| Резюме: | <p>All elliptic curves defined over Q are modular. This is the statement of the
modularity theorem that relates arithmetic properties of an elliptic curve to
a modular form for a subgroup of SL(2,Z). This modularity has been (and
still is) the subject of much research in number theory. At first, one might
suspect that this kind of modularity is irrelevant for physics. Nevertheless, we
will show in this thesis that modular Calabi-Yau threefolds are distinguished
by certain physical processes. Most notably, by the attractor mechanism of IIB
supergravity.</p>
<p>We will see that the zeta-function associated to an attractor variety of rank two
will factor in a specified manner and that a search for such factorisations leads
to an effective strategy for identifying examples of rank two attractor varieties
with small h<sup>2,1</sup> . We will also find and study a number of attractor varieties
of rank two by taking the Hadamard product of two Picard-Fuchs equations of
families of elliptic curves. As we will see, in many cases, the Hadamard product
admits an involution with fixed points that are attractor points of rank two. We
are able to identify the associated modular forms in both types of examples.</p>
<p>In the remainder of this thesis, we will explore the physical implication of mod-
ularity. We will show that certain arithmetically interesting quantities such as
critical values of the associated L-functions determine the area of the horizon of
certain dyonic N = 2 black holes. Similarly, we find that topological string free
energies, when evaluated at the mirror of a rank two attractor point, may also
be expressed in terms of arithmetically interesting quantities.</p>
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