Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Arithmetic and geometry of a K3 surface emerging from virtual corrections to Drell--Yan scattering

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The two-loop Drell–Yan square root sits on a K3 surface whose Picard lattice has rank 19 and discriminant 24, and whose Shioda–Inose structure is tied to the level-160 weight-2 modular form—so the root cannot be rationalised by a rational…

desk verdict The Picard lattice computation is solid and citable, but the Shioda-Inose section contains a numerical inconsistency that makes Theorem 5.11 false as stated. read the letter →

arxiv 1908.01079 v3 pith:3LGSRK7Z submitted 2019-08-02 math.AG hep-thmath.NT

Marco Besier , Dino Festi , Michael Harrison , Bartosz Naskrecki This is my paper · ORCID
classification math.AGhep-thmath.NT MSC 14J2814J2714G1011F4111G40
keywords K3surfacesPicardlatticeDrell–YanscatteringFeynmanintegralssquare-rootrationalisationellipticfibrationsShioda–Inosestructuremodularforms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies the surface $X_{DY}: w^2 = 4xy^2z(x-z)^2 + (x+y)^2(xy+z^2)^2$, the hypersurface attached to the square root that appears in the two-loop mixed electroweak–QCD virtual corrections to Drell–Yan scattering. It proves that $X_{DY}$ is birationally equivalent to a K3 surface and that the geometric Picard lattice of its desingularisation has rank $19$, discriminant $24$, and discriminant group $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$. The paper then builds an explicit Shioda–Inose structure and shows that it is governed by the classical modular form of level $160$ and weight $2$. The physical payoff is negative but useful: since $X_{DY}$ is birational to a K3 surface, the associated square root cannot be rationalised by any rational change of variables, so the Drell–Yan master integrals cannot be reduced to ordinary multiple polylogarithms by this route; instead, the paper points toward elliptic fibrations and elliptic multiple polylogarithms as the natural function class.

What carries the argument

The argument is carried by three connected structures. The first is the double-cover description of $X_{DY}$ as $w^2 = f(x,y,z)$ with $f$ a sextic, which makes the desingularisation $S$ a K3 surface once the five $A$-type singularities are resolved. The second is the geometric Picard lattice: the sublattice $\Lambda$ generated by twenty-four explicit divisors is computed by intersection theory (in part with computer algebra), and the Galois action of $\mathbb{Q}(\sqrt{5})$ is used to rule out an index-two overlattice, forcing $\Lambda = \mathrm{Pic}\, S$. The third is the elliptic-fibration machinery: a combinatorial search for Kodaira fibres supported on thirty-four known $(-2)$-curves produces over a hundred thousand genus-one fibrations, three of which are written out explicitly; the third fibration, pulled back by $t \mapsto t^2$, identifies $S$ with the Inose fibration of the Kummer surface of $E_1 \times E_2$. This Shioda–Inose structure is the hinge that transfers the arithmetic of $S$ to the elliptic curves $E_1,E_2$, and the modular forms of level $160$ enter when those curves are shown to be $\mathbb{Q}$-curves whose Hilbert modular form is a base change of the classical newform.

What would settle it

Compute the intersection matrix of the twenty-four divisors $\Sigma$ by an independent implementation: if its rank is not $19$ or its determinant is not $24$, then $\mathrm{Pic}\,S \ne \Lambda$ and the main lattice theorem fails. Separately, to test the Shioda–Inose step, compute the $\mathbb{F}_p$-point counts of $S$ for $p = 31$ and $p = 71$ directly from the elliptic fibration and compare with the formulas of Theorem 5.11; any disagreement would show that the quotient-Kummer assumption does not hold or the modular identification is wrong.

Watch

Extended reading notes

Core claim

The central claim is that the desingularisation $S$ of $X_{DY}$ is a K3 surface with geometric Picard lattice exactly equal to the sublattice spanned by an explicit list of $24$ divisors: the hyperplane class, seven lines, two conics, and the exceptional curves over the five singular points. The intersection matrix of these divisors has rank $19$ and discriminant $24$, and the discriminant group is $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$; the same lattice is recomputed in Section 6 from an explicit elliptic fibration via the Shioda–Tate formula and height pairings, giving an independent verification. In Section 5 the authors exhibit a quadratic base change that identifies $S$ with the Inose fibration of the Kummer surface $\mathrm{Kum}(E_1 \times E_2)$, where $E_1$ and $E_2$ are $3$-isogenous elliptic curves over $\mathbb{Q}(\sqrt{2},\sqrt{5})$. From the modularity of this pair they obtain, for every prime $p \ge 7$, explicit formulas $|S(\mathbb{F}_p)| = 1 + 17p + (1+(5/p))p + \mu(p) + p^2$ with $\mu(p) = a_p(f)^2 - (10/p)p$, and the analogous formula over $\mathbb{F}_{p^2}$. The stated consequence is that the square root in (2.10) is not rationalisable by a rational change of variables, and that the Drell–Yan surface is neither birational nor isogenous to the K3 surface coming from the two-loop Bhabha-scattering correction.

Load-bearing premise

The load-bearing premise is that the desingularised quotient of $S$ by the explicit Nikulin involution is a Kummer surface, as asserted in Remark 4.14; Section 5.1 invokes the Shioda–Inose structure conditionally ('Assume that $S$ admits a Shioda–Inose structure'), and without a complete proof of that quotient step the point-count and modular-form conclusions of Theorem 5.11 are unsupported, though the Picard-lattice claims of Sections 3 and 6 stand independently.

Editorial extensions

If this is right

  • The square root (2.10) cannot be rationalised by any rational change of variables; hence the two-loop mixed EW-QCD Drell–Yan master integrals cannot be solved in ordinary multiple polylogarithms by rationalising this root.
  • The Drell–Yan K3 surface is neither birationally equivalent nor isogenous to the K3 surface of the two-loop Bhabha-scattering correction, nor to the deformations of the latter studied there, because the geometric Picard ranks differ (19 versus 20).
  • For every prime $p \ge 7$, the number of $\mathbb{F}_p$-points of the Drell–Yan K3 is given by the closed formula $|S(\mathbb{F}_p)| = 1 + 17p + (1+(5/p))p + \mu(p) + p^2$ with $\mu(p) = a_p(f)^2 - (10/p)p$, where $f$ is the level-160 weight-2 newform; the formula over $\mathbb{F}_{p^2}$ involves the Frobenius trace of $E_1$.
  • The three explicit elliptic fibrations put the Drell–Yan integrals within reach of elliptic multiple polylogarithms, the function class recently used for the two-loop Bhabha master integrals.
  • The Shioda–Inose structure yields an effective algorithm for listing primes of supersingular reduction; the paper lists all such primes below $104729$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The conditional step in Section 5.1 ('Assume that $S$ admits a Shioda–Inose structure') can be turned into a checkable statement by verifying directly that the desingularised quotient of $S$ by the explicit Nikulin involution of Remark 4.14 is a Kummer surface; the modular-form and point-count story would then be unconditional.
  • The same pipeline—resolve the singular surface, find $(-2)$-curves, search for Kodaira fibres, pull back a fibration, and identify the Kummer partner—should transfer to other two-loop amplitudes whose square roots put K3 geometries into play.
  • If the modular relation is correct, the Drell–Yan master integrals inherit the arithmetic of the symmetric square of the level-160 newform; one testable consequence is that certain combinations of the integrals should have periods lying in the field generated by the eigenvalues of that newform.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the surface XDY defined by w^2 = 4xy^2z(x-z)^2 + (x+y)^2(xy+z^2)^2 in the weighted projective space P(1,1,1,3), a hypersurface associated with a square root appearing in two-loop mixed electroweak-QCD corrections to Drell-Yan scattering. The main mathematical claims are that the desingularization S is a K3 surface with geometric Picard lattice of rank 19, discriminant 24, and discriminant group Z/2Z x Z/2Z x Z/6Z; that the original square root cannot be rationalized by a rational change of variables; that S is neither birational nor isogenous to the Bhabha-scattering K3 surface; and that S admits an explicit Shioda-Inose structure related to a classical modular form of weight 2 and level 160. The Picard-lattice part is supported by a van Luijk upper bound, by an explicit sublattice computation, and by an independent elliptic-fibration computation with Magma code supplied. The Shioda-Inose and modular-form part is, however, not proven as written and contains an internal numerical inconsistency.

Significance. If the Picard-lattice theorems stand, the paper gives a clean, well-documented proof that the Drell-Yan square root is non-rationalizable and that the Drell-Yan K3 surface is distinct from the Bhabha K3 surface; the two independent lattice computations and the accompanying executable code are concrete strengths. The claimed Shioda-Inose structure and the modular-form description of point counts would be a substantial addition, but in the current form that part is conditional and, in one displayed formula, incompatible with the paper's own reduction data. The significance of the arithmetic/motivic conclusions is therefore conditional on a substantial repair of Section 5.

major comments (3)
  1. [Section 5.1 and Theorem 5.11] The claimed Shioda-Inose structure is assumed rather than proved. After constructing four candidate abelian surfaces A(P_i), the text says "Assume that S admits a Shioda-Inose structure with the abelian variety A(P_j) for some j," and the proof of Theorem 5.11 later uses the phrase "From the existence of the Shioda-Inose structure on S" to conclude the point-count formulas. This is circular: the existence of the Nikulin involution and the identification of the quotient S/<iota> with the Kummer surface of E_1 x E_2 is exactly what needs to be established. Remark 4.14 asserts that there is a Nikulin involution swapping the two II* fibres and that the quotient is a Kummer surface, but the remark gives no proof, only an explicit-looking formula and a reference to ancillary code. Since Theorem 5.11, Corollary 5.12, and the modular-form relation all depend on this quotient-Kummer step, the arithmetic conclusions of Section 5 are unsupported unless this step is proved in the text.
  2. [Section 5.1, Eq. (5.3) and Theorem 5.11] The numerical values in (5.3) are impossible for the Hecke eigenvalues of a classical weight-2 newform, and the resulting point-count formula contradicts Proposition 3.5. Corollary 5.10 states a_31(f) = -16 and a_71(f) = 48 for a classical newform f of weight 2 and level 160; Deligne's bound gives |a_p(f)| <= 2 sqrt(p), so |a_31(f)| <= 11.14 and |a_71(f)| <= 16.85. Thus those two values cannot be Hecke eigenvalues. Moreover, inserting a_31(f) = -16 into Theorem 5.11 gives mu(31) = 256 - 31 = 225 and therefore #S(F_31) = 1 + 17*31 + 2*31 + 225 + 31^2 = 1776, i.e., a Frobenius trace on H^2 equal to 814. But Proposition 3.5 computed rho(S_31) = 20, so 20 of the 22 Frobenius eigenvalues on H^2 are equal to +/-31 and the remaining two also have absolute value 31; the maximum possible total trace is 22*31 = 682. The value 814 is therefore impossible. The formula in Theorem 5.11 and the coefficient list (5.3) are internally inconsistent as written; the relation between Hecke eigenvalues, traces on the four-dimensional abelian variety, and the trace mu(p) needs to be stated correctly and recomputed.
  3. [Section 5.2 and Corollary 5.12] The supersingular-reduction results inherit the unresolved Shioda-Inose assumption. Corollary 5.12 asserts that the Picard rank over F_p is 22 exactly when the reductions of E_1 and E_2 are supersingular, and that there are infinitely many such primes; the proof invokes the Shioda-Inose structure through the rank formula rho(S_{F_p}) = 18 + rank Hom(tilde E_1, tilde E_2). Since the existence of the Shioda-Inose structure and the identification of the Kummer quotient are not established, the supersingular-reduction claims and the accompanying prime list are not justified by the present argument.
minor comments (4)
  1. [Theorem 4.7] The line "n <= 4 <= 10" appears to be a typo for "4 <= n <= 10".
  2. [Propositions 4.10, 4.12, and Theorem 5.11] The symbol E2 is used both for the generic fibre of the second elliptic fibration and, in Theorem 5.11, for an elliptic curve in the Kummer construction. This notational collision makes Section 5 unnecessarily hard to read.
  3. [Section 2.4 and Corollary 2.1] The proof of Corollary 2.1 invokes the Enriques-Kodaira classification to conclude that a K3-birational surface is not unirational. This is correct in characteristic zero, but the surface XDY is singular; it would help to state explicitly that birationality to a smooth K3 surface rules out rationality of the original model.
  4. [Section 3.2, Proposition 3.5] The discriminant comparison is sound, but the sentence "which is impossible, as the discriminants ... are not equivalent up to squares and therefore cannot be equal" is slightly compressed: the equality of lattices would imply equality of their discriminants as integers, not merely up to squares. The argument works because the displayed reductions imply different square classes, but the wording could be clarified.

Circularity Check

1 steps flagged · score 4.0 of 10

The Shioda–Inose existence is assumed rather than derived; the point-count formula in Theorem 5.11 is conditional on that assumption, while the Picard-lattice and non-rationalisability claims remain independent.

  1. other [Section 5.1 (p. 24) and proof of Theorem 5.11 (p. 28)]
    "Assume that S admits a Shioda-Inose structure with the abelian variety A(Pj) for some j∈{1,2,3,4}. — From the existence of the Shioda–Inose structure on S we know that the structure is determined by two elliptic curves Ea,b and Ec,d."

    The theorem whose content includes the existence of a Shioda–Inose structure is proved by feeding that same existence back in as a hypothesis. All the arithmetic conclusions of Theorem 5.11, including |S(Fp)| = 1 + 17p + (1+(5/p))p + μ(p) + p^2 with μ(p)=a_p(f)^2−ε(p)p, are consequently conditional on the assumed structure rather than derived from the explicit divisors or fibrations of S. This is not a fit-to-data reduction, but it is a circular proof step for the Shioda–Inose and modular-form claims.

full rationale

The core lattice geometry is self-contained: the Picard lattice rank, discriminant and discriminant group are computed from explicit divisors with a van Luijk upper bound at p=31 and 71 in Section 3, and recomputed from an explicit elliptic fibration via Shioda–Tate and height pairings in Section 6. The modular form f is taken from LMFDB, not fitted to the surface point counts. The non-rationalisability corollary depends only on the independent K3 birationality result. The only circularity-adjacent step is in the Shioda–Inose part: Remark 4.14 states without a complete proof that the quotient by the Nikulin involution is a Kummer surface, and Section 5.1 explicitly says 'Assume that S admits a Shioda-Inose structure' before the proof of Theorem 5.11 invokes that same existence. Thus the point-count formula of Theorem 5.11 is conditional rather than a fully derived prediction. Separate numerical concerns about the values in (5.3) relative to the Deligne bound are correctness issues, not additional circularity. The Picard-lattice and birationality claims stand on independent evidence, so the overall circularity is only partial.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

Central lattice claims do not introduce free parameters; the surface and divisors are explicit over Q or Q(sqrt(5)). The main external assumptions are standard theorems from K3 surface theory, elliptic surface theory, and modularity, plus trust in Magma. The Shioda-Inose portion introduces auxiliary elliptic curves E1 and E2 and a Hilbert modular form; these are not new physical entities and are identified by explicit equations and LMFDB labels, but the step asserting the quotient is a Kummer surface is the least supported axiom.

assumptions (6)
  • standard math Weil and Tate conjectures for K3 surfaces over finite fields of characteristic p >= 5
    Used in Proposition 3.5 to derive the Picard rank upper bound from Weil polynomials modulo 31 and 71.
  • standard math Enriques-Kodaira classification of surfaces and equivalence of unirationality and rationality over characteristic 0
    Used in Corollary 2.1 and Corollary 3.4 to conclude the square root is not rationalizable from non-rationality of the K3 surface.
  • standard math Correspondence between genus-1 fibrations and nef primitive isotropic divisor classes (Piatetski-Shapiro-Shafarevich)
    Used in Theorem 4.2 and Lemma 4.4 to justify the elliptic-fibration construction method.
  • domain assumption Nikulin and Morrison theory of Shioda-Inose structures, including that the quotient by a Nikulin involution is a K3 surface and that two II* fibres can produce a Kummer quotient
    Section 5 and Remark 4.14 use this without fully proving the quotient-Kummer statement for this specific surface; the text even says "Assume that S admits a Shioda-Inose structure..." in Section 5.1.
  • standard math Modularity of elliptic curves over totally real fields and LMFDB identification of Hilbert newform 4.4.1600.1-256.1-i and classical newform 160.2.f.a
    Used in Lemma 5.7 and Corollary 5.10 to attach the modular form and derive the trace formulas.
  • domain assumption Correctness of Magma computations in the accompanying code
    Intersection matrices, Riemann-Roch computations, Weil polynomials, and isogeny checks are delegated to Magma; no independent formal verification is provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Arithmetic and geometry of a K3 surface emerging from virtual corrections to Drell--Yan scattering." pith.science (2026). https://pith.science/paper/3LGSRK7Z

@misc{pith2026190801079,
  author       = {Pith},
  title        = {Pith review of: Arithmetic and geometry of a K3 surface emerging from virtual corrections to Drell--Yan scattering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3LGSRK7Z}},
  note         = {Machine review of arXiv:1908.01079}
}
read the original abstract

We study a K3 surface, which appears in the two-loop mixed electroweak-quantum chromodynamic virtual corrections to Drell--Yan scattering. A detailed analysis of the geometric Picard lattice is presented, computing its rank and discriminant in two independent ways: first using explicit divisors on the surface and then using an explicit elliptic fibration. We also study in detail the elliptic fibrations of the surface and use them to provide an explicit Shioda--Inose structure. Moreover, we point out the physical relevance of our results.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Special Fano geometry from Feynman integrals

    hep-th 2024-12 conditional novelty 5.0 of 10

    Special Fano varieties, which include Calabi-Yau spaces as the Q=1 case, arise from the Symanzik polynomials of several families of Feynman integrals.

Reference graph

Works this paper leans on

93 extracted references · 77 canonical work pages · cited by 1 Pith paper

  1. [1]

    Altarelli, R

    G. Altarelli, R. K. Ellis, M. Greco, and G. Martinelli. Vector Boson Production at Colliders: A Theoretical Reappraisal.Nucl. Phys., B246:12–44, 1984. 5 – 33 –

    1984

  2. [2]

    Altarelli, R

    G. Altarelli, R. K. Ellis, and G. Martinelli. Large Perturbative Corrections to the Drell-Yan Process in QCD.Nucl. Phys., B157:461–497, 1979. 5

    1979

  3. [3]

    Artin and H

    M. Artin and H. P. F. Swinnerton-Dyer. The Shafarevich-Tate conjecture for pencils of elliptic curves onK3 surfaces. Invent. Math., 20:249–266, 1973. 10

    1973

  4. [4]

    Barbieri, J

    R. Barbieri, J. A. Mignaco, and E. Remiddi. Electron form-factors up to fourth order. - I. Nuovo Cim., A11:824–864, 1972. 6

    1972

  5. [5]

    W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven.Compact complex surfaces. Springer-Verlag, Berlin, second edition, 2004. 9, 10, 16

    2004

  6. [6]

    C. W. Bauer, A. Frink, and R. Kreckel. Introduction to theGiNaC framework for symbolic computation within theC++ programming language. J. Symb. Comput., 33:1, 2002. 2

    2002

  7. [7]

    U. Baur, S. Keller, and W. K. Sakumoto. QED radiative corrections toZ boson production and the forward backward asymmetry at hadron colliders.Phys. Rev., D57:199–215, 1998. 5

    1998

  8. [8]

    Z. Bern, L. J. Dixon, and D. A. Kosower. Dimensionally regulated pentagon integrals. Nucl.Phys., B412:751–816, 1994. 5

    1994

Show all 93 references
  1. [9]

    M. J. Bertin, A. Garbagnati, R. Hortsch, O. Lecacheux, M. Mase, C. Salgado, and U. Whitcher. Classifications of elliptic fibrations of a singular K3 surface. InWomen in numbers Europe, volume 2 ofAssoc. Women Math. Ser., pages 17–49. Springer, Cham, 2015. 15

    2015

  2. [10]

    Besier, D

    M. Besier, D. Festi, M. Harrison, and B. Naskręcki. Accompanying MAGMA code. https://bnaskrecki.faculty.wmi.amu.edu.pl/doku.php/drell_yan_k3s, 2019. 3, 12, 13, 17, 19, 20, 21, 22

    2019

  3. [11]

    Besier, D

    M. Besier, D. van Straten, and S. Weinzierl. Rationalizing roots: an algorithmic approach. Commun. Num. Theor. Phys., 13:253–297, 2019. 2, 7

    2019

  4. [12]

    Blümlein, A

    J. Blümlein, A. De Freitas, C. G. Raab, and K. Schönwald. TheO(α2) Initial State QED Corrections toe+e− Annihilation to a Neutral Vector Boson Revisited.Phys. Lett., B791:206–209, 2019. 5

    2019

  5. [13]

    Bonciani, S

    R. Bonciani, S. Di Vita, P. Mastrolia, and U. Schubert. Two-Loop Master Integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering.JHEP, 09:091, 2016. 5, 7

    2016

  6. [14]

    Bosma, M

    J. Bosma, M. Sogaard, and Y. Zhang. Maximal Cuts in Arbitrary Dimension.JHEP, 08:051,

  7. [15]

    Bosma, J

    W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993). 3

    1997

  8. [16]

    J. L. Bourjaily, Y.-H. He, A. J. Mcleod, M. von Hippel, and M. Wilhelm. Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms.Phys. Rev. Lett., 121(7):071603, 2018. 9

    2018

  9. [17]

    J. L. Bourjaily, A. J. McLeod, M. von Hippel, and M. Wilhelm. Rationalizing Loop Integration. JHEP, 08:184, 2018. 2, 7

    2018

  10. [18]

    J. L. Bourjaily, A. J. McLeod, M. von Hippel, and M. Wilhelm. Bounded Collection of Feynman Integral Calabi-Yau Geometries.Phys. Rev. Lett., 122(3):031601, 2019. 9

    2019

  11. [19]

    A. P. Braun, Y. Kimura, and T. Watari. On the Classification of Elliptic Fibrations modulo Isomorphism on K3 Surfaces with large Picard Number. arXiv:1312.4421, 2013. 15, 16 – 34 –

    2013 arXiv

  12. [20]

    Broedel, C

    J. Broedel, C. Duhr, F. Dulat, and L. Tancredi. Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism.JHEP, 05:093, 2018. 7

    2018

  13. [21]

    F. Brown. On the periods of some Feynman integrals. arXiv:0910.0114, 2009. 9

    2009 arXiv

  14. [22]

    Brown and A

    F. Brown and A. Levin. Multiple Elliptic Polylogarithms. arXiv:1110.6917, 2011. 7

    2011 arXiv

  15. [23]

    Brown and O

    F. Brown and O. Schnetz. A K3 inφ4. Duke Math. J., 161(10):1817–1862, 2012. 9

    2012

  16. [24]

    Caron-Huot, L

    S. Caron-Huot, L. J. Dixon, M. von Hippel, A. J. McLeod, and G. Papathanasiou. The Double Pentaladder Integral to All Orders.JHEP, 07:170, 2018. 7

    2018

  17. [25]

    F. Charles. The Tate conjecture forK3 surfaces over finite fields.Invent. Math., 194(1):119–145, 2013. 10

    2013

  18. [26]

    Chaubey and S

    E. Chaubey and S. Weinzierl. Two-loop master integrals for the mixed QCD-electroweak corrections forH→b¯b through aHt¯t-coupling. JHEP, 05:185, 2019. 2

    2019

  19. [27]

    Chicherin, T

    D. Chicherin, T. Gehrmann, J. M. Henn, P. Wasser, Y. Zhang, and S. Zoia. All master integrals for three-jet production at NNLO.Phys. Rev. Lett., 123:041603, 2019. 6

    2019

  20. [28]

    Cornell and J

    G. Cornell and J. H. Silverman.Arithmetic Geometry. Springer, 1986. 15

    1986

  21. [29]

    P. Deligne. La conjecture de Weil pour les surfacesK3. Invent. Math., 15:206–226, 1972. 10

    1972

  22. [30]

    Derickx, F

    M. Derickx, F. Najman, and S. Siksek. Elliptic curves over totally real cubic fields are modular, 2019. 27

    2019

  23. [31]

    Dolgachev

    I. Dolgachev. Weighted projective varieties. InGroup actions and vector fields (Vancouver, B.C., 1981), volume 956 ofLecture Notes in Math., pages 34–71. Springer, Berlin, 1982. 2

    1981

  24. [32]

    S. D. Drell and T.-M. Yan. Massive Lepton Pair Production in Hadron-Hadron Collisions at High-Energies. Phys. Rev. Lett., 25:316–320, 1970. [Erratum: Phys. Rev. Lett.25,902(1970)]. 4

    1970

  25. [33]

    N. D. Elkies. Supersingular primes for elliptic curves over real number fields.Compositio Math., 72(2):165–172, 1989. 29

    1989

  26. [34]

    Elsenhans and J

    A.-S. Elsenhans and J. Jahnel. The Picard group of aK3 surface and its reduction modulo p. Algebra Number Theory, 5(8):1027–1040, 2011. 11

    2011

  27. [35]

    D. Festi. A practical algorithm to compute the geometric Picard lattice of K3 surfaces of degree 2. arXiv:1808.00351, 2018. 10

    2018 arXiv

  28. [36]

    Festi and D

    D. Festi and D. van Straten. Bhabha Scattering and a special pencil of K3 surfaces. Commun. Num. Theor. Phys., 13(2), 2019. 3, 7, 8

    2019

  29. [37]

    L. R. A. Finotti. A formula for the supersingular polynomial.Acta Arith., 139(3):265–273,

  30. [38]

    Freitas, B

    N. Freitas, B. V. Le Hung, and S. Siksek. Elliptic curves over real quadratic fields are modular. Invent. Math., 201(1):159–206, 2015. 27

    2015

  31. [39]

    Frellesvig and C

    H. Frellesvig and C. G. Papadopoulos. Cuts of Feynman Integrals in Baikov representation. JHEP, 04:083, 2017. 6

    2017

  32. [40]

    Gehrmann, J

    T. Gehrmann, J. M. Henn, and N. A. Lo Presti. Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD.Phys. Rev. Lett., 116(6):062001, 2016. [Erratum: Phys. Rev. Lett.116,no.18,189903(2016)]. 2, 7 – 35 –

    2016

  33. [41]

    Gehrmann and E

    T. Gehrmann and E. Remiddi. Differential equations for two loop four point functions. Nucl.Phys., B580:485–518, 2000. 5

    2000

  34. [42]

    González, J.-C

    J. González, J.-C. Lario, and J. Quer. Arithmetic ofQ-curves. InModular curves and abelian varieties, volume 224 ofProgr. Math., pages 125–139. Birkhäuser, Basel, 2004. 27

    2004

  35. [43]

    Hamberg, W

    R. Hamberg, W. L. van Neerven, and T. Matsuura. A complete calculation of the order α−s2 correction to the Drell-YanK factor. Nucl. Phys., B359:343–405, 1991. [Erratum: Nucl. Phys.B644,403(2002)]. 5

    2002

  36. [44]

    Harley, F

    M. Harley, F. Moriello, and R. M. Schabinger. Baikov-Lee Representations Of Cut Feynman Integrals. JHEP, 06:049, 2017. 6

    2017

  37. [45]

    Hartshorne

    R. Hartshorne. Algebraic geometry. Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. 8, 15, 28

    1977

  38. [46]

    Heller, A

    M. Heller, A. von Manteuffel, and R. M. Schabinger. Multiple polylogarithms with algebraic arguments and the two-loop EW-QCD Drell-Yan master integrals. arXiv:1907.00491, 2019. 2, 3, 5

    1907 arXiv

  39. [47]

    J. M. Henn. Multiloop integrals in dimensional regularization made simple.Phys. Rev. Lett., 110:251601, 2013. 6

    2013

  40. [48]

    J. M. Henn and V. A. Smirnov. Analytic results for two-loop master integrals for Bhabha scattering I. JHEP, 1311:041, 2013. 7

    2013

  41. [49]

    A. Hodges. Eliminating spurious poles from gauge-theoretic amplitudes.JHEP, 05:135, 2013. 7

    2013

  42. [50]

    Huybrechts

    D. Huybrechts. Complex geometry. Springer, 2005. 31

    2005

  43. [51]

    Huybrechts

    D. Huybrechts. Lectures on K3 surfaces, volume 158 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, 2016. 3, 10, 22

    2016

  44. [52]

    Kloosterman

    R. Kloosterman. EllipticK3 surfaces with geometric Mordell-Weil rank 15.Canad. Math. Bull., 50(2):215–226, 2007. 10

    2007

  45. [53]

    A. V. Kotikov. Differential equations method: New technique for massive Feynman diagrams calculation. Phys. Lett., B254:158–164, 1991. 5

    1991

  46. [54]

    A. V. Kotikov. The Property of maximal transcendentality in theN = 4 Supersymmetric Yang-Mills. InDiakonov, D. (ed.): Subtleties in quantum field theory, pages 150–174, 2010. 6

    2010

  47. [55]

    Kumar and M

    A. Kumar and M. Kuwata. Elliptic K3 surfaces associated with the product of two elliptic curves: Mordell-Weil lattices and their fields of definition.Nagoya Math. J., 228:124–185,

  48. [56]

    Lang and H

    S. Lang and H. Trotter.Frobenius distributions inGL2-extensions. Lecture Notes in Mathematics, Vol. 504. Springer-Verlag, Berlin-New York, 1976. Distribution of Frobenius automorphisms in GL2-extensions of the rational numbers. 29

    1976

  49. [57]

    R. P. Langlands.Base change forGL(2), volume 96 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. 27

    1980

  50. [58]

    Laporta and E

    S. Laporta and E. Remiddi. Analytic treatment of the two loop equal mass sunrise graph. Nucl. Phys., B704:349–386, 2005. 7

    2005

  51. [59]

    R. N. Lee and K. T. Mingulov. DREAM, a program for arbitrary-precision computation of dimensional recurrence relations solutions, and its applications. arXiv:1712.05173, 2017. 6 – 36 –

    2017 arXiv

  52. [60]

    R. N. Lee and V. A. Smirnov. The Dimensional Recurrence and Analyticity Method for Multicomponent Master Integrals: Using Unitarity Cuts to Construct Homogeneous Solutions. JHEP, 12:104, 2012. 6

    2012

  53. [61]

    The L-functions and modular forms database

    The LMFDB Collaboration. The L-functions and modular forms database. http://www.lmfdb.org, 2019. [Online; accessed 18 December 2019]. 25, 27

    2019

  54. [62]

    Matsuura, S

    T. Matsuura, S. C. van der Marck, and W. L. van Neerven. The Calculation of the Second Order Soft and Virtual Contributions to the Drell-Yan Cross-Section.Nucl. Phys., B319:570–622, 1989. 5

    1989

  55. [63]

    D. R. Morrison. OnK3 surfaces with large Picard number.Invent. Math., 75(1):105–121,

  56. [64]

    Naskręcki

    B. Naskręcki. Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples. Acta Arith., 160(2):159–183, 2013. 31

    2013

  57. [65]

    Naskręcki

    B. Naskręcki. Distribution of Mordell-Weil ranks of families of elliptic curves. InAlgebra, logic and number theory, volume 108 ofBanach Center Publ., pages 201–229. Polish Acad. Sci. Inst. Math., Warsaw, 2016. 31

    2016

  58. [66]

    Naskręcki

    B. Naskręcki. On a certain hypergeometric motive of weight 2 and rank 3. arXiv:1702.07738,

    arXiv

  59. [67]

    V. V. Nikulin. Finite groups of automorphisms of Kählerian surfaces of typeK3. Uspehi Mat. Nauk, 31(2(188)):223–224, 1976. 23

    1976

  60. [68]

    V. V. Nikulin. Elliptic fibrations on K3 surfaces.Proc. Edinburgh Math. Soc., 57(1):253–267,

  61. [69]

    Nishiyama

    K. Nishiyama. The Jacobian fibrations on someK3 surfaces and their Mordell-Weil groups. Japan. J. Math. (N.S.), 22(2):293–347, 1996. 15

    1996

  62. [70]

    Nishiyama

    K. Nishiyama. A remark on Jacobian fibrations onK3 surfaces. Saitama Math. J., 15:67–71,

  63. [71]

    K. Oguiso. On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves. J. Math. Soc. Japan, 41(4):651–680, 1989. 15

    1989

  64. [72]

    B. Poonen. Rational points on varieties, volume 186 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2017. 14

    2017

  65. [73]

    Primo and L

    A. Primo and L. Tancredi. On the maximal cut of Feynman integrals and the solution of their differential equations.Nucl. Phys., B916:94–116, 2017. 6

    2017

  66. [74]

    I. I. Pyatetskii-Shapiro and I. R. Shafarevich. A Torelli theorem for algebraic surfaces of type K3. Math. USSR Izv., 5(3):547–588, 1971. 14, 15

    1971

  67. [75]

    E. Remiddi. Differential equations for Feynman graph amplitudes.Nuovo Cim., A110:1435–1452, 1997. 5

    1997

  68. [76]

    Saint-Donat

    B. Saint-Donat. Projective Models of K3 Surfaces.Am. J. Math., 96(4):602–639, 1974. 15

    1974

  69. [77]

    A. J. Scholl. Motives for modular forms.Invent. Math., 100(2):419–430, 1990. 29

    1990

  70. [78]

    M. Schütt. Two lectures on the arithmetic of K3 surfaces. InArithmetic and geometry of K3 surfaces and Calabi-Yau threefolds, volume 67 ofFields Inst. Commun., pages 71–99. Springer, New York, 2013. 8 – 37 –

    2013

  71. [79]

    Schütt and T

    M. Schütt and T. Shioda. Elliptic surfaces. InAlgebraic geometry in East Asia—Seoul 2008, volume 60 ofAdv. Stud. Pure Math., pages 51–160. Math. Soc. Japan, Tokyo, 2010. 32, 33

    2008

  72. [80]

    Schütt and T

    M. Schütt and T. Shioda.Mordell-Weil lattices. Springer, Singapore, 2019. 29, 33

    2019

  73. [81]

    I. Shimada. Transcendental lattices and supersingular reduction lattices of a singularK3 surface. Trans. Amer. Math. Soc., 361(2):909–949, 2009. 29

    2009

  74. [82]

    T. Shioda. On the Mordell-Weil lattices.Comment. Math. Univ. St. Paul., 39(2):211–240,

  75. [83]

    Shioda and H

    T. Shioda and H. Inose. On singular K3 surfaces. InComplex Analysis and Algebraic Geometry: A Collection of Papers Dedicated to K. Kodaira, pages 119–136. Cambridge University Press, 1977. 15, 16

    1977

  76. [84]

    J. H. Silverman.The Arithmetic of Elliptic Curves. Springer, 2nd edition, 1986. 30

    1986

  77. [85]

    J. H. Silverman.Advanced topics in the arithmetic of elliptic curves, volume 151 ofGraduate Texts in Mathematics. Springer, 1994. 32

    1994

  78. [86]

    Tancredi

    L. Tancredi. Feynman integrals and higher-genus surfaces. Talk at Amplitudes Conference,

  79. [87]

    J. Tate. Algorithm for determining the type of a singular fiber in an elliptic pencil.Lecture Notes in Math., Vol. 476, pages 33–52, 1975. 31, 32

    1975

  80. [88]

    van Luijk

    R. van Luijk. K3 surfaces with Picard number one and infinitely many rational points. Algebra Number Theory, 1(1):1–15, 2007. 10, 31

    2007

  81. [89]

    Voisin.Hodge theory and complex algebraic geometry

    C. Voisin.Hodge theory and complex algebraic geometry. I, volume 76 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002. Translated from the French original by Leila Schneps. 23

    2002

  82. [90]

    Vollinga and S

    J. Vollinga and S. Weinzierl. Numerical evaluation of multiple polylogarithms.Comput. Phys. Commun., 167:177, 2005. 2

    2005

  83. [91]

    von Manteuffel and R

    A. von Manteuffel and R. M. Schabinger. Numerical Multi-Loop Calculations via Finite Integrals and One-Mass EW-QCD Drell-Yan Master Integrals.JHEP, 04:129, 2017. 5

    2017

  84. [92]

    von Manteuffel and L

    A. von Manteuffel and L. Tancredi. A non-planar two-loop three-point function beyond multiple polylogarithms. JHEP, 06:127, 2017. 2

    2017

  85. [93]

    Wackeroth and W

    D. Wackeroth and W. Hollik. Electroweak radiative corrections to resonant charged gauge boson production. Phys. Rev., D55:6788–6818, 1997. 5 – 38 –

    1997

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.