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Viazovska proved that the [Formula: see text] lattice sphere packing is the densest sphere packing in [Formula: see text] dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.

In this work, Ramanujan type congruences modulo powers of primes p ≥ 5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t -core partitions known to satisfy Ramanujan type congruences for p = 5 , 7 , 11. The vectors of exponents corresponding to products that are modular forms for Γ 1 (p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ 1 (p) of weights 1 ≤ k ≤ 5 for primes 5 ≤ p ≤ 19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including p > 19. Canonical bases for modular forms of level 5 ≤ p ≤ 19 are constructed by summing weight one Hecke Eisenstein series of levels 5 ≤ p ≤ 19 and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for Γ 1 (p) and Γ (p) are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight 1 for Γ 1 (p) as a direct sum of subspaces of modular forms for Γ (p) of the form q r / p Z [ [ q ] ]. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences. [ABSTRACT FROM AUTHOR]

In a 2021 paper, M. Just and the second author defined a class of "semi-modular forms" on C \ R , in analogy with classical modular forms, that are "half modular" in a particular sense; and constructed families of such functions as Eisenstein-like series using symmetries related to integer partitions. Looking for further natural examples of semi-modular behavior, here we construct a family of Eisenstein-like series to produce semi-modular forms, using symmetries related to Fibonacci numbers instead of partitions. We then consider other Lucas sequences that yield semi-modular forms. [ABSTRACT FROM AUTHOR]

We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic points of modular curves and dispenses us of the need for equations of modular curves and for q-expansion computations in the construction of models of modular Jacobians. The resulting algorithm compares very favourably to our complex-analytic method. [ABSTRACT FROM AUTHOR]

In recent years, Rogers and Zudilin developed a method to write L -values attached to elliptic curves as periods. In order to apply this method to a broader collection of L -values, we study Eisenstein series and determine their Fourier series at cusps. Subsequently, we write the L -values of an elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate the value at k > 1 explicitly as a period. As a side result, we give simple integral expressions for the generating functions of L (E , k) when even (or odd) k runs over positive integers. [ABSTRACT FROM AUTHOR]

We prove that for a graded algebra A with a derivation D A satisfying certain conditions, and a bi-graded algebra A [ q ] with an extended derivation D of D A , there are only finitely many D A - and D-invariant (or differential with respect to D A and D) principal prime ideals of A and of A [ q ] , respectively. As its application, we prove the algebraic independence of the values of certain Eisenstein series for the arithmetic Hecke groups H(m) for m = 4 , 6 , ∞ , which is an extension of Nesterenko's result for the Eisenstein series for SL 2 (Z) . [ABSTRACT FROM AUTHOR]

Two level 17 modular functions r = q 2 ∏ n = 1 ∞ (1 - q n) n 17 , s = q 2 ∏ n = 1 ∞ (1 - q 17 n) 3 (1 - q n) 3 are used to construct a new class of Ramanujan–Sato series for 1 / π . The expansions are induced by modular identities similar to those level of 5 and 13 appearing in Ramanujan's Notebooks. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived. [ABSTRACT FROM AUTHOR]

Properties of theta functions and Eisenstein series dating to Jacobi and Ramanujan are used to deduce differential equations associated with McKay Thompson series of level 20. These equations induce expansions for modular forms of level 20 in terms of modular functions. The theory of singular values is applied to derive expansions for 1 / π of level 20 analogous to those formulated by Ramanujan. [ABSTRACT FROM AUTHOR]

Using explicit constructions of the Weierstrass mock modular form and Eisenstein series coefficients, we obtain closed formulas for the generating functions of values of shifted convolution L-functions associated to certain elliptic curves. These identities provide a surprising relation between weight 2 newforms and shifted convolution L-values when the underlying elliptic curve has modular degree 1 with conductor N such that $$\text {genus}(X_0(N)) = 1$$ . [ABSTRACT FROM AUTHOR]