Let \(F\) be a global function field over the finite field \({\mathbb F}_q\) of characteristic \(p\), and fix a place ``\(\infty\)'' of \(F\). It is known that any elliptic curve \(E/F\) with split multiplicative reduction at \(\infty\) is a factor of the Jacobian of some Drinfeld modular curve \(M\) associated with these data (some sort of modularity conjecture over \(F\)). The uniformization map \(M \rightarrow E\) and the Tate period \(q_E\) of \(E\) are described in [the reviewer and \textit{M. Reversat}, J. Reine Angew. Math. 476, 27-93 (1996; Zbl 0848.11029)] through theta functions on the Drinfeld upper half-plane \(\Omega\). In particular, \(E\) determines a Drinfeld modular form \(f_E\) of weight 2, eigenform for the Hecke algebra, and thus a \(C_{\infty}\)-valued measure, say \(\nu_E\), on the boundary \(\partial \Omega = {\mathbb P}_1(F_{\infty})\). Here \(C_{\infty}\) is the completed algebraic closure of \(F_{\infty}\). Unlike the ``classical'' case, \(E \mapsto f_E\) loses much of the information about \(E\), in that knowledge of \(f_E\) is equivalent with knowledge of the reduction mod \(p\) of \(\varphi_E\), the normalized automorphic form attached to \(E\). As results from the theory of theta functions on \(\Omega\) and the formalism of integration on \(\partial \Omega\), that loss can be circumvented by working with measures with values in the multiplicative group \(C_{\infty}^{\ast}\). Viz., the theta functions yield a \(C_{\infty}^{\ast}\)-valued measure \(\mu_E\), which is a ``lift'' for the measure \(\nu_E\) (actually \({\mathbb F}_p\)-valued) according to the diagram \[ C_{\infty}^{\ast} \rightarrow {\mathbb Q}\supset {\mathbb Z} \rightarrow {\mathbb F}_p \hookrightarrow C_{\infty}, \] where the first arrow is the logarithm to base \(q\) of the absolute value and the second arrow is reduction. By construction, \(\mu_E\) encodes the Tate period \(q_E\) of \(E\). By choosing an embedding of a quadratic extension \(K/F\) unramified at \(\infty\) into the matrix algebra \(M_2(F)\), \(\mu_E\) is pushed forward to a measure on a \(p\)-adic group \(G\) isomorphic to an anticyclotomic Galois group over the Hilbert class field \(H\) of \(K\). Distinguishing cases, this yields a Heegner point on \(E\) (if \(\infty\) is inert in \(K\)) or an analogue of the \({\mathcal L}\)-invariant (if \(\infty\) splits). Both of these can be controlled to some extent (Theorems 14 and 17). Similar considerations for the split quadratic algebra \(K := F \times F\) yield an analogous result (Theorem 26) for a certain cyclotomic Galois group.