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The global SARS-CoV-2 immune landscape and population protection against emerging variants is largely unknown. We assessed SARS-CoV-2 antibody changes in the Dominican Republic and implications for immunological protection against variants of concern. Between March 2021 and August 2022, 2,300 patients with undifferentiated febrile illnesses were prospectively enrolled. Sera was tested for total anti-spike antibodies and simultaneously collected nasopharyngeal samples for acute SARS-CoV-2 infection with RT-PCR. Geometric mean anti-spike titers increased from 6.6 BAU/ml (95% CI 5.1–8.7) to 1,332 BAU/ml (1055–1,682). Multivariable binomial odds ratios for acute SARS-CoV-2 infection were 0.55 (0.40–0.74), 0.38 (0.27–0.55), and 0.27 (0.18–0.40) for the second, third, and fourth versus the first anti-S quartile, with similar findings by viral strain. Integrated serological and virological screening can leverage existing acute fever surveillance platforms to monitor population-level immunological markers and concurrently characterize implications for emergent variant transmission in near real-time.

A Dirichlet series is defined whose coefficients are determined by counting certain integral points on the quadratic twists of an elliptic curve. The function defined by this series has a meromorphic continuation with at most a simple pole at a certain distinguished point. In certain cases, the residue there gives a class number formula for positive definite binary quartic forms.

In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz's formula and indicate some extensions., Journal of the European Mathematical Society, 23 (12), ISSN:1435-9855, ISSN:1435-9863

The majority of JAK2V617F-negative myeloproliferative neoplasms (MPNs) have disease-initiating frameshift mutations in calreticulin (CALR), resulting in a common carboxyl-terminal mutant fragment (CALRMUT), representing an attractive source of neoantigens for cancer vaccines. However, studies have shown that CALRMUT-specific T cells are rare in patients with CALRMUTMPN for unknown reasons. We examined class I major histocompatibility complex (MHC-I) allele frequencies in patients with CALRMUTMPN from two independent cohorts. We observed that MHC-I alleles that present CALRMUTneoepitopes with high affinity are underrepresented in patients with CALRMUTMPN. We speculated that this was due to an increased chance of immune-mediated tumor rejection by individuals expressing one of these MHC-I alleles such that the disease never clinically manifested. As a consequence of this MHC-I allele restriction, we reasoned that patients with CALRMUTMPN would not efficiently respond to a CALRMUTfragment cancer vaccine but would when immunized with a modified CALRMUTheteroclitic peptide vaccine approach. We found that heteroclitic CALRMUTpeptides specifically designed for the MHC-I alleles of patients with CALRMUTMPN efficiently elicited a CALRMUTcross-reactive CD8+T cell response in human peripheral blood samples but not to the matched weakly immunogenic CALRMUTnative peptides. We corroborated this effect in vivo in mice and observed that C57BL/6J mice can mount a CD8+T cell response to the CALRMUTfragment upon immunization with a CALRMUTheteroclitic, but not native, peptide. Together, our data emphasize the therapeutic potential of heteroclitic peptide–based cancer vaccines in patients with CALRMUTMPN.

Although somatic mutations influence the pathogenesis, phenotype, and outcome of myeloproliferative neoplasms (MPNs), little is known about their impact on molecular response to cytoreductive treatment. We performed targeted next-generation sequencing (NGS) on 202 pretreatment samples obtained from patients with MPN enrolled in the DALIAH trial (A Study of Low Dose Interferon Alpha Versus Hydroxyurea in Treatment of Chronic Myeloid Neoplasms; #NCT01387763), a randomized controlled phase 3 clinical trial, and 135 samples obtained after 24 months of therapy with recombinant interferon-alpha (IFNα) or hydroxyurea. The primary aim was to evaluate the association between complete clinicohematologic response (CHR) at 24 months and molecular response through sequential assessment of 120 genes using NGS. Among JAK2-mutated patients treated with IFNα, those with CHR had a greater reduction in the JAK2 variant allele frequency (median, 0.29 to 0.07; P < .0001) compared with those not achieving CHR (median, 0.27 to 0.14; P < .0001). In contrast, the CALR variant allele frequency did not significantly decline in those achieving CHR or in those not achieving CHR. Treatment-emergent mutations in DNMT3A were observed more commonly in patients treated with IFNα compared with hydroxyurea (P = .04). Furthermore, treatment-emergent DNMT3A mutations were significantly enriched in IFNα–treated patients not attaining CHR (P = .02). A mutation in TET2, DNMT3A, or ASXL1 was significantly associated with prior stroke (age-adjusted odds ratio, 5.29; 95% confidence interval, 1.59-17.54; P = .007), as was a mutation in TET2 alone (age-adjusted odds ratio, 3.03; 95% confidence interval, 1.03-9.01; P = .044). At 24 months, we found mutation-specific response patterns to IFNα: (1) JAK2- and CALR-mutated MPN exhibited distinct molecular responses; and (2) DNMT3A-mutated clones/subclones emerged on treatment.

The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's generalization (the Minkowski chain) to give criteria for a real linear form to be either badly approximable or singular. We also give a variant of Dirichlet's approximation theorem for a real linear form that produces a whole basis of approximating integral vectors rather than a single one. This result holds if and only if the form is badly approximable. The proofs rely on properties of successive minima and reduced bases of lattices., 18 pages, 1 figure

We give examples of completely multiplicative arithmetic functions that assume only the values ±1 and that have bounded first Cesaro means. The method of proof also yields some interesting identities involving special values of Dirichlet L-functions. In particular, we present some new class number formulas for quadratic fields.

Modular graph functions associate to a graph an S L ( 2 , Z ) -invariant function on the upper half plane. We obtain the Fourier series of modular graph functions of arbitrary weight w and two-loop order. The motivation for this work is to develop a deeper understanding of the origin of the algebraic identities between modular graph functions which have been discovered recently, and of the relation between the existence of these identities and the occurrence of cusp forms. We show that the constant Fourier mode, as a function of the modulus τ, consists of a Laurent polynomial in y = π Im τ of degree ( w , 1 − w ) , plus a contribution which decays exponentially as y → ∞ . The Laurent polynomial is a linear combination with rational coefficients of the top term y w , and lower order terms ζ ( 2 k + 1 ) y w − 2 k − 1 for 1 ≤ k ≤ w − 1 , as well as terms ζ ( 2 w − 2 l − 3 ) ζ ( 2 l + 1 ) y 2 − w for 1 ≤ l ≤ w − 3 . The exponential contribution is a linear combination of exponentials of y and incomplete Γ-functions whose coefficients are Laurent polynomials in y with rational coefficients.

Se realizó un estudio observacional, descriptivo y transversal de recolección de datos tipo prospectivo, cuyo objetivo fue determinar la correlación entre bilirrubina sérica total y bilirrubina transcutánea en los recién nacidos ingresados en el área de neonatología del Hospital Universitario Maternidad Nuestra Señora de La Altagracia. El grupo de estudio estuvo constituido por un total de 30 neonatos con solicitud de examen de laboratorio de bilirrubinas séricas, a los que conjuntamente se les tomó la bilirrubina transcutánea torácica (Bttc) y frontal (Bttf) con no más de 30 minutos de diferencia. En este se obtuvo que la media de los valores de Bttf, Bttc y el estándar de oro fueron 15.56 mg/dl, 14.53 mg/dl y 12.89 g/dl, respectivamente; y las correlaciones de Pearson fueron las siguientes: r=0.71 para las medidas transcutáneas, r=0.91 para la medición frontal y bilirrubina sérica, y r=0.78 para la Bttc y la bilirrubina en sangre. La discrepancia entre Bttf/sérica fue de 2.67 mg/dl y entre Bttc/sérica de 1.64 mg/dl. Por lo que se concluyó que los valores de bilirrubinas séricas y transcutáneas presentan una buena correlación y pueden ser utilizados como método diagnóstico.

We investigate the asymptotic distribution of integrals of the $j$-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums of half-integral weight that is uniform in every parameter. To establish this estimate we prove a variant of Kuznetsov's formula where the spectral data is restricted to half-integral weight forms in the Kohnen plus space, and we apply Young's hybrid subconvexity estimates for twisted modular $L$-functions., Comment: 31 pages

BackgroundThe majority of JAK2V617F-negative myeloproliferative neoplasms (MPN) have disease-initiating frameshift mutations in calreticulin (CALR) resulting in a common novel C-terminal mutant fragment (CALRMUT), representing an attractive source of neoantigens for cancer vaccines. However, studies have shown that CALRMUT-specific T cells are rare in CALRMUT MPN patients, but the underlying reasons for this phenomenon are unknown. We speculate that this is due to an increased chance of immune-mediated tumor rejection by individuals expressing one of these MHC-I alleles such that the disease never clinically manifests. As a consequence of this MHC-I allele restriction, we reasoned that CALRMUT MPN patients would not efficiently respond to cancer vaccines composed of the CALRMUT neoantigen, but could do so when immunized with a properly epitope-optimized CALRMUT heteroclitic peptide vaccine approach.MethodsWe examined MHC-I allele frequency in CALRMUT MPN patients from two independent cohorts to identify under-represented MHC-I alleles. These MHC-I alleles were assessed for their ability to bind to CALRMUT-derived peptides using NetMHC and were subsequently validated experimentally in healthy donors and in CALRMUT MPN patients having received a CALRMUT cancer vaccine (clinical trial NCT03566446) to determine if these MHC-I were potentiating immunogenicity against the CALRMUT antigen. Epitope-optimized heteroclitic variants of the CALRMUT neoantigen were identified and tested experimentally in vitro in human PBMCs and in vivo in mice for their ability to mount an immune response against the non-modified CALRMUT neoantigen.ResultsWe observed that MHC-I alleles that present CALRMUT neoepitopes with high affinity are under-represented in CALRMUT MPN patients. Heteroclitic CALRMUT peptides specifically designed for CALRMUT MPN patient MHC-I alleles efficiently elicited a cross-reactive CD8+ T cell response in human PBMC samples otherwise unable to respond to the matched weakly immunogenic CALRMUT native peptides. We also modeled this effect in mice and observed that C57BL/6J mice, which are unable to mount an immune response to the human CALRMUT fragment, can mount a cross-reactive CD8+ T cell response against a CALRMUT-derived peptide upon heteroclitic peptide immunization and this was further amplified by combining the heteroclitic peptide vaccine with blockade of the immune checkpoint molecule PD-1.ConclusionsOur study shows that MHC-I alleles able to present CALRMUT neoepitopes are under-represented in CALRMUT MPN patients, demonstrating that MHC-I haplotype is a major mechanism of passive immune-evasion in CALRMUT MPN. However, we show that a cancer vaccine composed of heteroclitic variants of the CALRMUT antigen could overcome this limitation.Ethics ApprovalApproval was obtained for the use of patient-derived specimens and access to clinical data extracted from patient charts by the Institutional Review Boards at Memorial Sloan Kettering Cancer Center, the Dana-Farber Cancer Institute and the Massachusetts General Hospital, as well as by the Danish Regional Science Ethics Committee.

This work is motivated by a paper of Davenport and Schmidt, which treats the question of when Dirichlet's theorems on the rational approximation of one or of two irrationals can be improved and if so, by how much. We consider a generalization of this question in the simplest case of a single irrational but in the context of the geometry of numbers in $\mathbb R^2$, with the sup-norm replaced by a more general one. Results include sharp bounds for how much improvement is possible under various conditions. The proofs use semi-regular continued fractions that are characterized by a certain best approximation property determined by the norm., 30 pages, 7 figures

Background: To investigate the role of genomics in determining response and resistance to front line treatment in MPN, we performed somatic mutational profiling of the DALIAH trial, a randomized controlled phase III trial of interferon versus hydroxyurea in newly diagnosed MPN patients. Methods: We performed genomic analyses on 202 pre-treatment primary MPN samples obtained from patients enrolled on the DALIAH trial (NCT01387763) and 135 samples obtained after 24 months of treatment. Genomic profiling comprised targeted next generation sequencing (NGS) of 100 genes, selected on the basis of their known or suspected involvement in the pathogenesis of myeloid malignancies, and 1609 informative single nucleotide polymorphisms (SNPs) on chromosome 9p. Clinicohematological response was determined by central review using ELN 2009 (ET, PV and pre-MF) and EUMNET 2005 (PMF) criteria. We evaluated the association of somatic mutations with clinical parameters and with attainment of clinicohematological complete response (CR) at 24 months. Results: Prior to treatment, 191 of 202 (95%) patients had somatic mutations, including 93% with canonical MPN phenotypic drivers: JAK2 (74%), CALR (14%), and MPL (5%). Among those with JAK2 mutations 37% had JAK2 copy-neutral loss of heterozygosity (JAK2 CN-LOH). Patients with PV were more likely to have JAK2 CN-LOH (p = 0.0001) as compared with patients with other MPN subtypes. At baseline, patients with JAK2 CN-LOH had significantly higher hemoglobin (p = 0.0001), higher white blood cell count (WBC, p = 0.002) and lower platelet count (p=0.0001) than patients without JAK2 CN-LOH. Mutations in TET2 (24%), DNMT3A (16%), and ASXL1 (10%) were the most frequent co-occurring non-MPN phenotypic driver mutations and they occurred across all MPN subtypes. In addition, 5% of patients had spliceosome gene mutations, and 6% had mutations in genes involved in RAS/MAPK signaling. Patients with TET2, DNMT3A or ASXL1 mutations were significantly older than patients without these mutations (p= 0.0001) and there was a significant association between the presence of a TET2, DNMT3A or ASXL1 mutation and prior stroke (p = 0.004). There were no other significant associations between somatic mutation status and baseline clinical characteristics. The probability of attaining clinicohematological CR at 24 months was independent of baseline somatic mutations. Among patients with JAK2 mutations who remained on interferon treatment at 24 months, those with CR had a greater reduction in mean variant allele fraction (VAF) (28% to 8%, p Among patients who remained on treatment for 2 years, 44 mutations in 35 patients were newly detected or expanded on serial sampling. DNMT3A mutations were the most frequently acquired, accounting for 41% of new mutations. ASXL1, TET2, PPM1D, TP53, IDH2, and CBL mutations were also recurrently acquired. 97% of patients who acquired new mutations were JAK2-mutant. Among those with acquired DNMT3A mutations, 85% were treated with interferon, and 23% had CR at 24 months. Among those that acquired non-DNMT3A mutations, 38% were treated with interferon and 47% had CR at 24 months. The VAF of newly detected mutations was low (median 1.5%), and half of the patients with newly acquired mutations had at least 50% reduction in JAK2V617F VAF, suggesting that new mutations could either have arisen independently or be subclonal to the dominant JAK2-mutant clone. Conclusions: Using sequential genomic analyses of a phase III clinical trial of interferon versus hydroxyurea in MPN patients, we found mutation-specific and treatment-specific patterns of response. We uncovered distinct patterns of response to interferon in JAK2-mutant MPN as compared with CALR-mutant MPN. We found that DNMT3A mutations were the most frequent acquired mutations at 24 months and that these were enriched in patients treated with interferon. In aggregate, these results provide insights into molecular response and resistance to interferon and inform the clinical use of interferon in MPN patients. Disclosures Hansen: Alexion: Research Funding. Neuberg:Pharmacyclics: Research Funding; Madrigal Pharmaceuticals: Equity Ownership; Celgene: Research Funding. Hasselbalch:Novartis: Research Funding; AOP Orphan Pharmaceuticals: Other: Data monitoring board. Lindsley:Jazz Pharmaceuticals: Research Funding; Takeda Pharmaceuticals: Consultancy; Medlmmune: Research Funding. Mullally:Janssen: Research Funding.

We give some new applications of Kronecker’s first limit formula to real quadratic fields. In particular, we give a surprising geometrical relationship between the CM points associated with two imaginary quadratic fields with discriminants d and $$d^{\prime }$$ and certain winding number functions coming from the closed geodesics associated with the real quadratic field of discriminant $$d^{\prime }d$$ .

We introduce some analogues of the Markov spectrum defined in terms of modular billiards and consider the problem of characterizing that part of the spectrum below the lowest limit point., 19 pages, 18 figures

We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behaviour near roots of unity.

It is known that the $3$ -manifold $\operatorname{SL}(2,\mathbb{Z})\backslash\operatorname{SL}(2,\mathbb{R})$ is diffeomorphic to the complement of the trefoil knot in $S^{3}$ . E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind’s eta function under $\operatorname{SL}(2,\mathbb{Z})$ . In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It can be computed in terms of a special value of a certain Dirichlet series and satisfies a reciprocity law. The homogenization of this symbol, which generalizes the Rademacher symbol, gives the linking number between two distinct symmetric links formed from modular knots.

In this note, we give an explicit basis for the harmonic weak forms of weight two. We also show that their holomorphic coefficients can be given in terms of regularized inner products of weight zero weakly holomorphic forms.

In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are given in terms of cycle integrals of the modular j-function. Their shadows are weakly holomorphic forms of weight 3/2. These new mock modular forms occur as holomorphic parts of weakly harmonic Maass forms. We also construct a generalized mock modular form of weight 1/2 having a real quadratic class number times a regulator as a Fourier coefficient. As an application of these forms we study holomorphic modular integrals of weight 2 whose rational period functions have poles at certain real quadratic integers. The Fourier coefficients of these modular integrals are given in terms of cycle integrals of modular functions. Such a modular integral can be interpreted in terms of a Shimura-type lift of a mock modular form of weight 1/2 and yields a real quadratic analogue of a Borcherds product.

OBJECTIVES: To strengthen prevention of mother-to-child HIV transmission (pMTCT) program implementation in La Romana (LR) province, by estimating HIV prevalence and identifying characteristics associated with HIV infection in parturients. METHODS: Umbilical cord blood samples were collected at seven obstetrical sites where over 95% of LR's deliveries occur during four phases (pilot, expanded pilot, full study, and pMTCT program monitoring) from 2 August 2002 to 30 September 2006. Results were linked to data abstracted from delivery records. RESULTS: HIV seroprevalence was 2.6% (263/10 040 overall; 114/4 452, full-study phase (95% confidence interval = 2.1%-3.1%)). Most HIV-infected parturients were Dominican (68.9%) and urban (64.0%). However, prevalence was higher among Haitians (3.7%) than Dominicans (2.3% (p < 0.001)), especially those aged 21-25 years (5.2% vs. 2.3% (p < 0.001)), and among rural, batey, and peri-urban (vs. urban) parturients (3.4% vs. 2.3%, (p = 0.003)). HIV prevalence was associated with commercial sex work (reported by only 0.4%), and prior pregnancy. In logistic regression analysis, commercial sex work, Haitian nationality, and prior pregnancy were independently associated with HIV infection. Caesarean deliveries were more frequent, and rose in the last years of the study, among HIV seropositives; however, most deliveries among seropositives (57.5%) were vaginal. CONCLUSIONS: HIV prevalence among LR parturients was higher than the estimated prevalence in the Dominican Republic (0.8%-1.0%) and, in contrast to past studies, predominantly affected urban Dominicans. HIV prevalence among LR Haitian parturients was higher than among Dominican counterparts and prenatal clinic attendees in Haiti (who had a rate of 3.1%). Consistently implemented, targeted pMTCT interventions are needed.OBJETIVOS: Fortalecer el programa de prevención de la transmisión de la infección por el VIH madre-hijo (PPTIMH) en la provincia de La Romana (LR), mediante la estimación de la prevalencia y la identificación de las características asociadas con esta infección en las mujeres parturientas. MÉTODOS: Se tomaron muestras de sangre del cordón umbilical en siete puntos de atención obstétrica, que concentran más de 95% de los partos de LR, durante las cuatro fases de estudio (piloto, piloto extendido, estudio completo y monitoreo del PPTIMH) entre el 2 de agosto de 2002 y el 30 de septiembre de 2006. Los resultados se enlazaron con los datos extraídos de los registros de parto. RESULTADOS: La seroprevalencia al VIH fue de 2,6% (263/10 040 general; 114/4 452 en la fase de estudio completo; intervalo de confianza de 95%: 2,1% a 3,1%). La mayoría de las parturientas infectadas eran dominicanas (68,9%) y de zonas urbanas (64,0%). No obstante, la prevalencia fue mayor en las haitianas (3,7%) que en las dominicanas (2,3%; P < 0,001) - especialmente en las de 21 a 25 años (5,2% frente a 2,3%; P < 0,001)- y de bateyes y zonas rurales y periurbanas con respecto a las de zonas urbanas (3,4% frente a 2,3%; P = 0,003). La prevalencia de VIH se asoció con actividad sexual profesional (aunque solamente 0,4% manifestó realizarlo) y embarazos previos. Según el análisis de regresión logística, ser profesional del sexo, haitiana y haber tenido algún embarazo previo se asociaron independientemente con la infección por el VIH. Los partos por cesárea fueron más frecuentes en las seropositivas al VIH, y aumentaron en el último año del estudio; no obstante, la mayoría de los partos de las mujeres seropositivas fueron vaginales (57,5%). CONCLUSIONES: La prevalencia de la infección por el VIH en las parturientas de LR fue mayor que la estimada para la República Dominicana (0,8-1,0%) y, en contraste con estudios anteriores, afectó principalmente a las residentes en zonas urbanas. La prevalencia de esta infección en las parturientas haitianas de LR fue mayor que en las dominicanas y las que asistieron a clínicas prenatales en Haití (3,1%). Se necesitan intervenciones y PPTIMH dirigidos y consistentes.

We define traces associated to a weakly holomorphic modular form [math] of arbitrary negative even integral weight and show that these traces appear as coefficients of certain weakly holomorphic forms of half-integral weight. If the coefficients of [math] are integral, then these traces are integral as well. We obtain a negative weight analogue of the classical Shintani lift and give an application to a generalization of the Shimura lift.

Some of Ramanujan's original discoveries about hypergeometric functions and their relation to modular integrals, especially Eisenstein series ofnegative weight, are still not very well understood. These discoveries take the form of identities that he recorded, without proof, as entries in his notebooks. In the following sections I will introduce some of these entries, discuss their status, give new proofs of several of them and also provide new results of a similar nature.

We express the zeros of the Weierstrass $$\wp$$ -function in terms of generalized hypergeometric functions. As an application of our main result we prove the transcendence of two specific hypergeometric functions at algebraic arguments in the unit disc. We also give a Saalschützian 4 F 3-evaluation

We give a generalization of a result of Myerson on the asymptotic behavior of norms of certain Gaussian periods. The proof exploits properties of the Mahler measure of a trinomial.

Nous donnons une formule de type Chowla-Selberg qui relie une generalisation de la fonction eta a GL(n) avec les fonctions gamma multiples. Nous presentons egalement quelques identites de produit infinis pour certaines valeurs speciales de la fonction gamma multiple.

(1) zd = { i √ d 2 if d ≡ 0 (mod 4), −1+i √ d 2 if d ≡ 3 (mod 4). The j-function has the remarkable property that j(zd) is an algebraic integer of degree h(−d), the class number of K = Q(zd). In fact, K(j(zd)) is the Hilbert class field of K [4]. The first few values of j(zd) are: j(z3) = 0, j(z4) = 12 , j(z7) = −153, j(z8) = 20, j(z11) = −323 j(z15) = −191025−85995√5 2 , j(z19) = −963, j(z20) = 632000+282880 √ 5

(2005). Quadratic Reciprocity in a Finite Group. The American Mathematical Monthly: Vol. 112, No. 3, pp. 251-256.

It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures.” Here I will explore another part of his work whose position within number theory seems to be less well understood, even though it is more elementary, namely that related to continued fractions. I will concentrate on the special values of continued fractions that represent modular functions, especially the Rogers-Ramanujan continued fraction. These give analogues of the simple continued fraction expansions of units in real quadratic fields. My primary motivation is to furnish a coherent treatment of this topic, around which an air of mystery seems to linger. Another is to provide an inviting and non-standard introduction to the classical theory of modular functions. This is largely an expository paper; most of the ideas I discuss are well known. Yet it is hoped that the elaboration given here combines these ideas in a novel way. Although this paper is not intended to be comprehensive, its later sections contain more material than is likely needed to gain a clear impression of the main themes, which the first six sections should provide. These will take the general reader through a proof of the first main result, Theorem 1, introducing the needed concepts along the way. The sections that follow these assume a somewhat greater background in number theory.