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8 - Modular derivatives and vector-valued modular forms
from Part I - Modular forms and their variants
Eric D'Hoker, University of California, Los Angeles, Justin Kaidi, Kyushu University
Book:

Modular Forms and String Theory

Published online:

28 November 2024

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12 December 2024, pp 155-167
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Summary

In this chapter, we construct differential equations in the modular parameter and find solutions to these equations in simple cases. The solutions can generically be assembled into vector-valued modular forms, which have proven fruitful in recent works in mathematics and physics. We will establish that, in general, each component of a vector-valued modular form is a modular form for a congruence subgroup.

6 - Congruence subgroups and modular curves
from Part I - Modular forms and their variants
Eric D'Hoker, University of California, Los Angeles, Justin Kaidi, Kyushu University
Book:

Modular Forms and String Theory

Published online:

28 November 2024

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12 December 2024, pp 121-136
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Summary

Congruence subgroups form a countable infinite class of discrete non-Abelian subgroups of SL(2,Z) and play a particularly prominent role in deriving the arithmetic properties of modular forms. In this chapter, we study various aspects of congruence subgroups, including their elliptic points, cusps, and topological properties of the associated modular curve. Jacobi theta-functions, theta-constants, and the Dedekind eta-function are used as examples of modular forms under congruence subgroups that are not modular forms under the full modular group SL(2,Z).

CONGRUENCE SUBGROUPS OF BRAID GROUPS AND CRYSTALLOGRAPHIC QUOTIENTS. PART I
Part of
- Other groups of matrices
- Special aspects of infinite or finite groups
PAOLO BELLINGERI, CELESTE DAMIANI, OSCAR OCAMPO, CHARALAMPOS STYLIANAKIS
Journal:

Journal of the Australian Mathematical Society , First View

Published online by Cambridge University Press:

13 September 2024, pp. 1-19
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This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.

Stability in the high-dimensional cohomology of congruence subgroups
Part of
Jeremy Miller, Rohit Nagpal, Peter Patzt
Journal:

Compositio Mathematica / Volume 156 / Issue 4 / April 2020

Published online by Cambridge University Press:

24 March 2020, pp. 822-861

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April 2020
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We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

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