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Oct
16
comment Is the Cassels-Tate pairing defined for elliptic curves over function fields?
There are though known problems with the proof that the kernels are the maximal divisible subgroup. See Harari-Szamuely and Gonzalez-Aviles for the corrections, both in Crelle 632 in 2009 with a title containing "arithmetic duality theorems".
Sep
19
comment What are all positive integers n for which the congruence $a^{n+1} \equiv a (mod n)$ holds?
Moreover this is a duplicate from mathoverflow.net/questions/37097/…
Sep
19
comment p-adic L-function of curves
The complex $L(C,s)$ can be written in terms of the Jacobian of $C$. So you might also ask if for all abelian varieties $A/\mathbb{Q}$ there is a $p$-adic $L$-function. Certainly there should be one, but we do not know how to construct it in general, I fear. For instance, we may not know anything about the "modularity" of $A$ and hence not even that $L(C,s)$ extends to the complex plane and in particular to the places where we would like to interpolate $p$-adically.
Sep
9
comment Alternate Definition(s) for (Ir)regular Primes
It seems to me that, using the summation formula for consecutive powers (Faulhaber's formula), your "conjectured example" is equivalent to the definition of the divisibility of Bernoulli numbers. I would not call this a new definition.
Aug
10
comment padic BSD vs. BSD for algorithm to compute rank
For BSD, we only know $r_{an}=0\Rightarrow r=0$ and $r_{an}=1\Rightarrow r=1$. For $p$-adic BSD we know this plus $r_{an} \geq r$.
Aug
7
comment $p$-adic Regulators
What $p$ does not matter at all. The quotient is independent of $p$. In fact any quadratic form $h$ on $E(K)$ with $h(\varphi(P)) = \deg(\varphi) \cdot h(P)$ will give the same result.
Aug
7
comment $p$-adic Regulators
... and as a consequence: I don't think there is a simple relation to the quotient of the minimal modular degree. Take non-semistable examples and examples with non-trivial Sha to find counter-examples.
Aug
7
comment $p$-adic Regulators
Maybe I just should have said that both the p-adic and the real BSD formula are invariant under isogenies.
Aug
7
answered $p$-adic Regulators
Aug
5
comment Rank of Elliptic Curves
Probably you mean "size" of the 2-Selmer group or "dimension" as $\mathbb{F}_2$-vector space, not "rank" since it is finite. The corank of the 2-primary Selmer group is equal to the rank of the curve unless Sha is infinite.
Aug
5
answered Rank of Elliptic Curves
Jul
30
comment BSD leading-term coefficient in terms of places without distinction
Ok, I saw 3 on the linked post. I would disagree with that "canonical volume" interpretation to be very good for understanding the regulator. There is an symmetric bilinear form on $E(\mathbb{Q})$ with values in $\mathbb{R}$ defined for instance in Silverman's book VIII.9. The target here is the completion where the $L$-series takes values. In the $p$-adic BSD conjecture, the leading term of the $p$-adic $L$-series is linked to the canonical $p$-adic regulator coming from a bilinear form with values in $\mathbb{Q}_p$.
Jul
30
comment BSD leading-term coefficient in terms of places without distinction
? Sorry, I don't understand this. If the rank is 1, you have a "stack" $\mathbb{R}//\mathbb{Z}$. What sort of measure to you put on this to get the canonical height of the generator ?
Jul
30
comment BSD leading-term coefficient in terms of places without distinction
Even this formulation is uniform in the different completions. Sha is the kernel to all completions; the regulator is the determinant of the height, which is a sum over all completions; the Tamagawa numbers and the archemedian periods are gathered as an adelic measure over all places (in the number field case, the product does not split canonically).
Jul
30
comment BSD leading-term coefficient in terms of places without distinction
Another small correction: Sha also compares with the infintite place, not just finite primes.
Jul
30
comment BSD leading-term coefficient in terms of places without distinction
What would be your uniform interpretation for the class number formula ?
Jul
30
comment BSD leading-term coefficient in terms of places without distinction
Yet, in the end, whatever formulation you will see, there will always be the input of several sides. The Bloch Kato conjecture will also put together comparison isomorphisms etc to get to a formulation. So I don't think there is a single expression that covers the full quotient.
Jul
30
comment BSD leading-term coefficient in terms of places without distinction
Tate's original forumlation archive.numdam.org/ARCHIVE/SB/SB_1964-1966__9_/… using adelic integration uniformises $\Omega_E\cdot \prod c_p$ into one expression. Since the regulator measures something on the Mordell-Weil group modulo torsion, one could also consider the quotient of $\mathrm{Reg}_E/(\#E(\mathbb{Q})_{\mathrm{tors}}))^2$ as one thing.
Jul
29
comment If the $L$-series does not vanish
Probably, that sort of question is best asked to the authors directly :)
Jul
29
comment If the $L$-series does not vanish
"good" implies "semistable", so we are using a stronger hypothesis than the theorem we use, no ?