(with T. Ibukiyama and H. Katsurada) Some Siegel modular standard L-values, and Shafarevich-Tate groups (dvi) . Submitted.
(with B. Heim) Triple product L-values and dihedral congruences for cusp forms (dvi) . To appear in I.M.R.N. Revised version posted 6/10/09.
(with B. Heim) Symmetric square L-values and dihedral congruences for cusp forms (dvi) . Submitted.
Symmetric square L-functions and Shafarevich-Tate groups, II (dvi) . To appear in Int. J. Number Theory. Final version posted 13/5/09.
(with P. Martin and M. Watkins) Euler factors and local root numbers for symmetric powers of elliptic curves (dvi) . Revised version posted 18/12/07. Pure and Appl. Math. Qu. 5, no. 4, J. Tate special issue (2009), 1311--1341.
Rational points of order 7 (dvi) . Bull. London Math. Soc. 40 (2008), 1091--1093. Final version posted 4/11/08.
(with M. Watkins) Critical values of symmetric power L-functions (dvi) . New version posted 20/11/07. Pure and Appl. Math. Qu. 5, no. 1, J.-P. Serre special issue (2009), 127--161.
Level-lowering for higher congruences of modular forms (dvi) . New version posted, 17/3/05. Awaiting revision and resubmission.
Eisenstein primes, critical values and global torsion (dvi) . New version posted 24/5/07. Pacific J. Math 233 (2007), 291--308.
In the proof of Proposition 2.1, for some cusps u is zero rather than a unit.
On a conjecture of Watkins (dvi) . New version posted 17/3/06. J. Theorie de Nombres de Bordeaux 18 (2006), 345--355.
In Section 5, the reduced tangent space to the deformation problem is possibly slightly larger than the Selmer group that I claimed it was equal to. The local subgroup at odd p dividing N should be the kernel of restriction to I_p *combined with inclusion of ad^0 in ad *. In odd characteristic this would just be the same thing, but when l=2 it makes a difference. However, if 2^R divides a smaller number then it divides a bigger number, so the main conclusion is unaffected.
Rational torsion on optimal curves (dvi) . New version posted, 12/12/05. Int. J. Number Theory 1 (2005), 513--532.
Tamagawa factors for certain semi-stable representations (dvi) . New version posted, 27/9/04. Bull. London Math. Soc. 37 (2005), 835--845.
In the numerical example at the end, the period is out by a power of 2 (which is OK since we are looking at the 11-part). Actually, in several of my papers, the statement of the Bloch-Kato conjecture is out by a power of 2 because I have neglected torsion in the real points, i.e. the Tamagawa factor at infinity. Here the reason is a little more involved. See "Critical values of symmetric power L-functions" (above) for the truth about the power of 2
Values of a Hilbert modular symmetric square L-function and the Bloch-Kato conjecture (dvi) . J. Ramanujan Math. Soc. 20 (2005), 167--187.
In Section 10, our construction was conditional on the expected non-triviality of H^1_f(F,V_lambda(k/2)). I am grateful to Jan Nekovar for pointing out that this non-triviality follows from his latest results on the parity of ranks of Selmer groups. In his preprint "Selmer Complexes", available here , see Theorem 12.2.3, with F=F", noting that condition (1) holds because [F:Q] is odd. In the case F=Q (that examined in "Symmetric square L-functions and Shafarevich-Tate groups" below) this also provides an alternative to the quoted theorem of Skinner and Urban.
Congruences of modular forms and tensor product L-functions (dvi) . Bull. London Math. Soc. 36 (2004), 205--215.
The no-congruences condition of Proposition 3.1 should also be in Proposition 4.4 .
Tamagawa factors for symmetric squares of Tate curves (dvi)
. Math. Res. Lett. 10 (2003), 747--762.
Correction: p.4,l.21, replace "its maximal ideal" by "the maximal ideal of its subring". Just before 6.8, the ith term in the
filtration of S should be the intersection of S with (u-p)^i S[1/p]. If E has non-split multiplicative reduction at p then in Lemma 3.1.3
E should be replaced by a quadratic twist with split multiplicative reduction at p.
(with W. Stein and M. Watkins) Constructing elements in Shafarevich-Tate groups of modular motives (dvi)
. From Swinnerton-Dyer birthday volume "Number
Theory and Algebraic Geometry", M. Reid, A. Skorobogatov, eds.,
London Math. Soc. Lecture Note Series 303, 91--118, Cambridge
University Press, 2003.
Remark 5.2, that the sign is the same for f and g, is correct, but the reason given
only works when N is squarefree (and a plus or minus is inserted). More generally, one
considers the action of W_N on delta_f and delta_g.
In Theorem 6.1, the condition that, for p|N, p is not -w_p(mod q), is not necessary. The argument from the good reduction case applies once we have divisibility of the inertia-fixed part of A_q. Three useful remarks about this paper may be found in the review by J. Nekovar.
Symmetric squares of elliptic curves: rational points and Selmer
groups (pdf)
. Experiment. Math. 11:4 (2002), 457--464. (Almost final version.)
Published in Experimental Mathematics and placed by permission from the publisher A K Peters, Ltd.
Please note: in 6.4, ``analytic rank'' is really ``apparent analytic rank''. If E has non-split multiplicative reduction at p then in Lemma 3.1
E should be replaced by a quadratic twist with split multiplicative reduction at p. The ``i.e." in Lemma 4.3 is misleading. Due to the possibility
of rational cyclic l^2-isogenies, the implication is only one-way. For both parts of Lemma 4.1 to be true when l=3, we need E[l] to be irreducible
even when restricted to Q(sqrt(-3)), otherwise it is possible for E[l] and E[l](1) to be isomorphic (even though the twist is non-trivial).
Symmetric square L-functions and Shafarevich-Tate groups (ps)
. Experiment. Math. 10:3 (2001), 383--400.
Published in Experimental Mathematics and placed by permission from the publisher A K Peters, Ltd.
Correction to Section 7: I am grateful to Masataka Chida for pointing out errors in
my description of the computation. In fact the correct answer for my mod p calculation
is zero. But by doing a more refined calculation mod p^2, Chida has confirmed that
the p-adic dL/ds-values (for k=22 and p=131 or 593) are non-zero.
Actually, this whole calculation has been rendered unnecessary, by a theorem of
Skinner and Urban.
Correction to Table 1: I am grateful to H. Katsurada for pointing out that the entry for k=16, r=3 is incorrect. The correct value is given in the introduction to "Symmetric square L-functions and Shafarevich-Tate groups, II, above.
Correction to Section 10: The functional equation of the symmetric cube L-function is due to Shahidi (Compositio Math. 37(1978)), while the entirety of its meromorphic continuation is due to Kim and Shahidi (Ann. Math. 150 (1999)).
Again regarding Section 10, the following Pari programs give some additional numerical evidence for symmetric 4th power L-functions. sym4wt12, sym4wt16, sym4wt20. These read Tim Dokchitser's program ComputeL, which you will therefore need. They give decimal approximations to some ratios of L-values (divided also by a power of pi), which should be rational, with certain primes (like p=691 when k=12) in the numerators. Type "contfrac(%)" to get the continued fraction, truncate it in the obvious place, use"contfracpnqn([...])" to convert it to a rational number, then "factor(%)", and watch the expected primes appear.
Congruences of modular forms and Selmer groups (dvi) . Math. Res. Lett. 8 (2001), 479--494. Published by International Press.
Rather than starting from twists with vanishing L-functions, then appealing to the Birch-Swinnerton-Dyer conjecture to get rational points, one can start from the existence of many twists of rank at least 2, then use Kolyvagin's theorem to get the vanishing of the L-functions. Thus one may obtain an unconditional result in support of the Bloch-Kato conjecture. This was pointed out by McGraw and Ono, see J.L.M.S. 67 (2003), 302--318.
Period ratios of modular forms (dvi)
. Math. Ann. 318 (2000), 621--636.
Published by Springer-Verlag.Springer Link.
Contains some minor errors and
misleading remarks. Sorry.