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18h
answered Are ranks of Jacobians over number fields unbounded?
19h
answered endomorphisms of the Jacobian of a curve
2d
answered Generalizing Ramanujan's “1729 story”
Apr
25
comment A number theory question
Please make the titles of your questions more specific. The title you chose conveys no more information than the tag. Also, you seem to have used both $P$ and $p$ for the same quantity. Finally, the LaTeX macro for a plus-or-minus sign is \pm. Also, as Carlo noted, you shouldn't be posting duplicate questions. Please delete one copy and edit the other to correct it.
Apr
21
answered The best possible density in Hilbert's Irreducibility Theorem
Apr
6
comment 3-7 primes in base 10
More generally, for those who are interested, Maynard proves that for any single chosen digit $a_0\in\{0,1,\ldots,9\}$, there are infinitely many primes whose decimal expansion does not contain the digit $a_0$. He also proves that for any $s$, then for sufficiently large $q$ and any collection $B\subset\{0,1,\ldots,q-1\}$ with $|B|\le s$, there are infinitely many primes whose base $q$ expansion does not contain any of the digits in $B$. All quite spectacular.
Apr
3
comment Field of definition of a point in $[p]^{-1}E(K)$
Do you mean $m$-division points or $p^m$-division points? You say the former, but your definition of $P_m$ seems to imply you mean the latter. The multiplication-by-$p$ map factors as $p=F\circ G$ with $F$ the Frobenius map and $G$ separable (since you specify that $E$is ordinary). So doesn't your question just come down to the kernel of $F^m$?
Mar
20
comment On elliptic curves, $\sqrt{x^2-101y^2} ,\sqrt{x^2+101y^2}$, and their ilk
@NoamD.Elkies Indeed there are such families. But since the OP didn't construct his family with an eye toward skewing the FE signs, it's likely that it's a 50-50 family (although, as you note, proving it might be hard or currently impossible).
Mar
19
comment On elliptic curves, $\sqrt{x^2-101y^2} ,\sqrt{x^2+101y^2}$, and their ilk
"... is birationally equivalent to an elliptic curve, call it $E_1$, and generally has infinitely many rational points." Really? Current perceived wisdom seems to be that unless your family has a section, then the probability that a random element in your family has infinitely many rational points (i.e., has positive rank) is 50%. So maybe "frequently" would be a better word than "generally".
Mar
6
comment Analogy between Lagrange's Theorem and Rank-Nullity Theorem?
This is all good, but I think that the Lagrange's theorem (in the category of finite groups) is a bit more subtle, because it is more general that there being a homomorphism $M\to N$.. Note that $M/N$ in this context is the set of cosets of $N$, so it's just a set, not a group.
Mar
3
awarded  Enlightened
Mar
3
awarded  Nice Answer
Mar
2
answered Weak Mordell-Weil over number fields
Feb
25
answered Which groups are Galois over some p-adic field?
Feb
22
comment Find special elliptic curves from j-invariant
@NoamD.Elkies I know the OP asked for "deterministic", but I assumed that he/she wanted "practical deterministic". If one doesn't specify $j$, is there a practical way to find an $\mathbb F_p$ point on $X_1(\ell)$ if, say, $\ell$ is in the $10^3$ to $10^4$ range?
Feb
22
comment Find special elliptic curves from j-invariant
@DavidLampert Good point, I did know that, but it slipped my mind. Of course, the main point is that the likely answer is that there are no such curves.
Feb
22
answered Find special elliptic curves from j-invariant
Feb
17
comment A search for theorems which appear to have very few, if any hypotheses
Would it be possible to state the theorems (succinctly) instead of providing PDF links?
Feb
16
awarded  Nice Answer
Feb
15
answered Open access journals in number theory