Abstract: We construct regular integral canonical models for Shimura varieties attached to Spin and orthogonal groups at (possibly ramified) primes . Work of Vasiu–Zink then shows that the classical Kuga–Satake construction extends over the integral models and that the integral models we construct are canonical in a very precise sense. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla’s program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms.

Abstract: We prove the KKV conjecture expressing Gromov–Witten invariants of surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov–Witten/Pairs correspondence for -fibered hypersurfaces of dimension 3 to reduce the KKV conjecture to statements about stable pairs on (thickenings of) surfaces. Using degeneration arguments and new multiple cover results for stable pairs, we reduce the KKV conjecture further to the known primitive cases. Our results yield a new proof of the full Yau–Zaslow formula, establish new Gromov–Witten multiple cover formulas, and express the fiberwise Gromov–Witten partition functions of -fibered 3-folds in terms of explicit modular forms.

Abstract: This work lies at an intersection of three subjects: quantum field theory, algebraic geometry and number theory, in a situation where dialogue between practitioners has revealed rich structure. It contains a theorem and 7 conjectures, tested deeply by 3 optimized algorithms, on relations between Feynman integrals and L-series defined by products, over the primes, of data determined by moments of Kloosterman sums in finite fields. There is an extended introduction, for readers who may not be familiar with all three of these subjects. Notable new results include conjectural evaluations of non-critical L-series of modular forms of weights 3, 4 and 6, by determinants of Feynman integrals, an evaluation for the weight 5 problem, at a critical integer, and formulas for determinants of arbitrary size, tested up to 30 loops. It is shown that the functional equation for Kloosterman moments determines much but not all of the structure of the L-series. In particular, for problems with odd numbers of Bessel functions, it misses a crucial feature captured in this work by novel and intensively tested conjectures. For the 9-Bessel problem, these lead to an astounding compression of data at the primes.

Abstract: We describe the construction of a database of genus 2 curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated L-function. This data has been incorporated into the L-Functions and Modular Forms Database (LMFDB).

Abstract: We show that the Euler system associated with Rankin–Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in p-adic Coleman families. We prove an explicit reciprocity law for these families and use this to prove cases of the Bloch–Kato conjecture for Rankin–Selberg convolutions.

Abstract: The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters on the one hand and its interpretation of modules as objects in a modular tensor category on the other one. Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and $C_2$-cofinite vertex operator algebras. We suggest that their modular pillar are trace functions with insertions corresponding to intertwiners of the projective cover of the vacuum, and that the categorical pillar are finite tensor categories $\mathcal C$ which are ribbon and whose double is isomorphic to the Deligne product $\mathcal C\otimes \mathcal C^{opp}$. Overarching these pillars is then a logarithmic variant of Verlinde's formula. Numerical data realizing this are the modular $S$-matrix and modified traces of open Hopf links. The representation categories of $C_2$-cofinite and logarithmic conformal field theories that are fairly well understood are those of the $\mathcal W_p$-triplet algebras and the symplectic fermions. We illustrate the ideas in these examples and especially make the relation between logarithmic Hopf links and modular transformations explicit.

Abstract: Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a $$ \mathbb{C}{P}^2 $$ space. We show that for the simplest integrand, namely the n − gon, our proposal indeed reproduces the expected result. By using the recently formulated Λ−algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.

Abstract: We describe the construction of a database of genus- curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated -function. This data has been incorporated into the -Functions and Modular Forms Database (LMFDB).

Abstract: This paper investigates the relations between modular graph forms, which are generalizations of the modular graph functions that were introduced in earlier papers motivated by the structure of the low energy expansion of genus-one Type II superstring amplitudes. These modular graph forms are multiple sums associated with decorated Feynman graphs on the world-sheet torus. The action of standard differential operators on these modular graph forms admits an algebraic representation on the decorations. First order differential operators are used to map general non-holomorphic modular graph functions to holomorphic modular forms. This map is used to provide proofs of the identities between modular graph functions for weight less than six conjectured in earlier work, by mapping these identities to relations between holomorphic modular forms which are proven by holomorphic methods. The map is further used to exhibit the structure of identities at arbitrary weight.

Abstract: 3 Overconvergent modular forms for the group G∗ 7 3.1 Hilbert modular varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 The canonical subgroup theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.1 Strict neighborhoods of the ordinary locus . . . . . . . . . . . . . . . . . 8 3.2.2 The canonical subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 The sheaf F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 The modular sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.5 Families of modular sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.6 The specialization map for cusp forms . . . . . . . . . . . . . . . . . . . . . . . 14 3.7 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Abstract: We prove an explicit version of Weiss' bound on the least norm of a prime ideal in the Chebotarev density theorem, which is itself a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. In order to accomplish this, we prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon for Hecke $L$-functions. As an application, we prove the first explicit nontrivial upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also present applications to the group of $\mathbb{F}_p$-rational points of an elliptic curve and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.

Abstract: The general idea behind the proof of this theorem is as follows. Consider the congruence (1. 2) as an example. By applying a certain linear operatordenoted by U5-to the right member of (1. 1) one obtains the series with v replaced by 5v. The same operation performed on the left member yields a modular function belonging to a subgroup of the full modular group. We shall try, therefore, to express that modular function in terms of the basic function of the subgroup, which is known to have integral coefficient Fourier expansions. The identity which results is of the form

Abstract: We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail the double-well potential and the cubic and quartic oscillators, and we calculate the WKB expansion of their quantum free energies by using the direct integration of the anomaly equations. We reproduce in this way all known results about the quantum periods of these models, which we express in terms of modular forms on the WKB curve. As an application of our results, we study the large order behavior of the WKB expansion in the case of the double well, which displays the double factorial growth typical of string theory.

Abstract: We describe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight one-half that appears in work of Borcherds and Zagier on Borcherds products and traces of singular moduli. We conjecture the existence of an infinite dimensional graded module for the Thompson group and provide evidence for our conjecture by constructing McKay–Thompson series for each conjugacy class of the Thompson group that coincide with weight one-half modular forms of higher level. We also observe a discriminant property in this moonshine for the Thompson group that is closely related to the discriminant property conjectured to exist in Umbral Moonshine.

Abstract: Given an odd, semisimple, reducible, 2-dimensional mod l Galois representation, we investigate the possible levels of the modular forms giving rise to it. When the representation is the direct sum of the trivial character and a power of the mod l cyclotomic character, we are able to characterize the primes that can arise as levels of the associated newforms. As an application, we determine a new explicit lower bound for the highest degree among the fields of coefficients of newforms of trivial Nebentypus and prime level. The bound is valid in a subset of the primes with natural (lower) density at least 3/4.

Abstract: We study N = 2 superconformal theories with gauge group SU(N) and 2N fundamental flavours in a locus of the Coulomb branch with a ZN symmetry. In this special vacuum, we calculate the prepotential, the dual periods and the period matrix using equivariant localization. In the conformal limit, we find that the period matrix is completely specified by [ N 2 ] effective couplings. On each of these, we show that the Sduality group acts as a generalized triangle group and that its hauptmodul can be used to write a non-perturbatively exact relation between each effective coupling and the bare one. For N = 2, 3, 4 and 6, the generalized triangle group is an arithmetic Hecke group which contains a subgroup that is also a congruence subgroup of the modular group PSL(2,Z). For these cases, we introduce mass deformations that respect the symmetries of the special vacuum and show that the constraints arising from S-duality make it possible to resum the instanton contributions to the period matrix in terms of meromorphic modular forms which solve modular anomaly equations.

Abstract: This article can be read as a companion and sequel to the authors’ earlier article on Stark points and p-adic iterated integrals attached to modular forms of weight one, which proposes a conjectural expression for the so-called p-adic iterated integrals attached to a triple (f, g, h) of classical eigenforms of weights (2, 1, 1). When f is a cusp form, this expression involves the p-adic logarithms of so-called Stark points: distinguished points on the modular abelian variety attached to f, defined over the number field cut out by the Artin representations attached to g and h. The goal of this paper is to formulate an analogous conjecture when f is a weight two Eisenstein series rather than a cusp form. The resulting formula involves the p-adic logarithms of units and p-units in suitable number fields, and can be seen as a new variant of Gross’s p-adic analogue of Stark’s conjecture on Artin L-series at \(s=0\).

Abstract: We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet $L$-functions.

Abstract: The two Rogers-Ramanujan q-series where σ 0; 1, play many roles in mathematics and physics. By the Rogers- Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers- Ramanujan identities to doubly infinite families of q-series identities. If a ∈ (1, 2) and m, n ≥ 1, then we have [infinite product modular function], where the P λ(x, x, ...; q) are Hall-Littlewood polynomials. These q-series are specialized characters of affine Kac-Moody algebras. Generalizing the Rogers- Ramanujan continued fraction, we prove in the case of A that the relevant q-series quotients are integral units.

Abstract: This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic Zp-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson-Flach elements.

Abstract: We study N = 2 superconformal theories with gauge group SU(N) and 2N fundamental flavours in a locus of the Coulomb branch with a Z_N symmetry. In this special vacuum, we calculate the prepotential, the dual periods and the period matrix using equivariant localization. When the flavours are massless, we find that the period matrix is completely specified by [N/2] effective couplings. On each of these, we show that the S-duality group acts as a generalized triangle group and that its hauptmodul can be used to write a non-perturbatively exact relation between each effective coupling and the bare one. For N = 2, 3, 4 and 6, the generalized triangle group is an arithmetic Hecke group which contains a subgroup that is also a congruence subgroup of the modular group PSL(2,Z). For these cases, we introduce mass deformations that respect the symmetries of the special vacuum and show that the constraints arising from S-duality make it possible to resum the instanton contributions to the period matrix in terms of meromorphic modular forms which solve modular anomaly equations.

Abstract: We determine the convolution sums ∑l+27m=nσ(l)σ(m) and ∑l+32m=nσ(l)σ(m) for all positive integers n. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of n by the octonary quadratic forms x12 + x 1x2 + x22 + x 32 + x 3x4 + x42 + 9(x 52 + x 5x6 + x62 + x 72 + x 7x8 + x82) and x12 + x 22 + x 32 + x 42 + 8(x 52 + x 62 + x 72 + x 82). A modular form approach is used.

Abstract: In a previous work, exact formulae and differential equations were found for traces of powers of the zero mode in the W 3 algebra. In this paper we investigate their modular properties, in particular we find the exact result for the modular transformations of traces of W 0 for n = 1, 2, 3, solving exactly the problem studied approximately by Gaberdiel, Hartman and Jin. We also find modular differential equations satisfied by traces with a single W 0 inserted, and relate them to differential equations studied by Mathur et al. We find that, remarkably, these all seem to be related to weight 0 modular forms with expansions with non-negative integer coefficients.

Abstract: In this paper, we prove the existence of an infinite-dimensional graded supermodule for the finite sporadic Thompson group Th whose McKay–Thompson series are weakly holomorphic modular forms of weight $$\frac{1}{2}$$ 1 2 satisfying properties conjectured by Harvey and Rayhaun.

Abstract: We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler–Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms with paramodular level structures.

Abstract: This article is a continuation of our previous work on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate the Iwasawa main conjecture for suitable twists of a newform $f$ that is non-ordinary at $p$, over the cyclotomic $\mathbb{Z}_p$-extension, the anticyclotomic $\mathbb{Z}_p$-extensions (in both the \emph{definite} and the \emph{indefinite} cases) as well as the maximal $\mathbb{Z}_p^2$-tower of an imaginary quadratic field $K$ where $p$ splits. In order to do so, we define Kobayashi-Sprung-style doubly-signed Coleman maps, which we use to define doubly signed Selmer groups. In the same spirit, we construct doubly-signed (integral) Beilinson-Flach elements (out of the collection of unbounded Beilinson-Flach elements of Loeffler-Zerbes), which we use to define doubly-signed $p$-adic $L$-functions. The main conjecture then relates these two set of objects. Furthermore, we show that the integral Beilinson-Flach elements form a locally restricted Euler system that allows us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here.

Abstract: Let $p>2$ be a prime. Under mild assumptions, we prove the Iwasawa main conjecture of Kato, for modular forms with general weight and conductor prime to $p$. This generalizes an earlier work of the author on supersingular elliptic curves.