(with J. Bergstroem and T. Megarbane) Eisenstein congruences for SO(4,3), SO(4,4), spinor and triple product L-values (pdf), with an appendix by T. Ibukiyama and H. Katsurada. Slightly modified version posted 6th December 2016. Experimental Mathematics 2016, DOI 10.1080/10586458.2016.1251861. Link to published version.
Lifting puzzles and congruences of Ikeda and Ikeda-Miyawaki lifts (pdf). J. Math. Soc. Japan 69 (2017), 801--818. The ``as yet unproved equivalence'' referred to in the introduction has been proved by Arancibia, Moeglin and Renard, so the constructions in this paper are now unconditional. Link to published version.
(with S. Krishnamoorthy) Lifting congruences to weight 3/2 (pdf). To appear in Journal of the Ramanujan Mathematical Society. Revised version posted 14th November, 2016.
Eisenstein congruences for odd orthogonal groups (pdf). Modified version posted 17th April 2015.
Eisenstein congruences for unitary groups (pdf). Replaces `A U(2,2) analogue of Harder's conjecture'. Modified version posted 22nd July 2015.
(with J. Bergstroem) Eisenstein congruences for split reductive groups (pdf). Replaces `Harder's conjecture and its analogues'. Modified version posted 14th October 2015. The + at the top of p.11 should be -. Selecta Mathematica 22 (2016), 1073--1115. Link to published version.
(with V. Golyshev) Quadratic Q-curves, units and Hecke L-values (pdf). New version posted 1st April 2015. Math. Zeitschrift 280 (2015), 1015--1029. Link to published version.
(with D. Fretwell) Ramanujan-style congruences of local origin (pdf). J. Number Theory 143 (2014), 248--261. New version posted 1st July, 2014. Link to published version. Correction: Following Conjecture 3.2, the condition for triviality of the Tamagawa factor should be l>k+1, not l>k. Similarly, this should be the condition (same as Billerey and Menares) in Conjecture 4.1 and Proposition 4.2, whereas for Proposition 4.3 one needs only l>3. One more thing: Proposition 3.1 should have a condition l>k to get the local condition at l.
(with S. Farwa) Exact holomorphic differentials on a quotient of the Ree curve (pdf). New version posted 28th August, 2013. J. Algebra 400 (2014), 249--272. Link to published version.
A simple trace formula for algebraic modular forms (pdf). New version posted 3rd July 2012. Experimental Mathematics 22 (2013), 123--131. Link to published version.
(with S. Krishnamoorthy) Powers of 2 in modular degrees of modular abelian varieties (pdf). New version with a serious correction and revisions, posted 17th November, 2011. Another new version posted 5th June, 2012. J. Number Theory 133 (2013), 501--522. Link to published version.
(with S. Boecherer and R. Schulze-Pillot) Yoshida lifts and Selmer groups (pdf) . J. Math. Soc. Japan 64 (2012), 1353--1405. Final version posted 19th December 2011. Link to published version.
(with T. Ibukiyama and H. Katsurada) Some Siegel modular standard L-values, and Shafarevich-Tate groups (pdf) . J. Number Theory 131 (2011), 1296--1330. Link to published version. In the second paragraph of the proof of Proposition 4.3, a change of lattice may be required before the extension is non-trivial.
(with B. Heim) Triple product L-values and dihedral congruences for cusp forms (pdf) . I.M.R.N. (2010), 1792--1815. Link to published version. I am grateful to Victor Rotger for questioning Conjecture 5.2. I should have excluded the central point, where the L-function vanishes and the conjecture (now for the leading term) would involve a regulator. (Similar comment on Conjecture 2.1 in the paper immediately above.)
(with B. Heim) Symmetric square L-values and dihedral congruences for cusp forms (dvi) . J. Number Theory 130 (2010), 2078--2091. Link to published version.
Symmetric square L-functions and Shafarevich-Tate groups, II (pdf) . Int. J. Number Theory 5 (2009), 1321--1345. Link to published version. I am grateful to Masataka Chida for bringing to my attention that fact that the hypothesis on the Galois representation being symplectic can be removed, since this is now a known fact, following from a theorem in R. Weissauer ``Existence of Whittaker models related to four dimensional symplectic Galois representations". See, below, comments on the prequel for corrections to the data for (k,r)=(16,3), (16,11) and (20,11). Note also that in Proposition 4.4, the restrictions on r can be ameliorated to r geq 5 by proving the congruences starting from the pullback formula in Section 9, instead of using bracket operators. In fact, assuming a mild condition, this proof works without any condition about S_j+k being 1-dimensional. Please note that in Proposition 9.1, the condition k geq 3 should be k>5.
Selmer groups for tensor product $L$-functions, (pdf) . pp. 37--46 in Automorphic representations, automorphic $L$-functions and arithmetic, R.I.M.S. Kokyuroku 1659, July 2009. Partly an expository paper about more than what is in the title.
Critical values, congruences and moving between Selmer groups, (pdf) . Proceedings of Mathematical Science Colloquium 2008, Muroran Institute of Technology. Another expository paper.
(with P. Martin and M. Watkins) Euler factors and local root numbers for symmetric powers of elliptic curves (dvi) . Pure and Appl. Math. Qu. 5, no. 4, J. Tate special issue (2009), 1311--1341.
(with M. Watkins) Critical values of symmetric power L-functions (dvi) . Pure and Appl. Math. Qu. 5, no. 1, J.-P. Serre special issue (2009), 127--161.
Rational points of order 7 (dvi) . Bull. London Math. Soc. 40 (2008), 1091--1093.
Eisenstein primes, critical values and global torsion (pdf) . 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) . 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. But see `Powers of 2 in modular degrees of modular abelian varieties' above, for an important modification and correction.
Level-lowering for higher congruences of modular forms (dvi) . New version posted, 22/8/14 (after more than 9 years!). Then another one on 14/10/15.
Rational torsion on optimal curves (pdf) . Int. J. Number Theory 1 (2005), 513--532. Byeon and Yhee have removed two unnecessary conditions from Theorem 4.1, by proving that they always hold.
Tamagawa factors for certain semi-stable representations (pdf) . 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, the 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
. 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, and to Alex Ghitza for pointing out that the entries for (k,r)=(16,3), (16,11) and (20,11) are incorrect, and for checking that all the others are correct. These entries should be 2^20 / 3^7*5^3*7*11*13^2*17, 2^24*839 / 3^12*5^8*7^4*11^2*13^2*17*19*23 and 2^27*304477 / 3^19*5^8*7^4*11^2*13^2*17^2*19*23*29, respectively.
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.