but also that the isogeny is etale, i.e. that the Neron period lattice of $E$ is contained in the lattice of $E'$. That would be the analogous situation. But I am not sure if I expect more frequent surjectivity there... |
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Feb 28 |
answered | Do isogenies with rational kernels tend to be surjective? |
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Jan 30 |
awarded | ● Civic Duty |
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Jan 10 |
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Rank growth of elliptic curves after cubic extensions Yes, Noam, root number do not change in odd-degree Galois extensions. |
