# Publications

## since 2000

## 2024

Quinary forms and paramodular forms (PDF, 755KB)

(with A Pacetti, G Rama, G Tornaria)

Mathematics of Computation 93 (2024), pp. 1805–1858. Final author accepted version posted on 27th October 2023.

## 2023

Residual paramodularity of a certain Calabi-Yau threefold (PDF, 427KB)

(with G Tornaria)

Slightly revised version posted 3rd November 2023.

## 2022

Lifting congruences to half-integral weight (PDF, 356KB)

Research in Number Theory 8 (2022), 59, Published version. Final author accepted version posted on 22 March 2022.

Congruences of Saito-Kurokawa lifts and denominators of central spinor L-values (PDF, 479KB)

Glasgow Mathematical Journal 64 (2022), pp. 504–525. Published version.

Please note that the negative observation in Remark 2.4 is invalid, since the Riemann zeta function has a pole at s=1. cf. Theorem 3.13 in reference [16].

## 2021

Automorphic forms for some even unimodular lattices (PDF, 522KB)

(with D Fretwell)

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 91 (2021), pp. 29–67.

Published version.

Congruences of local origin and automorphic induction (PDF, 378KB)

(with D Spencer)

International Journal of Number Theory 17 (2021), pp. 1617–1629.

Published version.

## 2020

Twisted adjoint L-values, dihedral congruence primes and the Bloch-Kato conjecture (PDF, 361KB)

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 90 (2020), pp. 215–227.

Published version.

## 2019

GL2xGSp2 L-values and Hecke eigenvalue congruences (PDF, 439KB)

(with J Bergstroem, D Farmer and S Koutsoliotas)

Journal de Théorie des Nombres de Bordeaux 31 (2019), pp. 751–775.

Published version.

Automorphic forms on Feit's Hermitian lattices (PDF, 431KB)

(with S Schoennenbeck)

Experimental Mathematics, 2019, DOI 10.1080/10586458.2019.1581857.

Published version.

Kurokawa-Mizumoto congruences and degree-8 L-functions (PDF, 426KB)

(with B Heim and A Rendina)

Manuscripta mathematica 160 (2019), 217--237.

Published version.

## 2018

Eisenstein congruences for SO(4,3), SO(4,4), spinor and triple product L-values (PDF, 475KB)

(with J Bergstroem and T Megarbane)

with an appendix by T Ibukiyama and H Katsurada. Experimental Mathematics 27 (2018), pp. 230–250.

Published version.

## 2017

Eisenstein congruences and endoscopic lifts (PDF, 248KB)

in Automorphic representations, automorphic L-functions and related topics, S Hayashida (editor), RIMS Kokyuroku, 2036 (2017), pp. 128–134. The write-up of a conference talk based on the paper two places below.

Published version.

Lifting puzzles and congruences of Ikeda and Ikeda-Miyawaki lifts (PDF, 362KB)

Journal of the Mathematical Society of Japan 69 (2017), pp. 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.

Published version.

Lifting congruences to weight 3/2 (PDF, 352KB)

(with S Krishnamoorthy)

Journal of the Ramanujan Mathematical Society 32 (2017), pp. 431–440.

## 2016

Eisenstein congruences for split reductive groups (PDF, 520KB)

(with J Bergstroem)

Replaces `Harder's conjecture and its analogues'. Modified version posted on 14 October 2015.

The + at the top of p.11 should be -. Selecta Mathematica 22 (2016), pp. 1073–1115.

The conjectural lift from G2 to GSp3 referred to in the introduction is simply an instance of Langlands functoriality. The Gross-Savin lift is from a compact form of G2.

Published version.

## 2015

Eisenstein congruences for odd orthogonal groups (PDF, 366KB)

Modified version posted on 17 April 2015.

Eisenstein congruences for unitary groups (PDF, 446KB)

Replaces `A U(2,2) analogue of Harder's conjecture'. Modified version posted on 22 July 2015.

Quadratic Q-curves, units and Hecke L-values (PDF, 405KB)

(with V Golyshev)

Math. Zeitschrift 280 (2015), pp. 1015–1029.

Published version.

## 2014

Ramanujan-style congruences of local origin (PDF, 348KB)

(with D Fretwell)

Journal of Number Theory 143 (2014), pp. 248–261.

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.

Exact holomorphic differentials on a quotient of the Ree curve (PDF, 391KB)

(with S Farwa)

Journal of Algebra 400 (2014), pp. 249–272.

Published version.

## 2013

A simple trace formula for algebraic modular forms (PDF, 136KB)

Experimental Mathematics 22 (2013), pp. 123–131.

Published version.

Powers of 2 in modular degrees of modular abelian varieties (PDF, 230KB)

(with S Krishnamoorthy)

New version with a serious correction and revisions, posted on 17 November, 2011. Another new version posted on 5 June, 2012. J. Number Theory 133 (2013), pp. 501–522.

Published version.

## 2012

Yoshida lifts and Selmer groups (PDF, 619KB)

(with S Boecherer and R Schulze-Pillot)

Journal of the Mathematical Society of Japan 64 (2012), pp. 1353–1405.

Published version.

## 2011

Some Siegel modular standard L-values, and Shafarevich-Tate groups (PDF, 393KB)

(with T Ibukiyama and H Katsurada)

Journal of Number Theory 131 (2011), 1296--1330.

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.

## 2010

Triple product L-values and dihedral congruences for cusp forms (PDF, 283KB)

(with B Heim)

International Mathematics Research Notices (2010), pp. 1792–1815.

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.)

Symmetric square L-values and dihedral congruences for cusp forms (PDF, 295KB)

(with B Heim)

Journal of Number Theory 130 (2010), pp. 2078–2091.

Published version.

## 2009

Symmetric square L-functions and Shafarevich-Tate groups, II (PDF, 311KB)

International Journal of Number Theory 5 (2009), pp. 1321–1345.

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, 230KB)

Automorphic representations, automorphic L-functions and arithmetic, RIMS Kokyuroku 1659, July 2009, pp. 37–46. Partly an expository paper about more than what is in the title.

Critical values, congruences and moving between Selmer groups (PDF, 179KB)

Proceedings of Mathematical Science Colloquium 2008, Muroran Institute of Technology. Another expository paper.

Euler factors and local root numbers for symmetric powers of elliptic curves (dvi, 124KB)

(with P Martin and M Watkins)

Pure and Applied Mathematics Quarterly, 5, no. 4, J Tate special issue (2009), pp. 1311–1341.

Critical values of symmetric power L-functions (PDF, 463KB)

(with M Watkins)

Pure and Applied Mathematics Quarterly, 5, no. 1, J-P Serre special issue (2009), pp. 127–161.

## 2008

Rational points of order 7 (PDF, 168KB)

Bulletin of the London Mathematical Society, 40 (2008), pp. 109–1093.

## 2007

Eisenstein primes, critical values and global torsion (PDF, 271KB)

Pacific Journal of Mathematics, 233 (2007), pp. 291–308.

In the proof of Proposition 2.1, for some cusps u is zero rather than a unit.

## 2006

On a conjecture of Watkins (dvi, 54KB)

Journal de Théorie des Nombres de Bordeaux, 18 (2006), pp. 45–355.

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.

## 2005

Level-lowering for higher congruences of modular forms (PDF, 397KB)

New version posted, 22 August 2014 (after more than 9 years!). Then another one on 14 October 2015.

Rational torsion on optimal curves (PDF, 288KB)

International Journal of Number Theory 1 (2005), pp. 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, 342KB)

Bulletin of the London Mathematical Society, 37 (2005), pp. 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, ie 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 (PDF, 383KB)

Journal of the Ramanujan Mathematical Society, 20 (2005), pp. 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", 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.

## 2004

Congruences of modular forms and tensor product L-functions (dvi, 63KB)

Bulletin of the London Mathematical Society, 36 (2004), pp. 205–215.

The no-congruences condition of Proposition 3.1 should also be in Proposition 4.4.

## 2003

Tamagawa factors for symmetric squares of Tate curves (dvi, 85KB)

Mathematical Research Letters, 10 (2003), pp. 747–762.

Published version.

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.

Constructing elements in Shafarevich-Tate groups of modular motives (dvi, 120KB)

(with W Stein and M Watkins)

From Swinnerton-Dyer birthday volume "Number Theory and Algebraic Geometry", M Reid, A. Skorobogatov, eds., London Mathematical Society Lecture Note Series 303, pp. 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.

## 2002

Symmetric squares of elliptic curves: rational points and Selmer groups (PDF, 258KB)

Experimental Mathematics, 11, 4 (2002), pp. 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 "ie" 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).

## 2001

Symmetric square L-functions and Shafarevich-Tate groups (ps, 892KB)

Experimental Mathematics, 10, 3 (2001), pp. 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 (Annals of Mathematics 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 (PDF, 372KB)

Mathematical Research Letters 8 (2001), pp. 479–494. Published by International Press.

Published version.

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 Journal of the London Mathematical Society, 67 (2003), pp. 302–318.

## 2000

Period ratios of modular forms (dvi, 72KB)

Mathematische Annalen, 318 (2000), pp. 621–636. Published by Springer-Verlag.

Springer link.

Contains some minor errors and misleading remarks.