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| People: Berinder, M. Tip, Adam, Robert Miller, Robert Bradshaw, Chris Wuthrich | People: Barinder, M. Tip, Adam, Robert Miller, Robert Bradshaw, Chris Wuthrich |
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| == Computing root numbers project: notes == | |
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| References: 1. Silverman I Chapter VII (does not mention root numbers but gives background information to compute them) |
Main background reference: Silverman I Chapter VII (does not mention root numbers but gives background information to compute them) - reduction types of elliptic curves etc; plus Section of Silverman I Appendix C on Tate's algorithm. |
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| Elliptic Curves over \QQ | Elliptic Curves over Q |
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| root number w = \prod_p w_p * w_{\infinity} | root number $w = \prod_p w_p$ * $w_\infty$ |
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| p is a prime of good reduction iff p \nmid discriminant. If E has good reduction at p then w_p = +1. |
p is a prime of good reduction iff $p$ does not divide the discriminant. If E has good reduction at p then $w_p = +1$. |
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| p \mid \mid means p is a prime of multiplicative reduction If E has split multiplicative reduction then w_p = -1 If E has non-split multiplicative reduction then w_p +1 |
$p \mid \mid N$ means p is a prime of multiplicative reduction If E has split multiplicative reduction then $w_p = -1$ If E has non-split multiplicative reduction then $w_p = +1$ |
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| Step 1: implement w for E/\QQ with N square-free (already done in GP) If |
Step 1: implement w for $E/Q$ with N square-free (already done in GP) |
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| If p^2\mid N then p is a prime of additive reduction for E and w_p is more complicated. There are formulae to compute them, they rely on Tate's algorithm. | If $p^2\mid N$ then p is a prime of additive reduction for E and $w_p$ is more complicated. There are formulae to compute them, they rely on Tate's algorithm. |
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| w = \prod_p w_p \prod_{v\div \infinity} (-1) | w = $\prod_p w_p \prod_{v\div \infty} (-1)$ |
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| w_p for p \nmid 2,3 has been done by Rohrlich (somewhat hard to read) see Theorems 2 and 3 which are self-contained | $w_p for p$ not dividing $2,3$ has been done by Rohrlich (somewhat difficult) see [1] Theorem 2 which is self-contained |
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| for p \mid 3 this has been done Kobayashi | for $p \mid 3$ this has been done Kobayashi [2] |
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| for $p \mid 2$ T & V Dokchitser the formulae are really hard - ignore this (!!) | for $p \mid 2$ T & V Dokchitser [3] the formulae are really hard - ignore this (!!) |
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| Would perhaps be better to do for any p, or even determine w globally (T & V Dokshitser: page 1) | Would perhaps be better to do for any p, or even determine w globally (T & V Dokchitser: [4] in the introduction) |
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[1] D. Rohrlich, Galois Theory, elliptic curves, and root numbers, Compos. Math. 100 (1996), 311--349. [2] S. Kobayashi, The local root number of elliptic curves with wild ramification, Math. Ann. 323 (2002), 609--623; available online [3] http://arxiv.org/abs/math/0612054 [4] http://arxiv.org/abs/0906.1815 == Working groups == Implementation: * (Local) Armin, Charlie, Chris * (Global) Lola, Chris, Hatice, Charlie Reading (Kobayashi) * Lola, AJ, Thilina, MTip L-functions of elliptic curves over number fields * Adam |
Tim Dokchitser (Cambridge University): Complex L-functions and the Birch and Swinnerton-Dyer conjecture
Structure of the course
- Quick review of Elliptic curves over Q and the Mordell-Weil theorem
- Elliptic curves over finite fields, heuristics for their distribution and the naive version of BSD
- L-functions of elliptic curves and the BSD-conjecture
- Root numbers and how to compute them
- Parity predictions, Goldfeld's conjecture and ranks of elliptic curves over number fields
Prerequisites
Some familiarity with basic algebraic number theory (number fields, primes), and having seen elliptic curves
Background reading
J. H. Silverman, "The arithmetic of elliptic curves", Chapters 3, 7 and 8.
Sage Reference Manual on elliptic curves: http://sagemath.org/doc/reference/plane_curves.html, up to `Isogenies'.
Computational projects
There will be many small problems and larger assignments to play with, illustrating all the concepts and conjectures from the course.
A. Root Numbers over K for elliptic curves (implement)
- People: Armin, Charlie, Hatice, Christ, Lola, Robert Miller, Thilina, M. Tip, Robert Bradshaw
B. #III(E/K)_{an} function (L-functions, connection to Wuthrich)
- People: Barinder, M. Tip, Adam, Robert Miller, Robert Bradshaw, Chris Wuthrich
C. Parity Predictions
- People: Arijit, Anil, Adam
Computing root numbers project: notes
Main background reference: Silverman I Chapter VII (does not mention root numbers but gives background information to compute them) - reduction types of elliptic curves etc; plus Section of Silverman I Appendix C on Tate's algorithm.
Elliptic Curves over Q
root number w = \prod_p w_p * w_\infty
p is a prime of good reduction iff p does not divide the discriminant. If E has good reduction at p then w_p = +1.
p \mid \mid N means p is a prime of multiplicative reduction If E has split multiplicative reduction then w_p = -1 If E has non-split multiplicative reduction then w_p = +1
Step 1: implement w for E/Q with N square-free (already done in GP)
If p^2\mid N then p is a prime of additive reduction for E and w_p is more complicated. There are formulae to compute them, they rely on Tate's algorithm.
Elliptic Curves over general number fields
Root number classification
w = \prod_p w_p \prod_{v\div \infty} (-1)
Additive reduction w_p for p not dividing 2,3 has been done by Rohrlich (somewhat difficult) see [1] Theorem 2 which is self-contained
for p \mid 3 this has been done Kobayashi [2]
for p \mid 2 T & V Dokchitser [3] the formulae are really hard - ignore this (!!)
Would perhaps be better to do for any p, or even determine w globally (T & V Dokchitser: [4] in the introduction) There is a decision to be made as to which methods to use.
[1] D. Rohrlich, Galois Theory, elliptic curves, and root numbers, Compos. Math. 100 (1996), 311--349.
[2] S. Kobayashi, The local root number of elliptic curves with wild ramification, Math. Ann. 323 (2002), 609--623; available online
[3] http://arxiv.org/abs/math/0612054
[4] http://arxiv.org/abs/0906.1815
Working groups
Implementation:
- (Local) Armin, Charlie, Chris
- (Global) Lola, Chris, Hatice, Charlie
Reading (Kobayashi)
- Lola, AJ, Thilina, MTip
L-functions of elliptic curves over number fields
- Adam