Differences between revisions 2 and 4 (spanning 2 versions)

Christian Wuthrich (Nottingham): p-adic L-series and Iwasawa theory

Description

Artin and Tate have shown a large part of the conjecture of Birch and Swinnerton-Dyer in the function field case in the 60s. Iwasawa theory for elliptic curves as initiated by Mazur tries to use similar tools to approach the p-adic version of the Birch and Swinnerton-Dyer conjecture.

Let E/Q be an elliptic curve. The traditional conjecture by Birch and Swinnerton-Dyer states that there is a link between the arithmetic invariance, like the Mordell-Weil group E(Q), and the analytically defined complex L-function. In the p-adic BSD, we work with an analytic function Lp(E,s) taking values in the p-adic numbers. It is built on the values of the complex L-function and can be described explicitly using modular symbols. The p-adic conjecture says again that the order of vanishing of Lp(E,s) at s=1 is equal to the rank of the Mordell-Weil group E(Q). (Except in one special case, namely when the curve has split multiplicative reduction at p.)

The big advantage of the p-adic setting is that we actually know something about it. The p-adic L-function has a natural link to the arithmetic side via the so called "main conjecture" of Iwasawa theory about which we know quite a lot. Iwasawa theory deals with the question how so the arithmetic objects vary as one climbs up the tower of fields K∞/Q obtained by adjoining the p-power roots of unity. Similarily one can ask how does the mysterious Tate-Shafarevich group grow (or shrink). Much like the zeta-function for varieties over finite fields, there is a generating function that incodes this information. The main conjecture states that this generating function is equal to the p-adic L-function.

A big and difficult theorem by Kato shows half of this conjecture and Skinner and Urban claim they have shown the other half of it. As a consequence one gets that the order of vanishing of Lp(E,s) is at most the rank of E(Q). It even says something about the size of the mysterious Tate-Shafarevich group. It also implies that the rank of E(K∞) is finitely generated over the field K∞.

Projects

The p-adic L-function of E can be computed using modular symbols. And sage contains already code to do so. But this code could be improved in several direction.

* Allow to twist the function by Dirichlet characters. In particular with the Teichmüllers.

* Implement a function that extracts the λ and μ invariant and which decides it the growth of the Selmer group is due to the growth of the Tat-Shafarevich group or due to the increase of the rank.

Statistics on the values of these fundamental Iwasawa theoretic invariants. A question I was often asked by Iwasawa theorists is: Are the μ-invariants over Q(ζp) zero, too.

* Can we compute the modular symbols using complex integration ?

* Look at overconvergent modular symbols

* What happens for primes of additive reduction ?

References

* Mazur, Tate, Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer.

Invent. Math. 84 (1986), no. 1, 1--48. At mathscinet or gdz.

* Greenberg Ralph, Introduction to Iwasawa Theory for Elliptic Curves, (paper) on his web page full of Iwasawa theory.

Also there is the more advanced Iwasawa Theory for Elliptic Curves (paper).

* Stein and Wuthrich, Computations About Tate-Shafarevich Groups Using Iwasawa Theory, preprint .

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-Artin and Tate have shown a large part of the conjecture of Birch and Swinnerton-Dyer in the function field case in the 60s. Iwasawa theory for elliptic curves as initiated by Mazur tries to use similar to tools to approach the p-adic version of the Birch and Swinnerton-Dyer conjecture.
+== Description ==
 Line 5:
-Let E/Q be an elliptic curve. Now, we work with an analytic function L_p(E,s) taking values in the p-adic numbers. It is built on the values of the complex L-function and can be described explicitly using modular symbols. The conjecture says again that the order of vanishing of L_p(E,s) at s=1 is equal to the rank of the Mordell-Weil group E(Q). The big advantage of the p-adic setting is that this p-adic L-function has a natural link to the arithmetic side via the so called "main conjecture" of Iwasawa theory about which we know quite a lot.
+Artin and Tate have shown a large part of the conjecture of Birch and Swinnerton-Dyer in the function field case in the 60s.
Iwasawa theory for elliptic curves as initiated by Mazur tries to use similar tools to approach the  $p$ -adic version of the Birch and Swinnerton-Dyer conjecture.

Let  $E / Q$  be an elliptic curve. The traditional conjecture by Birch and Swinnerton-Dyer states that there is a link between the arithmetic invariance, like the Mordell-Weil group  $E (Q)$ , and the analytically defined complex L-function.
In the  $p$ -adic BSD, we work with an analytic function  $L_{p} (E, s)$  taking values in the  $p$ -adic numbers.
It is built on the values of the complex L-function and can be described explicitly using modular symbols.
The  $p$ -adic conjecture says again that the order of vanishing of  $L_{p} (E, s)$  at  $s = 1$  is equal to the rank of the Mordell-Weil group  $E (Q)$ . (Except in one special case, namely when the curve has split multiplicative reduction at  $p$ .)

The big advantage of the  $p$ -adic setting is that we actually know something about it. The  $p$ -adic L-function has a natural link to the arithmetic side via the so called "main conjecture" of Iwasawa theory about which we know quite a lot.
Iwasawa theory deals with the question how so the arithmetic objects vary as one climbs up the tower of fields  $K_{\infty} / Q$  obtained by adjoining the  $p$ -power roots of unity. Similarily one can ask how does the mysterious Tate-Shafarevich group grow (or shrink). Much like the zeta-function for varieties over finite fields, there is a generating function that incodes this information. The main conjecture states that this generating function is equal to the  $p$ -adic L-function.
-Line 7:
+Line 16:
-A big and difficult theorem by Kato shows that the order of vanishing of L_p(E,s) is at most the rank of E(Q). It even says something about the size of the mysterious Tate-Shafarevich group. Furthermore one can generalize it to abelian extensions K/Q. It is suitable for explicit computations.
+A big and difficult theorem by Kato shows half of this conjecture and Skinner and Urban claim they have shown the other half of it. As a consequence one gets that the order of vanishing of  $L_{p} (E, s)$  is at most the rank of  $E (Q)$ . It even says something about the size of the mysterious Tate-Shafarevich group. It also implies that the rank of  $E (K_{\infty})$  is finitely generated over the field  $K_{\infty}$ . 

== Projects ==

The  $p$ -adic L-function of  $E$  can be computed using modular symbols. And sage contains already code to do so. But this code could be improved in several direction.

* Allow to twist the function by Dirichlet characters. In particular with the Teichmüllers.

* Implement a function that extracts the  $λ$  and  $μ$  invariant and which decides it the growth of the Selmer group is due to the growth of the Tat-Shafarevich group or due to the increase of the rank.
  Statistics on the values of these fundamental Iwasawa theoretic invariants. A question I was often asked by Iwasawa theorists is: Are the  $μ$ -invariants over  $Q (ζ_{p})$  zero, too.

* Can we compute the modular symbols using complex integration ?

* Look at overconvergent modular symbols

* What happens for primes of additive reduction ?


== References ==

* Mazur, Tate, Teitelbaum,  On  $p$ -adic analogues of the conjectures of Birch and Swinnerton-Dyer.
  Invent. Math. 84 (1986), no. 1, 1--48. At [[http://www.ams.org/mathscinet-getitem?mr=830037| mathscinet]] or [[http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=175497|gdz]].

* Greenberg Ralph, Introduction to Iwasawa Theory for Elliptic Curves, [[http://www.math.washington.edu/~greenber/Park.ps|(paper)]] on [[http://www.math.washington.edu/~greenber/research.html|his web page]] full of Iwasawa theory. 
  Also there is the more advanced Iwasawa Theory for Elliptic Curves [[http://www.math.washington.edu/~greenber/CIME.ps|(paper)]].

* Stein and Wuthrich, Computations About Tate-Shafarevich Groups Using Iwasawa Theory, [[http://wstein.org/papers/shark/|preprint]]  .

Diff for "days22/wuthrich"

Christian Wuthrich (Nottingham): p-adic L-series and Iwasawa theory

Description

Projects

References