Abstract: These notes give a brief introduction to a number of topics in the classical theory of modular forms. Some of theses topics are (planned) to be treated in much more detail in a book, currently in preparation, based on various courses held at the College de France in the years 2000–2004. Here each topic is treated with the minimum of detail needed to convey the main idea, and longer proofs are omitted.

Abstract: Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These non-holomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics. Here we outline the general facets of the theory, and we give several applications to number theory: partitions and $q$-series, modular forms, singular moduli, Borcherds products, extensions of theorems of Kohnen-Zagier and Waldspurger on modular $L$-functions, and the work of Bruinier and Yang on Gross-Zagier formulae. What is surprising is that this story has an unlikely beginning: the pursuit of the solution to a great mathematical mystery.

Abstract: In 1987 Serre conjectured that any mod l ("ell", not "1") two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where l is unramified. The hard work is in formulating an analogue of the "weight" part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a "mod l Langlands philosophy". Using ideas of Emerton and Vigneras, we formulate a mod l local-global principle for the group D^*, where D is a quaternion algebra over a totally real field, split above l and at 0 or 1 infinite places, and show how it implies the conjecture.

Abstract: For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2-k (resp. D k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are “dual” under ξ2-k to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of D k-1.

Abstract: These are the lecture notes of the lectures on Siegel modular forms at the Nordfjordeid Summer School on Modular Forms and their Applications. We give a survey of Siegel modular forms and explain the joint work with Carel Faber on vector-valued Siegel modular forms of genus 2 and present evidence for a conjecture of Harder on congruences between Siegel modular forms of genus 1 and 2.

Abstract: For integers $k\geq 2$, we study two differential operators on harmonic weak Maass forms of weight $2-k$. The operator $\xi_{2-k}$ (resp. $D^{k-1}$) defines a map to the space of weight $k$ cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are "dual" under $\xi_{2-k}$ to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of $D^{k-1}$.

Abstract: For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function Using the solution of the Nekrasov conjecture (33, 38, 3) and its refinement (34), we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with b+ = 1 in terms of modular forms This formula was proved earlier in (19) more generally for simply connected 4-manifolds with b+ = 1, assuming the Kotschick-Morgan conjecture and it was also derived by physical arguments in (31)

Abstract: Introduction Joe P. Buhler, Elliptic curves, modular forms, and applications: Preface Elliptic curves Points on elliptic curves Elliptic curves over $\mathbf C$ Modular forms of level 1 L-series Modular forms of higher level $l$-adic representations The rank of elliptic curves over $\mathbf Q$ Applications of elliptic curves Bibliography Alice Silverberg, Open questions in arithmetic algebraic geometry: Overview Torsion subgroups Ranks Conjectures of Birch and Swinnerton-Dyer ABC and related conjectures Some other conjectures Bibliography Kenneth A. Ribet and William A. Stein, Lectures on Serre's conjectures: Preface Introduction to Serre's conjecture Optimizing the weight Optimizing the level Exercises Appendix by Brian Conrad: The Shimura construction in weight 2 Appendix by Kevin Buzzard: A mod $\ell$ multiplicity one result Bibliography Fernando Q. Gouvea, Deformations of Galois representations: Introduction Galois groups and their representations Deformations of representations The universal deformation: existence The universal deformation: properties Explicit deformations Deformations with prescribed properties Modular deformations $p$-adic families and infinite ferns Appendix 1 by Mark Dickinson: A criterion for existence of a universal deformation ring Appendix 2 by Tom Weston: An overview of a theorem of Flach Appendix 3 by Matthew Emerton: An introduction to the $p$-adic geometry of modular curves Bibliography Ralph Greenberg, Introduction to Iwasawa theory for elliptic curves: Preface Mordell-Weil groups Selmer groups $\Lambda$-modules Mazur's control theorem Bibliography John Tate, Galois cohomology: Galois cohomology Bibliography Wen-Ching Winnie Li, The arithmetic of modular forms: Introduction Introduction to elliptic curves, modular forms, and Calabi-Yau varieties The arithmetic of modular forms Connections among modular forms, elliptic curves, and representations of Galois groups Bibliography Noriko Yui, Arithmetic of certain Calabi-Yau varieties and mirror symmetry: Introduction The modularity conjecture for rigid Calabi-Yau threefolds over the field of rational numbers Arithmetic of orbifold Calabi-Yau varieties over number fields $K3$ surfaces, mirror moonshine phenomenon Bibliography.

Abstract: Preface Contributors 1. On effective approximations to cubic irrationals A. Baker and C. L. Stewart 2. Applications of measure theory and Haudorff dimension to the theory of Diophantine approximation V. I. Bernik 3. Galois representations and transcendental numbers D. Bertrand 4. Some new results on algebraic independance of E-functions F. Beukers 5. Algebraic values of hypergeometric functions F. Beukers and J. Wolfart 6. Some new applications of an inequality of Mason B. Brindza 7. Aspects of the Hilbert Nullstellensatz W. D. Brownawell 8. On the irrationality of certain series: problems and results P. Erdos 9. S-unit equations and their applications J.-H. Evertse, K. Gyory, C. L. Stewart and R. Tijdeman 10. Decomposable form equations J.-H. Evertse and K. Gyory 11. On Gelfond's method N. I. Feldman 12. On effective bounds for certain linear forms A. I. Galochkin 13. Automata and transcendence J. H. Loxton 14. The study of Diphantine equations over function fields R. C. Mason 15. Linear relations on algebraic groups D. W. Masser 16. Estimates for the number of zeros of certain functions Yu. V. Nesterenko 17. An applications of the S-unit theorem to modular forms on T0(N) R. W. K. Odoni 18. Lower bounds for linear forms in logarithms P. Philippon and M. Waldschmidt 19. Reducibility of lacunary polynomials, IX A. Schinzel 20. The number of solutions of Thue equations W. M. Schmidt 21. On arithmetic properties of the values of E-functions A. B. Shidlovsky 22. Some exponential Deiophantine equations T. N. Shorey 23. Arithmetic specialisations theory V. G. Sprindzuk 24. On the transcendence methods of Gelfond and Scheider in several variables M. Waldschmidt 25. A new approach to Baker's theorem on linear forms in logarithms III G. Wustholz 26. Linear forms in logarithms in the p-adic case Kunrui Yu.

Abstract: We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of $n$-dimensional mod $p$ Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for $n$-dimensional automorphic Galois representations.

Abstract: As a generalization to Banach contraction principle, Ciric introduced the concept of quasi-contraction mappings. In this paper, we investigate these kinds of mappings in modular function spaces without the 2-condition. In particular, we prove the existence of fixed points and discuss their uniqueness.

Abstract: We study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms. We show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry, in particular the moduli of K3 surfaces. We give applications to modular forms and to computing the fundamental groups of some moduli spaces.

Abstract: The winter semester 2002/2003 was the last semester before my retirement from the university. It also happened that I was the chairman of the Colloquium and the speaker foreseen for February 7 had to cancel his visit. At about the same time I found some numerical support for a very general conjecture relating divisibilities of certain special values of L-functions to congruences between modular forms. I have been thinking about this kind of relationship for many years, but I never had any idea how one could find experimental evidence. But in the early 2003 C. Faber and G. van der Geer had written a program that produced lists of eigenvalues of Hecke operators on some special Siegel modular forms. After a few days of suspense we could compare their list with my list of eigenvalues of elliptic modular forms and verify the congruence in our examples.

Abstract: We study the Fourier coefficients of generalized modular forms $f(\tau)$ of integral weight $k$ on subgroups $\Gamma$ of finite index in the modular group. We establish two Theorems asserting that $f(\tau)$ is constant if $k = 0$, $f(\tau)$ has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that $f(\tau)$ has a cuspidal divisor, $k$ is arbitrary, and $\Gamma = \Gamma_{0}(N)$, where we show that $f(\tau)$ is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficients. We also explain how these results apply to the theory of orbifold vertex operator algebras.

Abstract: Let F be a totally real field and p ≥ 3 a prime. If ρ : Open image in new window is continuous, semisimple, totally odd, and tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which ρ is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, which required p to be unramified in F. We also prove a theorem that verifies one half of the conjecture in many cases and use Dembele’s computations of Hilbert modular forms over \(\mathbb{Q}(\sqrt 5 )\) to provide evidence in support of the conjecture.

Abstract: * Author writes in a clear and engaging style * Contains never before published elementary proofs * Author provides new results and detailed exposition * Self-contained, and suitable for use in a classroom setting or for self-study * A highly creative contribution to the theory of modular forms and dirichlet series The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books reiterate a single old result with a single old method. After a review of elementary Fourier analysis, the author presents completely new results with new methods, though old results will also be proved. No advanced knowledge of number theory is required up to this point. As applications, new formulas for the second factor of the class number of a cyclotomic field will be given. The second half of the book assumes familiarity with basic knowledge of modular forms. However, all definitions and facts are clearly stated, and precise references are given. The notion of nearly holomorphic modular forms is introduced and applied to the determination of the critical values of Hecke L-functions of an imaginary quadratic field. Other notable features of the book are: (1) some new results on classical Eisenstein series; (2) the discussion of isomorphism classes of elliptic curves with complex multiplication in connection with their zeta function and periods; (3) a new class of holomorphic differential operators that send modular forms to those of a different weight. The book will be of interest to graduate students and researchers who are interested in special values of L-functions, class number formulae, arithmetic properties of modular forms (especially their values), and the arithmetic properties of Dirichlet series. It treats in detail, from an elementary viewpoint, the simplest cases of a fundamental area of ongoing research, the only prerequisite being a basic course in algebraic number theory.

Abstract: This article is divided in two parts. In the first part we endow a certain ring of ``Drinfeld quasi-modular forms'' for $\GL_2(\FF_q[T])$ (where $q$ is a power of a prime) with a system of ``divided derivatives" (or hyperderivations). This ring contains Drinfeld modular forms as defined by Gekeler in \cite{Ge}, and the hyperdifferential ring obtained should be considered as a close analogue in positive characteristic of famous Ramanujan's differential system relating to the first derivatives of the classical Eisenstein series of weights $2$, $4$ and $6$. In the second part of this article we prove that, when $q ot=2,3$, if ${\cal P}$ is a non-zero hyperdifferential prime ideal, then it contains the Poincare series $h=P_{q+1,1}$ of \cite{Ge}. This last result is the analogue of a crucial property proved by Nesterenko \cite{Nes} in characteristic zero in order to establish a multiplicity estimate.

Abstract: We consider the action of Hecke operators on weakly holomorphic modular forms and a Hecke-equivariant duality between the spaces of holomorphic and weakly holomorphic cusp forms. As an application, we obtain congruences modulo supersingular primes, which connect Hecke eigenvalues and certain singular moduli.

Abstract: We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus two vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a question formulated during the conference "Black holes, Black Rings and Modular Forms" (ENS, Paris, August 2007). These modular forms generalize the classical Igusa form and the forms constructed by Gritsenko and Nikulin in 1998.

Abstract: We characterize the 2-line of the p-local Adams-Novikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p > 3. We give a similar characterization of the 1-line, reinterpreting some earlier work of A. Baker and G. Laures. These results are then used to deduce that, for l a prime which generates the p-adic units, the spectrum Q(l) detects the alpha and beta families in the stable stems.

Abstract: We show that the coefficients of Ramanujan's mock theta function f(q) are the first non-trivial coefficients of a canonical sequence of modular forms. This fact follows from a duality which equates coefficients of the holomorphic projections of certain weight 1/2 Maass forms with coefficients of certain weight 3/2 modular forms. This work depends on the theory of Poincare series, and a modification of an argument of Goldfeld and Sarnak on Kloosterman�Selberg zeta functions.

Abstract: We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of $q$. And we prove that certain linear sums of $q$-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.

Abstract: A main issue in superstring theory are the superstring measures. D'Hoker and Phong showed that for genus two these reduce to measures on the moduli space of curves which are determined by modular forms of weight eight and the bosonic measure. They also suggested a generalisation to higher genus. We showed that their approach works, with a minor modification, in genus three and we announced a positive result also in genus four. Here we give the modular form in genus four explicitly. Recently S. Grushevsky published this result as part of a more general approach.

Abstract: Let g = � c(D)q D and f = � anq n be modular forms of half-integral weight k + 1/2 and integral weight 2k respectively that are associated to each other under the Shimura–Kohnen correspondence. For suitable fundamental discriminants D ,at heorem of Waldspurger relates the coefficient c(D) to the central critical value L(f, D, k )o f the Hecke L-series of f twisted by the quadratic Dirichlet character of conductor D .T his paper establishes a similar kind of relationship for central critical derivatives in the special case k =1 , wheref is of weight 2. The role of c(D) in our main theorem is played by the first derivative in the weight direction of the Dth Fourier coefficient of a p-adic family of half-integral weight modular forms. This family arises naturally, and is related under the Shimura correspondence to the Hida family interpolating f in weight 2. The proof of our main theorem rests on a variant of the Gross–Kohnen–Zagier formula for Stark–Heegner points attached to real quadratic fields, which may be of some independent interest. We also formulate a more general conjectural formula of Gross–Kohnen–Zagier type for Stark– Heegner points, and present numerical evidence for it in settings that seem inaccessible to our methods of proof based on p-adic deformations of modular forms.

Abstract: Aspects of arithmetic and modular forms: Motives and mirror symmetry for Calabi-Yau orbifolds by S. Kadir and N. Yui String modular motives of mirrors of rigid Calabi-Yau varieties by S. Kharel, M. Lynker, and R. Schimmrigk Update on modular non-rigid Calabi-Yau threefolds by E. Lee Finite index subgroups of the modular group and their modular forms by L. Long Aspects of geometric and differential equations: Apery limits of differential equations of order 4 and 5 by G. Almkvist, D. van Straten, and W. Zudilin Hypergeometric systems in two variables, quivers, dimers and dessins d'enfants by J. Stienstra Some properties of hypergeometric series associated with mirror symmetry by D. Zagier and A. Zinger Ramanujan-type formulae for $1[LAMBDA]pi$: A second wind? by W. Zudilin Aspects of physics and string theory: Meet homological mirror symmetry by M. Ballard Orbifold Gromov-Witten invariants and topological strings by V. Bouchard Conformal field theory and mapping class groups by T. Gannon $SL(2,\mathbb{C})$ Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial by S. Gukov and H. Murakami Open strings and extended mirror symmetry by J. Walcher.

Abstract: We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and Kasten-Schmidt. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands functoriality one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blasius and Panchishkin on twisted L-values in some cases. We comment on the compatibility of our results with Deligne's conjecture on the critical values of motivic L-functions. These results, as in the above mentioned works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals.

Abstract: Modular Forms.- Dirichlet Series of Modular Forms.- Hecke-Shimura Rings of Double Cosets.- Hecke Operators.- Euler Factorization of Radial Series.